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2.11 Fundamentals of CT Reconstruction in 2D and 3DJA Fessler, University of Michigan, Ann Arbor, MI, USA
ã 2014 Elsevier B.V. All rights reserved.
2.11.1 Introduction 2642.11.2 Radon Transform in 2D 2642.11.2.1 Definition 2642.11.2.2 Signed Polar Forms 2652.11.2.3 Radon Transform Properties 2662.11.2.4 Sinogram 2672.11.2.5 Fourier-Slice Theorem 2672.11.3 Back Projection 2682.11.3.1 Image-Domain Analysis 2692.11.3.2 Frequency-Domain Analysis 2692.11.3.3 Summary 2702.11.4 Radon Transform Inversion 2702.11.4.1 Direct Fourier Reconstruction 2722.11.4.2 The BPF Method 2722.11.4.3 The FBP Method 2732.11.4.4 Ramp Filters and Hilbert Transforms 2742.11.4.5 Filtered Versus Unfiltered Back Projection 2752.11.4.6 The CBP Method 2752.11.4.7 PSF of the FBP Method 2772.11.4.8 Summary 2772.11.5 Practical Back Projection 2772.11.5.1 Rotation-Based Back Projection 2782.11.5.2 Ray-Driven Back Projection 2782.11.5.3 Pixel-Driven Back Projection 2782.11.5.4 Interpolation Effects 2792.11.5.5 Summary 2792.11.6 Sinogram Restoration 2792.11.7 Sampling Considerations 2792.11.7.1 Radial Sampling 2802.11.7.2 Angular Sampling 2802.11.8 Linogram Reconstruction 2802.11.9 2D Fan-Beam Tomography 2802.11.9.1 Fan-Parallel Rebinning Methods 2822.11.9.2 The FBP Approach for 360� Scans 2822.11.9.2.1 Equiangular case 2842.11.9.2.2 Equidistant case 2842.11.9.3 FBP for Short Scans 2852.11.9.4 The BPF Approach 2852.11.10 3D Cone-Beam Reconstruction 2862.11.10.1 Equidistant Case (Flat Detector) 2862.11.10.2 Equiangular Case (Third-Generation Multislice CT) 2872.11.10.3 Extensions (Data Truncation and Helical Scans) 2872.11.10.3.1 Rebinning 2872.11.10.3.2 Offset detectors 2872.11.10.3.3 Long object problem 2872.11.10.3.4 Helical scans 2872.11.10.3.5 Region of interest reconstruction 2872.11.10.3.6 Motion compensation 2872.11.10.3.7 Local tomography 2872.11.11 Iterative Image Reconstruction 2872.11.11.1 Object Model 2882.11.11.2 Measurement Model 2882.11.11.3 Algebraic Methods 2882.11.11.4 Statistical Models 288
mprehensive Biomedical Physics http://dx.doi.org/10.1016/B978-0-444-53632-7.00212-4 263
264 Fundamentals of CT Reconstruction in 2D and 3D
2.11.11.5 Regularized Weighted Least Squares 2892.11.11.6 Total Variation Regularization and Sparsity 2892.11.11.7 Optimization Algorithms 2892.11.11.8 Example 2902.11.12 Summary and Future Trends 290References 291
GlossaryBack projection Creating an image from a sinogram.
Sinogram Raw data measured by a tomographic imaging
system.
Tomography Forming cross-sectional images of a 3D
object.
AbbreviationsBPF Backproject-filter method for image reconstruction
FBP Filter-backproject method for image reconstruction
FT Fourier transform
PSF Point-spread function
WLS Weighted least squares
2.11.1 Introduction
Methods for tomographic image reconstruction can be cate-
gorized as either analytic methods or iterative methods.
Analytic image reconstruction methods typically are based
on idealized models for the imaging system, and these sim-
plifications lead to noniterative algorithms that are used rou-
tinely in x-ray CT clinical practice because they require only
modest computation times. (Other names are Fourier recon-
struction methods and direct reconstruction methods, a term
that emphasizes that these methods are noniterative.) Itera-
tive methods are based on more accurate models for the
imaging system, and these can lead to improved image quality
at the price of greatly increased computation. Many iterative
methods are also based on models for the measurement
statistics, and these methods can reduce image noise and
thus enable lower-dose scans.
This chapter begins with a review of classical analytic tomo-
graphic reconstruction methods. These analytic methods are
also useful for developing intuition and for initializing iterative
algorithms. See Chapter 2.03 for several x-ray CT examples.
Several books have been devoted to the subject of image
reconstruction (Deans, 1983; Helgason, 1980; Kak and Slaney,
1988; Natterer, 1986; Natterer and Wubbeling, 2001). The
treatment in this chapter slightly generalizes classical deriva-
tions by considering an angularly weighted back projection
that is described in Section 2.11.3. This weighted back-
projector is introduced here to facilitate analysis of weighted
least square (WLS) formulations of image reconstruction
methods.
There are several limitations of analytic reconstruction
methods that impair their performance. Analytic methods gen-
erally ignore measurement noise in the problem formulation
and treat noise-related problems as an ‘afterthought’ by post-
filtering operations. Analytic formulations usually assume con-
tinuous measurements and provide integral-form solutions.
Sampling issues are treated by discretizing these solutions
‘after the fact.’ Analytic methods require certain standard
geometries (e.g., parallel rays and complete sampling in radial
and angular coordinates). Statistical methods for image recon-
struction can overcome all of these limitations. The mathemat-
ical background needed to work with analytic reconstruction
methods is Fourier analysis, whereas iterative methods are
based primarily on tools from linear algebra.
2.11.2 Radon Transform in 2D
The foundation of analytic reconstruction methods is the
Radon transform that relates a 2D function f(x, y) to the col-
lection of line integrals of that function (Cormack, 1963;
Radon, 1917, 1986). (We focus initially on the 2D case.)
Emission and transmission tomography systems acquire mea-
surements that are something like blurred line integrals, so the
line-integral model represents an idealization of such systems.
Figure 1 illustrates the geometry of the line integrals associated
with the (ideal) 2D Radon transform.
2.11.2.1 Definition
Let L r; ’ð Þ denote the line in the Euclidean plane at angle ’
counterclockwise from the y-axis and at a signed distance r
from the origin:
L r; ’ð Þ ¼ x; yð Þ 2 R2 : x cos’þ y sin’ ¼ r� �
[1]
¼ r cos’� ‘ sin’, r sin’þ ‘ cos’ð Þ : ‘ 2 Rf g [2]
Let p’(r) denote the line integral through f(x, y) along the line
L r; ’ð Þ. There are several equivalent ways to express this line
integral, each of which has its uses:
f(x, y)
x
y
r
r0
r 0
pj(
r)
2ar0
Figure 2 Projection of a centered uniform disk object, illustrated at’¼p/2.
pj(r)
j
j
Objectf(x, y)
L(r,j)
r
r
y
x
Projection
Figure 1 Geometry of the line integrals associated with the Radontransform.
Fundamentals of CT Reconstruction in 2D and 3D 265
p’ rð Þ ¼ðL r;’ð Þ
f x; yð Þd‘
¼ð1�1
f r cos’� ‘ sin’, r sin’þ ‘ cos’ð Þ d‘[3]
¼ð1�1
ð1�1
f r0cos’� ‘ sin’, r0 sin’þ ‘ cos’ð Þd r0 � rð Þ dr0 d‘
[4]
¼ð1�1
ð1�1
f x; yð Þd x cos’þ y sin’� rð Þ dx dy [5]
¼1
cos’j jð1�1
fr � t sin’
cos’; t
� �dt, cos’ 6¼ 0
1
sin’j jð1�1
f t;r � t cos’
sin’
� �dt, sin’ 6¼ 0
8>><>>: [6]
where d(�) denotes the 1D Dirac impulse. (The last form came
from Edholm and Herman, 1987). The step between [4] and
[5] uses the following change of variables:
xy
� �¼ cos’ � sin’
sin’ cos’
� �r0
‘
� �� [7]
The Radon transform of f is the complete collection of line
integrals:
f $Radon p’ rð Þ : ’ 2 0; p½ �, r 2 �1,1ð Þ� �[8]
The function p’(�) is called the projection of f at angle ’.
Sometimes, one refers to values of ’ outside of the domain
given in [8]; this is possible using the ‘periodic extension’
described in [17]. Of course, a practical system has a finite
maximum radius that defines its circular field of view.
In its most idealized form, the 2D image reconstruction
problem is to recover f(x, y) from its projections {p’(�)}. To do
this, one must somehow return the data in projection space
back to object space, as described in Section 2.11.4.
Example 1 Consider the centered uniform disk object with
radius r0:
f x; yð Þ ¼ a rectr
2r0
� �, rect tð Þ≜1 jtj�1=2f g ¼
1, jtj � 1=2
0, otherwise
([9]
Using [3], the Radon transform of this object is
p’ rð Þ ¼ð1�1
f r cos’� ‘ sin’, r sin’þ ‘cos’ð Þ d‘
¼ð1�1
a rect
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir cos’� ‘ sin’ð Þ2 þ r sin’þ ‘cos’ð Þ2
q2r0
0@
1Ad‘
¼ að1�1
rect
ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ ‘2
p2r0
!d‘ ¼ a
ð‘:r2þ‘2�r20f g
d‘ ¼ aðþ ffiffiffiffiffiffiffiffiffi
r20�r2p
�ffiffiffiffiffiffiffiffiffir20�r2p d‘
¼ 2affiffiffiffiffiffiffiffiffiffiffiffiffiffir20 � r2
prect
r
2r0
� �[10]
which is a semicircle function as shown in Figure 2. These
projections are independent of ’ due to the circular symmetry
of f(x, y).
2.11.2.2 Signed Polar Forms
It can be useful to have a form of the Radon transformwhen f is
represented in a polar form. Throughout this chapter, we use a
‘signed polar form’ fo(r,’)¼ f(r cos’, r sin’), in which the
radial argument r can be both positive and negative. Usually,
we abuse notation slightly and write f(r, ’) without the
subscript.
For making changes of variables between Cartesian coordi-
nates and signed polar coordinates, we define
r� a; bð Þ≜ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2p
, b > 0f g or b ¼ 0 & a � 0f g� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2 þ b2p
, b < 0f g or b ¼ 0 & a < 0f g
[11]
∠p a; bð Þ≜
tan�1b
a
� �, ab > 0
0, b ¼ 0
p=2, a ¼ 0, b 6¼ 0
tan�1b
a
� �þ p, ab < 0
8>>>>>>>><>>>>>>>>:
[12]
These functions obey the following natural properties:
∠p a; bð Þ 2 0, p½ Þ∠p b; að Þ ¼ 0, a ¼ 0 & b ¼ 0
p=2� ∠p a; bð Þð Þmod p, else
jr� a; bð Þj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2
p
266 Fundamentals of CT Reconstruction in 2D and 3D
r� aa; abð Þ ¼ ar� a, bð Þ∠p aa; abð Þ ¼ ∠p a, bð Þ, a 6¼ 0
cos∠p a; bð Þ ¼ 1, b ¼ 0a sgn bð Þ= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2 þ b2p
, b 6¼ 0
sin∠p a; bð Þ ¼ 0, b ¼ 0jbj= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2 þ b2p
, b 6¼ 0
r� a; bð Þ cos∠p a, bð Þ ¼ a
r� a; bð Þ sin∠p a, bð Þ ¼ b [13]
Making a change of variables r¼ r�(x,y) and ’¼∠p(x,y) leads
to the following integral relationship:ð1�1
ð1�1
f x; yð Þdxdy ¼ðp0
ð1�1
fo r; ’ð Þjrjdrd’ [14]
In particular, substituting r0 ¼ r� (x,y) and ’0 ¼∠p(x,y) into the
Radon transform expression [5] leads to the following Radon
transform in polar coordinates:
p’ rð Þ ¼ðp0
ð1�1
fo r0; ’
0 �
d r0cos ’� ’
0 �
� r �
jr0 jdr0d’0 [15]
The properties [13] arise in several of the subsequent
derivations.
2.11.2.3 Radon Transform Properties
The following list shows a few of the many properties of the
Radon transform. This list is far from exhaustive; indeed, new
properties continue to be found, for example, Marzetta and
Shepp (1999) and Zhang et al. (2002). Throughout this list, we
assume f x; yð Þ $Radon p’ rð Þ.
• Linearity
If g x; yð Þ $Radon q’ rð Þ, thenaf þ bg $Radon apþ bq
• Shift/translation
f x� x0, y � y0ð Þ $Radon p’ r � x0 cos’� y0 sin’ð Þ [16]
• Rotation
f x cos’0 þ y sin’0, � x sin’0 þ y cos’0ð Þ $Radon p’�’0 rð Þ• Circular symmetry
fo r; ’ð Þ ¼ fo r, 0ð Þ 8’) p’ ¼ p0 8’• Symmetry/periodicity
p’ rð Þ ¼ p’�p �rð Þ ¼ p’�kp �1ð Þkr �
, 8k 2 Z [17]
• Affine scaling
f ax;byð Þ $Radonp∠p b cos’, a sin’ð Þ rjajbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b cos’ð Þ2 þ a sin’ð Þ2q
0B@
1CA
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib cos’ð Þ2 þ a sin’ð Þ2
q[18]
for a,b 6¼0, where r� and ∠p were defined in Section 2.11.2.2.
The following two properties are special cases of the
affine scaling property:
• Magnification/minification
f ax; ayð Þ $Radon 1
aj j p’ arð Þ, a 6¼ 0
• Flips
f x, �yð Þ $Radon pp�’ �rð Þf �x, yð Þ $Radon pp�’ rð Þ
• The projection-integral theorem
For a scalar function h : ℝ!ℝ,ðp’ rð Þh rð Þdr ¼
ð ðf r cos’� ‘ sin’, r sin’þ ‘ cos’ð Þd‘
� �h rð Þdr
¼ðð
f x; yð Þh x cos’þ y sin’ð Þdx dy[19]
by making the orthonormal coordinate rotation:
x ¼ r cos’� ‘ sin’, y ¼ r sin’þ ‘ cos’.
• Volume conservation (DC value)
F 0;0ð Þ ¼ð1�1
ð1�1
f x; yð Þdx dy ¼ð1�1
p’ rð Þdr, 8’ [20]
This is a corollary to the projection-integral theorem for
h(r)¼1. The volume conservation property is one of many
consistency conditions of the Radon transform
(Natterer, 1986).
The following example serves to illustrate some of these
properties.
Example 2 Determine the Radon transform of an ellipse
object centered at the origin having major axes of half lengths
rX and rY, that is, f x; yð Þ ¼ rect 12
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix=rXð Þ2 þ y=rYð Þ2
q� �, where
the function is unity within the ellipse and zero outside. Using
[10] and the affine scaling property [18] with a¼1/rX and
b¼1/rY,
p’ rð Þ ¼ rXrYffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirX cos’ð Þ2 þ rY sin’ð Þ2
q grffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rX cos’ð Þ2 þ rY sin’ð Þ2q
0B@
1CA
where g tð Þ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� t21p
jtj<1f g denotes the projection of a circle
of unity radius.
Example 3 Consider the object f(x,y)¼d2(x�x0, y�y0), the
2D Dirac impulse centered at (x0,y0). Informally, we can
think of this object as a disk function centered at (x0; y0) of
radius r0 and height 1/(pr02) (so that volume is unity) in the
limit as r0!0.
Let Cr0 rð Þ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffir20 � r2
prect r
2r0
�denote the projection of
centered uniform disk with radius r0 as derived in [10] in
Example 1. Then, by the shift property [16], the projections
of a disk centered at (x0; y0) are
p’ rð Þ ¼ Cr 0 r � x0 cos’þ y0 sin’½ �ð Þ(See Figure 4.) Thus, the projections of the 2D Dirac impulse
are found as follows:
p’ rð Þ ¼ 1
pr20Cr 0 r � x0 cos’þ y0 sin’½ �ð Þ
! d r � x0 cos’þ y0 sin’½ �ð Þ as r0 ! 0
Sinogram for disk0 40
x
yr0
φ
0
p/2
p
Figure 3 Left: cross-section of 2D object containing three Dirac impulses. Right: the corresponding sonogram consisting of three sinusoidal impulseridges.
Fundamentals of CT Reconstruction in 2D and 3D 267
An alternative derivation uses [5]. In summary, for a 2D Dirac
impulse object located at (x0; y0), the projection at angle ’ is a
1D Dirac impulse located at r¼x0 cos’þy0 sin’ (see
Figure 3).
r
φ
−60 −40 −20 0 20 40 60p 0
Figure 4 Sinogram for a disk object of radius r0¼20 centered at(x0; y0)¼ (40; 0).
2.11.2.4 Sinogram
Because p’(r) is a function of two arguments, we can display
p’(r) as a 2D grayscale picture where usually r and ’ are the
horizontal and vertical axes, respectively. If we make such a
display of the projections p’(r) of a 2D Dirac impulse, then the
picture looks like a sinusoid corresponding to the function
r¼x0 cos’þy0 sin’. Hence, this 2D function is called a sino-
gram and (when sampled) represents the raw data available for
image reconstruction. So, the goal of tomographic reconstruc-
tion is to estimate the object f(x, y) from a measured sinogram.
Each point (x, y) in object space contributes a unique sinu-
soid to the sinogram, with the ‘amplitude’ of the sinusoid
beingffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2
p, the distance of the point from the origin,
and the ‘phase’ of the sinusoid depending on ∠p(x,y).
A sinogram of an object f(x, y) is the superposition of all of
these sinusoids, each one weighted by the value f(x, y). Hence,
it seems plausible that there could be enough information in
the sinogram to recover the object f, if we can unscramble all of
those sinusoids.
Example 4 Figure 3 illustrates these concepts for the object
f(x,y)¼d2(x,y)þd2(x�1,y)þd2(x�1,y�1) with correspond-
ing projections p’(r)¼d(r)þd(r�cos ’)þd(r�cos ’� sin ’).
Example 5 Figure 4 shows the sinogram for a disk of radius
r0¼20 centered at position (x0, y0)¼(40, 0).
2.11.2.5 Fourier-Slice Theorem
Themost important corollary of the projection-integral theorem
[19] is the Fourier-slice theorem, also known as the central-slice
theorem, or central-section theorem, or projection-slice theo-
rem. Apparently the first publication of this result was in Brace-
well’s 1956 paper (Bracewell, 1956). In words, the statement of
this theorem is as follows. If p’(r) denotes the Radon transform
of f(x, y), then the 1D Fourier transform (FT) of p’(∙) equals theslice at angle ’ through the 2D FT of f(x, y).
Let P’(v) denote the 1D FT (we assume existence of the FTs
of all functions of interest here) of p’(r), that is,
P’ vð Þ ¼ð1�1
p’ rð Þe�i2pnrdr
Let F(u, v) denote the 2D FT of f(x, y), that is,
F u; vð Þ ¼ð1�1
ð1�1
f x; yð Þe�i2p uxþnyð Þdxdy [21]
Then, in mathematical notation, the Fourier-slice theorem is
simply
½22�where Fo(r,F)¼F(r cos F,r sin F) denotes the polar form of
F(u, v). (Again, we will frequently recycle notation and omit
the subscript.) The proof of the Fourier-slice theorem is
remarkably simple: merely set h(r)¼exp(�i2pvr) in the
projection-integral theorem [19].
268 Fundamentals of CT Reconstruction in 2D and 3D
It follows immediately from the Fourier-slice theorem that
the Radon transform [8] describes completely any (Fourier
transformable) object f(x, y), because there is a one-to-one corre-
spondence between the Radon transform and the 2D FT F(u, v),
and from F(u, v), we can recover f(x, y) by an inverse 2D FT
(see Section 2.11.4.1).
Example 6 The 2D the uniform rectangle object and its FT are
f x; yð Þ ¼ rectx
a
�rect
y
b
�$2D FT
F u; vð Þ ¼ a sinc auð Þb sinc bvð Þ
So, in polar form, Fo(r,F)¼a sinc(ar cos F)b sinc(br sin F).By the Fourier-slice theorem, the 1D FT of its projections are
given by
P’ vð Þ ¼ Fo v; ’ð Þ ¼ a sinc v a cos’ð Þb sinc v b sin’ð Þ [23]
Thus, by the convolution property of the FT, each projection is
the convolution of two rect functions:
p’ rð Þ ¼ 1
cos’j j rectr
a cos’
� � 1
sin’j j rectr
b sin’
� �[24]
where ‘*’ denotes 1D convolution with respect to r. This is a
trapezoid in general, as illustrated Figure 5. Specifically, defin-
ing a generic trapezoid by
trap t; t1; t2; t3; t4ð Þ≜
t � t1t2 � t1
t1 < t < t2
1 t2 � t � t3t4 � t
t4 � t3t3 < t < t4
0, otherwise
8>>>>>><>>>>>>:
[25]
the Radon transform of a rectangle object is given by
p’ rð Þ ¼ ‘max ð’Þtrapðr;�dmax ð’Þ, � dbreakð’Þ, dbreakð’Þ, dmax ð’ÞÞ
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2p
trir
ab=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2p
!, ja cos’j ¼ jb sin’j
b rectr
a
�, ’ ¼ 0, � p, . . .
a rectr
b
�, ’ ¼ �p=2, � 3p=2, . . .
1
cos’ sin’j jdmax ’ð Þtri r
dmax ’ð Þ� �
�dbreak ’ð Þtri r
dbreak ’ð Þ� �
26664
37775, otherwise
8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:
[26]
where the unit triangle function is defined by
tri xð Þ ¼ trap x;�1, 0, 0, 1ð Þ ¼ 1� jxjð Þrect x
2
�¼ 1� jxjð Þ1 jxj<1f g [27]
and we define
rdbreak(j)-dbreak(j)-dmax(j) dmax(j)
pj(r)
lmax(j)
Figure 5 The trapezoidal projection at angle ’ of a rectangular object.
dmax ’ð Þ ¼ a cos’��þ ��b sin’
�� ��2
dbreak ’ð Þ ¼ ja cos’j � jb sin’jj j2
‘max ’ð Þ ¼ abj jmax a cos’
��, ��b sin’�� �� �
[28]
At angles ’ that are multiples of p/2, the trapezoid degenerates
to a rectangle, and at angles where |a cos ’|¼ |b sin ’|, the
trapezoid degenerates to a triangle.
Example 7 The 2D FT of a uniform disk object
f x; yð Þ ¼ rect r2r0
�is F(r)¼ r0
2jinc(r0r).
Thus, P’ nð Þ ¼ r20 jinc r0nð Þ ¼ r20J1 pr0nð Þ2r0n
, where J1 denotes the
first-order Bessel function of the first kind. Because J1(2pn)/2v
andffiffiffiffiffiffiffiffiffiffiffiffiffi1� t2p
rect t=2ð Þ are 1D FT pairs (Bracewell, 2000, p. 337),
we see that the projections of a uniform disk are given by
p’ rð Þ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffir20 � r2
prect r
2r0
�. This agrees with the result shown
in [10] by integration.
Example 8 Consider the 2D Gaussian object f x; yð Þ ¼f rð Þ ¼ 1
w2 exp �p r=wð Þ2 �, with corresponding 2D FT F(r)¼
exp(�p(wr)2). By the Fourier-slice theorem, P’(n)¼exp
(�p(wn)2), the inverse 1D FT of which is p’ rð Þ ¼ 1w exp
�p r=wð Þ2 �. (Note the slight change in the leading constant.)
Thus, the projections of a Gaussian object are Gaussian, which
is a particularly important relationship. This property is related
to the fact that two jointly Gaussian random variables have
Gaussian marginal distributions.
The following corollary follows directly from the Fourier-
slice theorem.
Corollary 9 Convolution property If f $Radon p and g $Radon q, then
f x; yð Þg x; yð Þ$Radon p’ rð Þq’ rð Þ [29]
Example 10 In particular, it follows from Example 8 that 2D
Gaussian smoothing of an object is equivalent to 1D radial
Gaussian smoothing of each projection:
f x; yð Þ 1
w2e�p r=wð Þ2 $Radon p’ rð Þ 1
we�p r=wð Þ2
Expressions of the form f(x,y)**h(r) should be interpreted as
2D convolution in Cartesian coordinates as follows:
g x; yð Þ ¼ f x; yð Þh rð Þ ¼ðð
f x� s, y � tð Þhffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ t2
p �dsdt.
2.11.3 Back Projection
The Radon transform maps a 2D object f(x, y) into a sinogram
p’(r) consisting of line integrals through the object.One approach
to try to recover the object from p’(r) would be to take each
sinogram value and ‘smear’ it back into object space along the
corresponding ray, as illustrated in Figure 6. This type of opera-
tion is called back projection and is fundamental to tomographic
image reconstruction. Unfortunately, in its simplest form, this
procedure does not recover the object f(x, y), but instead yields a
L(r, j)
Projection
p j(r)
r
r
x
y
j
Figure 6 Illustrationofbackprojectionoperationforasingleprojectionview.
Fundamentals of CT Reconstruction in 2D and 3D 269
blurred version of the object fb(x, y). This blurred version fb(x, y) is
called a laminogram or layergram (Smith et al., 1973).
Recall from Example 3 that the projection of an impulse
object centered at (x0, y0) is the ‘sinusoidal impulse’ along
r¼x0 cos ’þy0 sin ’. Because each object point (x0, y0) con-
tributes its own sinusoid to the sinogram, it is natural to ‘sum
along the sinusoid’ to attempt to find f(x0, y0). (There are
analogous image formation methods in other modalities
such as ultrasound beamforming by delay and sum.)
When the sinogram of an asymmetric object is corrupted by
noise, it is conceivable that different views will have different
signal to noise ratios, so it may be useful to weight the views
accordingly while ‘summing along the sinusoid.’ (It could also
be useful to weight each ray differently, but such weighting is
more difficult to analyze.Most readers should probably consider
w(’)¼1 on a first pass anyway. See Davison and Grunbaum
(1981) for related analysis of tomography with arbitrary view
angles and view-dependent filters.) Therefore, we analyze the
following angularly weighted back projection operation:
fb x; yð Þ ¼ðp0
w ’ð Þp’ x cos’þ y sin’ð Þd’ [30]
where w(’) denotes the user-chosen weight for angle ’. In the
usual case where w(’)¼1, this operation is the adjoint of the
Radon transform (see Section 2.11.4.4).
2.11.3.1 Image-Domain Analysis
The following theorem shows that the laminogram fb(x, y) is a
severely blurred version of the original object f(x, y).
Theorem 11 If p’(r) denotes the Radon transform of f(x, y) in [3]
and fb(x, y) denotes the angularly weighted back projection of p’(r)
as given by [30], then
fb x; yð Þ ¼ h r; ’ð Þf x; yð Þ, where h r; ’ð Þ ¼ w ’þ p=2ð Þmod pð Þrj j
[31]
for ’2 [0,p] and r 2 R.
Proof:
It is clear from [3] and [30] that the operation f(x,y)!p’(r)! fb(x,y) is linear. Furthermore, this operation is shift-
invariant because
fb x� c, y � dð Þ ¼ðp0
w ’ð Þp’ x� cð Þ cos’þ y � dð Þ sin’ð Þd’
¼ðp0
w ’ð Þq’ x cos’þ y sin’ð Þd’
where, using the shift property [16], the projections q’(r)
≜p’(r� c cos ’�d sin ’) denote the Radon transform of
f(x� c, y�d).
Due to this shift invariance, it suffices to examine the
behavior of fb(x, y) at a single location, such as the center.
Using [3],
fb 0; 0ð Þ ¼ðp0
w ’0
�p’0 0ð Þd’
0
¼ðp0
w ’0
� ð1�1
f 0 cos’0 � ‘ sin’
0, 0 sin’
0 þ ‘ cos’0
�d‘
� �d’
0
¼ðp0
ð1�1
o ’þ p=2ð Þmod pð Þrj j f 0� r cos’, 0� r sin’ð Þjrjdr d’
[32]
making the variable changes ’0 ¼(’þp/2)mod p and
‘ ¼ r, ’0 2 p=2,p½ �
�r, ’0 2 0,p=2½ Þ
. Thus, using the shift-invariance
property noted earlier,
fb x; yð Þ ¼ðp0
ð1�1
w ’þ p=2ð Þmod pð Þrj j f x� r cos’, y � r sin’ð Þjrjdr d’
[33]
which is the convolution integral [31] in (signed) polar
coordinates.
An alternative proof uses the projection and back projec-
tion of a centered Dirac impulse based on Example 3.
In the usual case where w(’)¼1, we see from [31] that
unmodified back projection yields a result that is the original
object blurred by the 1/r function. This point-spread function
(PSF) has very heavy tails, so the laminogram is nearly useless
for visual interpretation. Figure 7 illustrates the 1/r function.
Thus far, we have focused on the parallel-ray geometry
implicit in [3]. For a broad family of other geometries, there
exist pixel-dependent weighted back projection operations that
also yield the original object convolved with 1/r (Gullberg and
Zeng, 1995). So, the nature of [31] is fairly general.
2.11.3.2 Frequency-Domain Analysis
Because the laminogram fb(x, y) is the object f(x, y) convolved
with the PSF h(r, ’) in [31], it follows that in the frequency
domain, we have
Fb r;Fð Þ ¼ H r;Fð ÞFo r;Fð Þwhere H(r,F) denotes the polar form of the 2D FT of h(r, ’).
It is well known that 1/|r| and 1¼|r| are 2D FT pairs
(Bracewell, 2000, p. 338). The following theorem generalizes
that result to the angularly weighted case.
Theorem 12 The PSF given in [31] has the following 2D FT for
F2 [0,p] and r 2 R:
h(r)
x
y
–5 –4 –3 –2 –1 0 1 2 3 4 5–5
–4
–3
–2
–1
0
1
2
3
4
5
0
1
2
3
4
5
6
7
8
9
10
00
0
xy –5
5
–5
5
Figure 7 Illustrations of 1/r function and its ‘heavy tails.’
270 Fundamentals of CT Reconstruction in 2D and 3D
h r; ’ð Þ ¼ 1
rj jw ’þ p=2ð Þmod pð Þ ���!2D FTH r;Fð Þ ¼ 1
rj jw Fð Þ[34]
Alternatively, we could write H r;Fð Þ ¼ 1rj jw Fmod pð Þ for
F2 [0,2p) and r�0.
Proof:
Evaluate the 2D FT of h :
H r;Fð Þ ¼ðp0
ð1�1
h r;’ð Þe�i2prrcos ’�Fð Þjrjdr d’
¼ðp0w ’þp=2ð Þmodpð Þ
ð1�1
e�i2prrcos ’�Fð Þdr� �
d’
¼ðp0w ’þp=2ð Þmodpð Þd rcos ’�Fð Þð Þd’
¼ 1
rj jðp0w ’0ð Þd sin ’0 �Fð Þð Þd’0 ¼ 1
rj jw Fð Þ
letting ’0 ¼(’þp/2)mod p and using the following Dirac
impulse property (Bracewell, 2000, p. 100):
d f tð Þð Þ ¼X
s:f sð Þ¼0
d t � sð Þ_f sð Þ��� ��� � [35]
In particular,
d sin tð Þð Þ ¼Xk
d t þ pkð Þ
Thus, the 2D FT of h(r, ’) in [31] is H(r,F)¼w(F)/|r|. This isnot a rigorous proof because the function 1/|r| is not square
integrable (in 2D), so its 2D FT exists only in the sense of
distributions, as used for other signals like Dirac impulses,
sinusoids, and step functions.
So, the frequency–space relationship between the lamino-
gram and the original object is
Fb r;Fð Þ ¼ w Fð Þrj j Fo r;Fð Þ [36]
High spatial frequencies are severely attenuated by the 1/
|r| term, so the laminogram is very blurry. However, the
relationship [36] immediately suggests a ‘deconvolution’
method for recovering f(x, y) from fb(x, y), as described in
the next section.
More generally, if q’(r) is an arbitrary sinogram to which we
apply a weighted back projection of the form [30], then the FT
of the resulting image is
Fb r;Fð Þ ¼ w Fð Þrj j Q’ nð Þ��n¼r,’¼F
¼ w Fð Þrj j
QF rð Þ, F 2 0,p½ ÞQF�p �rð Þ, F 2 p, 2p½ Þ
[37]
where Q’(v) is the 1D FT of q’(r) along r. The special case [36]
follows from the Fourier-slice theorem.
2.11.3.3 Summary
Figure 8 summarizes the various Fourier-transform relation-
ships described earlier, as well as the Fourier-slice theorem, and
the projection and back projection operations. Figure 9 shows
an example of an object f(x, y), its sinogram, and the resulting
laminogram.
2.11.4 Radon Transform Inversion
By manipulating the expressions derived in the preceding sec-
tions, one can find several methods for inverting the Radon
transform, that is, for recovering an object f(x, y) from its pro-
jections {p’(r)}. This section describes three alternatives: direct
Fourier reconstruction based on the Fourier-slice theorem, the
backproject-filter (BPF) method based on the laminogram, and
finally the convolve-backproject (CBP) method, also called the
filter-backproject (FBP) method. Each of these methods uses
some of the relationships shown in Figure 8.
In this section, we continue to treat the idealized version of
the tomography problem in which the entire continuum of
projections {p’(r)} is available. In practical tomography
systems, only a discrete set of projections and rays are avail-
able; these sampling considerations will be addressed in
Section 2.11.5.
Image
x
y
-64 64-64
64
0
1
Sinogram
r
ff
-64 64
0
p
p
-64 640
15
40
Top row of sinogram
r
Pro
ject
ion
Laminogram
x
y
-64 64-64
64
Ramp filtered sinogram
r-64 64
0
-64 64
-4
-2
2
4
Top row of filtered sinogram
r
FBP image
x
y
-64 64-64
64
0
1
Figure 9 Illustration of filter-backproject (FBP) method. Top row: an object f(x, y) consisting of two squares, the larger of which has severalsmall holes in it, its sinogram p’(r), its top row p0(r), and its laminogram fb(x, y). The laminogram is so severely blurred that the small holes arenot visible. Bottom row: the ramp-filtered sinogram p
^
’ rð Þ, its top row p^
0 rð Þ, and FBP image f x; yð Þ. Because of the ramp filtering described inSection 2.11.4.3, the small details are recovered.
Fb(u,v)
Projectionpj(r)
Sinogram
pj (r)
Filteredsinogram
Con
efil
ter
|r|
(con
volv
e)ra
mp
filte
r
Backprojection
1/r
**2D FT
1D FT
|n|
Ram
pfil
ter
2D FT 1D FT
Pj (n)
Pj(n)
Gridding
Slice
F(u, v)
Object
f (x, y)
fb(x, y)
Laminogram
Figure 8 Relationships between a 2D object f(x, y) and its projections and transforms. Left side of the figure is image domain; right side is projectiondomain. Inner ring is space domain; outer ring is frequency domain.
Fundamentals of CT Reconstruction in 2D and 3D 271
272 Fundamentals of CT Reconstruction in 2D and 3D
2.11.4.1 Direct Fourier Reconstruction
The direct Fourier reconstruction method is based on the
Fourier-slice theorem [22]. To reconstruct f(x, y) from {p’(r)}
by the direct Fourier method, one performs the following steps:
• Take the 1D FT of each p’(∙) to get P’(∙) for each ’.
• Create a polar representation Fo(r,F) of the 2D FT of object
F(u, v) using the Fourier-slice relationship:
Fo r; ’ð Þ ¼ P’ rð Þ• Convert from polar representation Fo(r,F) to Cartesian
coordinates F(u, v). This approach, first proposed in De
Rosier and Klug (1968), was ‘the first applicable method
for reconstructing pictures from their projections (Herman,
1972).’
For sampled data, this polar to Cartesian step, often called
gridding, requires very careful interpolation. Figure 10 illus-
trates the process. Numerous papers have considered this step
in detail (e.g., Alliney et al., 1993; Bellon and Lanzavecchia,
1997; Bracewell, 1956; Cheung and Lewitt, 1991; Choi and
Munson, 1998; Dusaussoy, 1996; Edholm and Herman, 1987;
Fourmont, 2003; Gottlieb et al., 2000; Lanzavecchia and
Bellon, 1997; Lewitt, 1983; Matej and Bajla, 1990; Mersereau,
1974, 1976; Mersereau and Oppenheim, 1974; Natterer, 1985;
O’Sullivan, 1985; Penczek et al., 2004; Potts and Steidl, 2001;
Schomberg and Timmer, 1995; Seger, 1998; Stark et al.,
1981a,b; Sweeney and Vest, 1973; Tabei and Ueda, 1992;
Walden, 2000). Of these, the nonuniform FFT methods
with good interpolation kernels are particularly appealing
(e.g., Fourmont, 2003; Matej et al., 2004; Schomberg and
Timmer, 1995).
• Take the inverse 2D FT of F(u, v) to get f(x, y).
In practice, this is implemented using the 2D inverse FFT,
which requires Cartesian samples, whereas the relationship
Fo(r,’)¼P’(r) is intrinsically polar, hence the need for
interpolation.
This method would work perfectly if given noiseless, con-
tinuous projections p’(r). Practical disadvantages of this
method are that it requires 2D FTs and gridding can cause
interpolation artifacts. An alternative approach uses a Hankel
transform rather than FTs (Higgins and Munson, 1988); this
method also uses interpolation.
v
u
vGridding
u
Figure 10 Illustration of polar samples of Fo(r,’)¼P’(r) that onemust interpolate onto Cartesian samples of F(u, v) for the direct Fourierreconstruction method.
Example 13 Consider the sinogram described by
p’ rð Þ ¼ rectr � x0 cos’� y0 sin’
w
� �
What is the object f(x, y) that has these projections?
First, taking the 1D FT yields
P’ nð Þ ¼ w sinc wvð Þe�i2pv x0 cos’þy0 sin’ð Þ
so by the Fourier-slice theorem, the spectrum of f(x, y) is
given by
Fo r;Fð Þ ¼ w sinc wrð Þe�i2pp x0 cosFþy0 sinFð Þ
or equivalently
F u; nð Þ ¼ w sinc wffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ n2
p �e�i2p x0uþy0nð Þ
Because (Bracewell, 2000, p. 338) w sinc wrð Þ ���!2D FT
1
prect r=wð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw=2ð Þ2 � r2
q , the corresponding object is
f x; yð Þ ¼ rect
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix� x0ð Þ2 þ y � y0ð Þ2
qw
0@
1A 1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw=2ð Þ2 � x� x0ð Þ2 � y � y0ð Þ2
q
In other words, the object that has ‘flat’ projections has a
circular singularity. Using this relationship, one can analyze
the deficiencies of simplistic pixel-driven forward projection
(De Man and Basu, 2002).
2.11.4.2 The BPF Method
Another reconstruction method is suggested by the Fourier
relationship [36] between the laminogram and the original
object. Solving [36] for the 2D FT of the object yields
F u; nð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ n2p
w ∠p u; nð Þð Þ Fb u; nð Þ [38]
where Fb(u, v) denotes the 2D FT of the laminogram and ∠p
was defined in [12]. The filter with frequency response��r�� ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ v2p
is called the cone filter due to its shape. (This
method is also called the r-filtered layergram approach
(Natterer, 1986, p. 153; Smith et al., 1973).)
The aforementioned relationship suggests the following
reconstruction method:
• Choose a nonzero angular weighting function w(’) (typi-
cally unity).
• Perform angularly weighted back projection of the sino-
gram p’(r) to form the laminogram fb(x, y) using [30].
• Take the 2D FT of fb(x, y) to get Fb(u, v).
• Apply the angularly modulated cone filter in the Fourier
domain using [38].
• The cone filter nulls the DC component of f(x, y). This
component can be recovered using the volume conserva-
tion property [20] of the Radon transform. For noisy sino-
gram data, one can compute such an integral for all
projections and take the average value to estimate the DC
component: F 0; 0ð Þ ¼ 1
p
ðp0
ðp’ rð Þdr
� �d’.
• Take the inverse 2D FT of F u; vð Þ to get f x; yð Þ.
-2 -1 0 1 2-2
0
2
4
6
8
10
12
14
r
h(r)
Figure 11 Impulse response of cone filter that is apodized by anexponential.
-8 -6 -4 -2 0 2 4 6 8
-0.5
0
0.5
1
1.5
2
r
h(r)
Figure 12 Impulse response of band-limited cone filter.
Fundamentals of CT Reconstruction in 2D and 3D 273
This approach is called the BPF method because we first
backproject the sinograms and then apply the cone filter to
‘deconvolve’ the 1/|r| effect of the back projection.
In practice, using the cone filter without modification
would excessively amplify high-frequency noise. To control
noise, the cone filter is usually apodized in the frequency
domain with a windowing function. Specifically, we replace
[38] by
F u; nð Þ ¼ A u; nð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ n2p
w ∠p u; nð Þð Þ Fb u; nð Þ
where A(u, v) is an apodizing low-pass filter. In the absence of
noise, the resulting reconstructed image satisfies
f x; yð Þ ¼ a x; yð Þf x; yð Þ
where a(x, y) is the inverse 2D FT of A(u, v). (See Chu
and Tam (1977) and Colsher (1980) for early 3D versions
of BPF.)
One practical difficulty with the BPF reconstruction method
is that the laminogram fb(x, y) has unbounded spatial support
(even for a finite-support object f ) due to the tails of the 1/|r|
response in [31]. In practice, the support of fb(x, y) must be
truncated to a finite size for computer storage, and such trun-
cation of tails can cause problems with the deconvolution step.
Furthermore, using 2D FFTs to apply the cone filter results in
periodic convolution, which can cause wrap-around effects
due to the high-pass nature of the cone filter. To minimize
artifacts due to spatial truncation and periodic convolution,
one must evaluate fb(x, y) numerically using a sampling grid
that is considerably larger than the support of the object f(x, y).
A large grid increases the computational costs of both the back
projection step and the 2D FFT operations used for the cone
filter. The FBP reconstruction method, described next, largely
overcomes this limitation. The FBP method has the added
benefit of only requiring 1D FTs, whereas the direct Fourier
and BPF methods require 2D transforms.
Example 14 For theoretical analysis, a convenient choice for
the apodizer is A(r)¼e�ar. Using Hankel transforms
(Bracewell, 2000, p. 338) and the following Laplacian
property,
�4p2r2h rð Þ $Hankel 1
rd
drH rð Þ þ d2
dr2H rð Þ
one can show that the corresponding impulse response of the
apodized cone filter is given by
h rð Þ ¼ 4pa2 � 2p2r2
a2 þ 4p2r2ð Þ5=2$Hankel
re�ar
Taking the limit as a!0 shows that h(r)¼�1/(4p2r3) for
r 6¼0 and that h(r) has a singularity at r¼0.
Figure 11 illustrates this impulse response for the case a¼1.
Example 15 If we band-limit the cone filter by choosing
A rð Þ ¼ rect r2rmax
�, then the resulting impulse response h(r)
has a complicated expression that depends on both Bessel
functions and the Struve function (Abramowitz and Stegun,
1964). Figure 12 illustrates this impulse response for the case
rmax¼1.
2.11.4.3 The FBP Method
We have seen that an unfiltered back projection yields a
blurry laminogram that must be deconvolved by a cone filter
to yield the original image. The steps involved look like the
following:
Because the cascade of the first two operations is linear and
shift-invariant, as shown in Section 2.11.3, in principle, we
could move the cone filter to be the first step to obtain the
same overall result:
Þ
274 Fundamentals of CT Reconstruction in 2D and 3D
where, assuming w(’)¼1 hereafter, the filtered object f∨
x; yð Þhas the following spectrum:
F∨
r;Fð Þ ¼ jrjFo r;Fð ÞOf course, in practice, we cannot filter the object before acquir-
ing its projections. However, applying the Fourier-slice theo-
rem to the scenario in the preceding text, we see that each
projection p^’ rð Þ has the following 1D FT:
p∨
’ rð Þ$FT P∨’ nð Þ ¼ F∨
r; ’ð Þ���r¼n¼ jrjFo r; ’ð Þjr¼n ¼ jnjFo n; ’ð Þ
¼ jnjP’ nð ÞThis relationship implies that we can replace the cone filter in
the preceding text with a set of 1D filters with frequency
response |n| applied to each projection p’(�). This filter is calledthe ramp filter due to its shape. The block diagram in the
preceding text becomes
This reconstruction approach is called the FBP method and is
used the most widely in tomography.
A formal derivation of the FBP method uses the Fourier-
slice theorem as follows:
f x; yð Þ ¼ðð
F u; vð Þei2p xuþyvð Þdudv
¼ðp0
ð1�1
F n cos’, n sin’ð Þei2pn x cos ’þy sin’ð Þjnjdn d’
¼ðp0
ð1�1
P’ nð Þei2pn x cos ’þy sin ’ð Þjnjdn d’
¼ðp0
p∨
’ x cos’þy sin’ð Þd’
where we define the filtered projection p∨’ rð Þ as follows:
p∨
’ rð Þ ¼ð1�1
P’ nð Þjnjei2pnrdn [39]
The steps of the FBP method are summarized as follows.
• For each projection angle ’, compute the 1D FT of the
projection p’(�) to form P’(n).
• Multiply P’(n) by |n| (ramp filtering) to get
P∨
’ nð Þ ¼ ��n��P’ nð Þ.• For each ’, compute the inverse 1D FT of P
∨
’ nð Þ to get the
filtered projection P∨
’ rð Þ in [39]. In practice, this filtering is
often done using an FFT, which yields periodic convolu-
tion. Because the space-domain kernel corresponding to |n|is not space-limited (see Figure 14), periodic convolution
can cause ‘wrap-around’ artifacts. With care, these artifacts
can be avoided by zero padding the sinogram. Sampling the
ramp filter can also cause aliasing artifacts. See Example 18
in the succeeding text for a preferable approach.
• The ramp filter nulls the DC component of each projection.
If desired, this can be restored using the volume conserva-
tion property [20]. The approach of Example 18 avoids the
need for any such DC correction. Discretizing the integrals
carefully avoids the need for empirical scale factors.
• Backproject the filtered sinogram P∨
’ rð Þn o
using [30] to get
f x; yð Þ, that is,
f x; yð Þ ¼ðp0
p∨
’ x cos’, y sin’ð Þd’ [40]
In practice, usually, the pixel-driven back projection
approach of Section 2.11.5.3 is used.
With some hindsight, the existence of such an approach
seems natural because the Fourier-slice theorem provides a
relationship between the 2D FT in object domain and the 1D
FT in projection domain.
2.11.4.4 Ramp Filters and Hilbert Transforms
It can be useful to relate the ramp filter |n| to a combination of
differentiation and a Hilbert transform.
The Hilbert transform of a 1D function f(t) is defined (using
Cauchy principal values) by Grafakos (2004, p. 248) (note that
some texts use the opposite sign, e.g., Barrett and Myers, 2003,
p. 194; Bracewell, 2000, p. 359):
fHilbert tð Þ ¼ 1
p
ð1�1
1
t � sf sð Þds ¼ 1
ptf tð Þ [41]
Note that this ‘transform’ returns another function of t. The
corresponding relationship in the frequency domain is
FHilbert nð Þ ¼ �i sgn nð ÞF nð Þ [42]
Example 16 The Hilbert transform of the rect function
rect tð Þ ¼ 1 jtj�1=2f g is (Grafakos, 2004, p. 249) 1p log tþ1=2
t�1=2��� ���.
Using the Hilbert transform frequency response [42], we
rewrite the ramp filter |n| in [39] as follows:
nj j ¼ 1
2pi2pnð Þ �isgn nð Þð Þ
The term (i2pn) corresponds to differentiation, by the dif-
ferentiation property of the FT. Therefore, another expression
for the FBP method [40] is
f x; yð Þ ¼ 1
2p
ðp0
d
drpHilbert r; ’ð Þ
����r¼x cos’þy sin’
d’ [43]
where pHilbert(r, ’) denotes the Hilbert transform of p’(r) with
respect to r. Combining [41] and [43] yields
f x; yð Þ ¼ 1
2p2
ðp0
ð1�1
@@r p’ rð Þ
x cos’þ y sin’� rdr d’ [44]
This form is closer to Radon’s inversion formula (Natterer,
1986, p. 21; Radon, 1917, 1986).
Example 17 Continuing Example 6, the spectrum of the pro-
jection at angle ’ of a rectangle object is given by [23], so its
ramp-filtered projections are given by (for sin ’ 6¼0)
P∨
’ nð Þ ¼ ��n��a sinc n acos’ð Þb sinc nb sin’ð Þ¼ 1
p cos’ sin’sin pn acos’ð Þ sgn nð Þ b sin’ð Þ sinc nb sin’ð
¼ 1
2p cos’ sin’e�ipna cos’ � eipna cos’Þð
i sgn nð Þ b sin’ð Þ sinc nb sin’ð Þ½ �
-2 -1 0 1 2-1
0
1
r
Pro
ject
ion
Projection of squareIdeal ramp-filtered projectionBand-limited ramp-filtered projection
Figure 13 Projection p’(r) of a unit square at angle ’¼p/9 and its filtered versions �p’ rð Þ for both ideal ramp filter |n| and band-limited ramp filter withcutoff frequency n0¼4.
Fundamentals of CT Reconstruction in 2D and 3D 275
Using the Hilbert transform in Example 16, the inverse 1D FT
of the bracketed term is 1pb sin’ log
x� 12 b sin’
xþ 12 b sin’
��������, so by the shift
property of the FT, the filtered projections are
�p’ rð Þ ¼ 1
2p2cos’sin’log
r2 � a cos ’þb sin ’2
�2r2 � a cos ’�b sin ’
2
�2�������
�������Compare with Servieres et al. (2004, eqn [14]). Figure 13
shows an example of the projection p’(r) of a unit square
and its filtered version p∨
’ rð Þ. The ramp filter causes singulari-
ties at each of the points of discontinuity in the projections
(cf. Figure 9).
2.11.4.5 Filtered Versus Unfiltered Back Projection
Recall that an unfiltered back projection of a sinogram gives an
image blurred by 1/|n|. This blurring is due to the fact that the
(all nonnegative) projection values ‘pile up’ in the laminogram
and there is no destructive interference. In contrast, after filter-
ing with the ramp filter, the projections have both positive and
negative values, so a destructive interference can occur, which
is desirable for the parts of the image that are supposed to be
zero, for example. Figure 9 illustrates these concepts.
2.11.4.6 The CBP Method
The ramp filter amplifies high-frequency noise; so in practice,
one must apodize it by a 1D low-pass filter A(n), in which case
[39] is replaced by
�p’ rð Þ ¼ð1�1
P’ nð ÞA nð Þjnjei2pnr dn [45]
Alternatively, one can perform this filtering operation in the
spatial domain by radial convolution:
�p’ rð Þ ¼ p’ rð Þha rð Þ ¼ðp’ r0ð Þha r � r0ð Þdr0 [46]
where the filter kernel ha(r) is the inverse FT of Ha(n)¼A(n)|n|,that is,
ha rð Þ ¼ð1�1
A nð Þjnjei2pnr dn [47]
Combining with [40] and [46] leads to the following CBP
method:
f x; yð Þ ¼ðp0
p’ha
�x cos’þ y sin’ð Þd’
¼ðp0
ðp’ rð Þha x cos’þ y sin’� rð Þdrd’ [48]
Although the convolution kernel ha(r) usually is not space-
limited, the object and hence its projections are space-limited,
so space-domain convolution is feasible. On the other hand,
the space-domain convolution requires more computation
than a frequency-space implementation using the FFT method,
so the FBP approach is often more attractive than the CBP
approach.
Example 18 As a concrete example, consider the case of a
rectangular band-limiting window
A nð Þ ¼ rectn2n0
� �[49]
which is a logical choice when the object and hence its
projections are band-limited to a maximum spatial frequency
n0. In this case, the band-limited ramp filter has the frequency
response shown in Figure 14. This is called the Ram–Lak filter
-8 -6 -4 -2 0 2 4 6 8
-0.4
0
1
Impulse response of band-limited ramp filter
r
h(r)
Figure 15 Convolution kernel for band-limited ramp filter ha(r) withn0¼1 and the sample values h[n]¼ha(n/(2n0)).
−6 0 6
−1
0
1
2
rh(
r)
e= 1e= 0
Figure 16 Impulse response ha(r) of ramp filter with exponentialapodization.
n
n0 tri n-Ha(n)
n0 n0 n0n n
n0 rect n2n0
n0=
Figure 14 Frequency response of band-limited ramp filter.
276 Fundamentals of CT Reconstruction in 2D and 3D
(Natterer and Wubbeling, 2001, p. 83) after Ramachandran
and Lakshminarayanan (1971).
To determine the corresponding convolution kernel,
observe that
jnjrect n2n0
� �¼ n0rect
n2n0
� �� n0tri
nn0
� �
where tri(�) was defined in [27]. Thus, the convolution kernel
is (Bracewell and Riddle, 1967)
ha rð Þ ¼ 2n20 sinc 2n0rð Þ � n20 sinc2 n0rð Þ [50]
as shown in Figure 15. The ringing is due to the implicit
assumption that the object is band-limited. In practice, one
usually uses an apodization filter A(n) that goes to zero grad-
ually to reduce ringing; see Section 2.11.4.7.
In practice, one uses samples of this impulse response.
Sampling it using the Nyquist rate 2n0 yields (Kak and Slaney,
1988, p. 72)
h n½ � ¼ han
2n0
� �¼ 2n20 sinc nð Þ � n20 sinc
2 n=2ð Þ
¼ n20
1, n ¼ 00, neven�1= pn=2ð Þ2, n odd
8<: [51]
Rarely is n0 given in practice, so one assumes that the
sampling is adequate, that is, n0¼1/(2rR), where rR is the
radial sample spacing. This approach is preferable to sampling
the ramp filter directly in the frequency domain (Crawford,
1991; Kak and Slaney, 1988, p. 69).
Although the filter [51] is infinitely long, given a sinogram
with a finite number nR of radial samples, we need only to
evaluate the filtered sinogram p^’ rð Þ at those same radial sam-
ple locations, so it suffices to compute h[n] for n¼�nR, . . .,
nR�1 and to zero-pad the sinogram radially with nR zeros
before computing the FFTs to perform the filtering. When
using this discrete-space filter h[n] to approximate the convo-
lution [46], one should include a scaling factor rR to account
for dr in that integral.
Example 19 In Example 17, Figure 13 showed the projections
of a square after filtering with an ideal ramp filter. Figure 13
also shows those same projections when filtered with the
rectangularly apodized ramp filter described in Example 18.
Example 20 For theoretical analysis, an alternative to the rect-
angular apodization considered in Example 18 is to use expo-
nential apodization A(n)¼e�e|n|, for some small e>0. One can
verify the following FT pair (Macovski, 1983, p. 127):
ha rð Þ ¼ 2 e2 � 4p2r2ð Þe2 þ 4p2r2ð Þ2 $
FTHa nð Þ ¼ e�e nj j nj j
Figure 16 shows examples of this impulse response. Taking
the limit as e!0 yields the following expression for the ramp
filter for r 6¼0:
h∗ rð Þ ¼ �12p2r2
[52]
and a singularity at r¼0. (See Gelfand and Shilov (1977) for
rigorous treatment of FTs of such functions.) This ramp filter
satisfies the following scaling property:
h∗ rð Þ ¼ a2h∗ arð Þ [53]
This is also known as the homogeneity property (Zeng, 2004).
Fundamentals of CT Reconstruction in 2D and 3D 277
2.11.4.7 PSF of the FBP Method
Apodizing the ramp filter will reduce amplification of high-
frequency noise but will also degrade spatial resolution in the
reconstructed object. To analyze the effects of apodization, we
again turn to the Fourier-slice theorem [22]. By that theorem,
multiplying the 1D FT P’(n) of each projection by A(n) is equiv-alent to premultiplying the object spectrum by A(r), that is,
F u; vð Þ↦Affiffiffiffiffiffiffiffiffiffiffiffiffiffiu2þv2
p �F u; vð Þ
(Note that for this simple relationship to hold, it is essential
that the same apodizer be used for every projection angle ’).
Thus, the reconstructed object f x; yð Þ is a blurred version of
the original:
f x; yð Þ ¼ f x; yð Þh x; yð Þ [54]
where
h x; yð Þ $2D FTH u; vð Þ≜A
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2þv2
p �[55]
Because H is circularly symmetrical, so is h; thus, h(r) is simply
the Hankel transform of H(r)¼A(r).
Example 21 For the rectangular apodizing window
A nð Þ ¼ rect n2n0
�, the corresponding PSF in the image domain
would be
h rð Þ ¼ n20 jinc n0rð ÞThus, the image would be blurred by a jinc function, which
has large side lobes that would cause undesirable ‘ringing.’
Example 22 A popular choice in nuclear medicine is a
Gaussian window: A(n)¼exp(�p(n/n0)2). The half-amplitude
cutoff frequency n1/2 for this window, that is, the point where
A(n1/2)¼A(0)/2, is n1=2 ¼ n0ffiffiffiffiffiffiffiffilog 2p
q n0
2 0:9394 n02 . Because
the Hankel transform of a Gaussian is Gaussian, in the image
domain, the PSF is
h rð Þ ¼ n20 exp �p n0rð Þ2 �To find the full-width half-maximum (FWHM) of this
Gaussian, find r such that h(r)¼h(0)/2 or exp(�p(n0r)2)¼1/2
so p(n0r)2¼ log 2. Thus,
FWHM¼ 2
n0
ffiffiffiffiffiffiffiffiffiffiffilog 2
p
r 0:9394
n0 1
n0 1
2n1=2
So, for a 5 mm FWHM PSF, we would use n0¼1/5¼0.2
cycles per centimeter.
Example 23 Other popular window functions include the
following:
• Hann or Hanning: A nð Þ ¼ 12þ 1
2 cos pn=n0ð Þ� �rect n
2n0
�• Hamming: A nð Þ ¼ 0:54þ 0:46 cos pn=n0ð Þ½ �rect n
2n0
�• Generalized Hamming:
A nð Þ ¼ aþ 1� að Þ cos pn=n0ð Þ½ �rect n2n0
�, for a 2 0; 1½ �
• Butterworth: A nð Þ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ n=n0ð Þ2np , for n � 0
• Parzen: A nð Þ ¼1� 6 n=n0ð Þ2 1� jnj=n0ð Þ jnj � n0=22 1� ��n��=n0 �3 n0=2 �
��n�� � n00, otherwise
8<:
• Shepp Logan (Shepp and Logan, 1974):
A nð Þ ¼ sinc n2n0
���� ��� or sinc n2n0
���� ���3• Modified Shepp Logan: A nð Þ ¼ sinc n
2n0
�0:4� 0:6 cos pn=n0ð Þ½ �
It is not always easy to find a closed-form expression for the
PSF h that results from apodization. But the general rule of
thumb, FWHM1/(2n1/2), is usually pretty close.
In light of the result [54], one might wonder why we apply
the window A(n) to the projections rather than just smooth
(postfilter) the reconstructed image. Themain reason is that we
have to apply the ramp filter anyway, so we can include A(n)essentially for free. In contrast, postsmoothing would require
either an ‘expensive’ convolution or a pair of 2D FFTs. How-
ever, if one wants to experiment with several different amounts
of smoothing, then it is preferable to smooth after a (ramp-
filtered) back projection so that only one back projection
operation is needed.
2.11.4.8 Summary
We have described three methods for inverting the Radon
transform, that is, for reconstructing a 2D object f(x, y) from
its projections {p’(r)}:
• Direct Fourier reconstruction (gridding)
• The BPF method (cone filter)
• The FBP method (ramp filter) and its cousin the CBP
method
The derivations of these methods all used the Fourier-slice
theorem. These methods would yield identical results for
noiseless continuous-space data but are based on different
manipulations of the formulas so they lead to different ways
of discretizing and implementing the equations, yielding very
different numerical algorithms in practice. We also analyzed
the PSF due to windowing the ramp filter. In practice, one must
choose the apodizing window to make a suitable compromise
between spatial resolution and noise.
Recently, other inversion formulas for the 2D Radon trans-
form have been discovered for objects with compact support,
for example, Clackdoyle and Noo (2004). These methods
include user-selectable parameters that allow one to avoid
corrupted or missing regions of the sinogram. An interesting
open problem is to determine whether the methods could be
extended to include some type of statistical weighting.
2.11.5 Practical Back Projection
The preceding sections considered the idealized case where
there is a continuum of projection views. In practice, sino-
grams have only finite angular samples, so each of the recon-
struction methods described in Section 2.11.4 requires
modification for practical implementations. (There are also
alternative methods for solving the problem for a finite
number of views, such as the minimum-norm approach
(Kazantsev, 1998; Logan and Shepp, 1975).)
A critical step in both the BPF and FBP reconstruction
methods is the back projection operation [30]. Given only a
finite number n’ of projection angles, we must approximate
278 Fundamentals of CT Reconstruction in 2D and 3D
the integral in [30]. Usually, the projection angles are uni-
formly spaced over the interval [0, p), that is,
’i ¼i� 1
n’
� �p, i¼1, . . . , n’
In such cases, the usual approach is to use the following
Riemann sum approximation to [30]:
fb x; yð Þ pn’
Xn’i¼1
p’ixcos’i þ y sin’ið Þ [56]
Whether more sophisticated approximations to this integral
would be beneficial is an open problem.
There are at least three distinct approaches to implementing
[56]: rotation-based back projection, ray-driven back projec-
tion, and pixel-driven back projection. If the available projec-
tions were continuous functions of the radial argument, then
these formulations would be identical. In practice, not only are
the projection angles discrete, but also we have only discrete
radial samples of p’(r). Ignoring noise and blur, we are given
the discrete sinogram
yi n½ � ¼ p’ rð Þ��’¼’i, r¼rc n½ �, i¼1, . . . , n’, n¼0, . . . , nR � 1 [57]
where the radial sample locations are given by
rc n½ � ¼ n� n0ð ÞDR [58]
and typically n0¼nR/2 or n0¼(nR�1)/2. For such sinograms,
the various back projection methods can produce different
results because they differ in how the equations are discretized.
If the true object ftrue can be assumed to be appropriately
band-limited, then its projections will also be band-limited (by
the Fourier-slice theorem), so in principle, we could recover p’i
from {yi[�]} using sinc interpolation:
p’irð Þ ¼
X1n¼�1
yi n½ �sinc r � rc n½ �DR
� �
In practice, this interpolation is inappropriate because real
objects are space-limited so they cannot be band-limited; sinc
interpolation expects an infinite number of samples, whereas
practical sinograms have only a finite number of samples; and
sinc interpolation is computationally impractical. Thus, sim-
pler interpolation methods are used in practice, such as linear
interpolation or spline interpolation (Horbelt et al., 2002a),
perhaps combined with oversampling of the FFT used for the
ramp filter.
2.11.5.1 Rotation-Based Back Projection
We can rewrite the back projection formula [56] as follows:
fb x; yð Þ ¼ pn’
Xn’i¼1
bi x; yð Þ [59]
where the back projection of the ith view is given by
bi x; yð Þ ¼ p’ixcos’iþy sin’ið Þ [60]
We can also write bi ¼ P ’ip’i
, where P ’iis the adjoint opera-
tor. This operator maps the ith 1D projection back into a 2D
image by ‘smearing’ that projection along the angle ’i. In this
approach, we form temporary images by backprojecting each
view and accumulating the sum of those temporary images.
To better understand bi(x, y), note that when i¼1, we have
’¼0, so
b1 x; yð Þ ¼ p0 xð Þ [61]
which is just a 2D version of the function p0(x).
For sampled sinograms, implementing [61] is trivial. (This
method assumes that n0¼(nR�1)/2, i.e., that the center of theimage projects onto the center of each projection. It further-
more assumes that the desired pixel size equals DR. Otherwise,
a more complicated approach is needed.) Simply replicate the
first row of the sinogram (a vector) to make a matrix. For other
angles, perform the following steps to implement [60]:
• Replicate the ith sinogram row to make an image, as if it
were the ’¼0 case.
• Rotate that image counterclockwise by ’. This rotation will
require an interpolation method, such as bilinear interpo-
lation or a more precise spline approach (Unser
et al., 1995).
• Accumulate these rotated images over all angles, as
described in [59].
In this approach, the ‘outer loop’ is over projection angles.
The first step (replication) inherently ‘accounts’ for discrete
radial samples.
The rotation approach is easily implemented but can be
somewhat slow because a high-quality rotation is a fairly
expensive operation. One of the faster methods uses three 1D
passes (Unser et al., 1995).
2.11.5.2 Ray-Driven Back Projection
For ray-driven back projection, one loops through all the rays
and for each ray one interpolates yi[n] onto the pixels whose
centers are nearest to the ray L rc n½ �,’ið Þ, as defined in [2].
Although this approach is somewhat popular for forward pro-
jection, it can produce significant artifacts when used for back
projection, so will not be considered further here. Figure 6
somewhat illustrates the approach.
When radial sample spacing equals image sample spacing,
ray-driven back projection is equivalent to rotation-based back
projection (Fessler, Unpublished technical report).
For a N�N image, ray-driven back projection requires
O(Nn’nR) operations. Usually, n’N and nRN, so we say
2D ray-driven back projection is O(N3).
2.11.5.3 Pixel-Driven Back Projection
For image display, we compute fb(x, y) on a finite grid of pixel
coordinate pairs {(xj, yj): j¼1, . . ., np}. For pixel-driven back
projection, we loop over the (xj, yj) pairs of interest and eval-
uate [56] for each of the grid points, thereby filling up an image
array. To implement, the outer loop is over pixel index j and
the inner loop is over angles ’i. In essence, for each pixel, we
are summing along the corresponding sinusoid (illustrated in
Figure 3) in the sinogram.
However, the radial argument xj cos’iþyj sin’i in [56]
rarely exactly equals one of the radial sample locations rc[n]
Fundamentals of CT Reconstruction in 2D and 3D 279
shown in [57]. Therefore, radial interpolation is required for
pixel-driven back projection. The usual approach is linear
interpolation, which is equivalent mathematically to the fol-
lowing approximation:
p’ irð Þ
Xn
yi n½ � tri r � rc n½ �DR
� �[62]
where the unit triangle function is denoted:
tri tð Þ ¼ 1� ��t��, ��t�� � 10, otherwise
Although [62] is a mathematically correct expression for linear
interpolation and is useful for theoretical analysis, it poorly
conveys how one would implement linear interpolation in
practice. Because support of the function tri(t) is two sample
units, for any given r, only two terms in the sum in [62] are
possibly nonzero. An alternative expression is
p’irð Þ yi n rð Þ½ �tri r � rc n rð Þ½ �
DR
� �þ yi n rð Þ þ 1½ �tri r � rc n rð Þ þ 1½ �
DR
� �
¼ yi n rð Þ½ � 1� r � rc n rð Þ½ �DR
� �þ yi n rð Þ þ 1½ � r � rc n rð Þ½ �
DR
� �
where we define
n rð Þ≜ r=DR þ n0b cOther interpolators, such as an oversampled FFT or spline
functions, are also used (Horbelt et al., 2002a).
For a N�N image, pixel-driven back projection requires
O(N2n’) operations. Usually, n’N so we say 2D pixel-driven
back projection is O(N3). Hierarchical methods requiring
O(N2 log N) operations have also been developed (Basu and
Bresler, 2000, 2001).
2.11.5.4 Interpolation Effects
Generalizing [62], suppose that we use an interpolation
method of the form
p’irð Þ ¼
Xn
yi n½ �h r � rc n½ �DR
� �
for some interpolation kernel h(�). Suppose furthermore that
p’(r) is band-limited with maximum frequency less than 12DR
.
Then, it follows from [57] and the sampling theorem that
(ignoring noise)
P’inð Þ ¼ P’i
nð ÞH nð Þ, ��n�� < 1
2DR0, otherwise
(
For example, when h is the linear interpolator in [62], we
have H(n)¼rR sinc2(rRn), which is strictly positive for��n�� < 1
2DR. Therefore, while we are applying the ramp filter |n|
in the discretized version of [39], we can also apply the inverse
filter 1/H(n) to compensate for the effects of interpolation
(Fessler, 1995, eqn [45]; Horbelt et al., 2002a,b).
2.11.5.5 Summary
Pixel-driven, rotation-based, and ray-driven back projections
are all used in practice, depending on the number of samples,
sample spacing, etc. The formulations are exactly identical in
continuous space but can yield slightly different results when
discretized.
2.11.6 Sinogram Restoration
Because a sinogram p’(r) has two coordinates (r and ’), one
can display it as a 2D picture or even treat it as a 2D ‘image’ and
apply any number of image processing methods to it. Numer-
ous linear and nonlinear filters have been applied to sinograms
in an attempt to reduce noise (Abella et al., 2009; Abidi and
Davis, 1990; August and Kanade, 2004; Duerinckx et al., 1978;
Fessler, 1993; Hebert, 1992; Hebert and Gopal, 1991; Hsieh,
1998; Hutchins et al., 1987, 1990; Kachelrieß et al., 2001; Kao
et al., 1998; King et al., 1983; La Riviere and Billmire, 2005; La
Riviere and Pan, 2000; La Riviere et al., 1998, 2006; Pawitan
et al., 2005; Penney et al., 1988; Sauer and Liu, 1991; Thibault
et al., 2006; Villain et al., 2003), to extrapolate missing data
(Karp et al., 1988; Riviere and Pan, 1999; Soumekh, 1986;
Zamyatin and Nakanishi, 2007), and to compensate for detec-
tor blur (Alessio et al., 2006; Huesman et al., 1989; Kao et al.,
2000; Karuta and Lecomte, 1992; Lewitt et al., 1989; Liang,
1994; Wernick and Chen, 1992; Xu et al., 1998) and/or single-
photon emission computed tomography (SPECT) attenuation
(Glick et al., 1994; Kao and Pan, 2000; Pan et al., 1996; Xia
et al., 1995). Some of these methods can even be called ‘statis-
tical’ methods because they include measurement noise
models.
A typical linear approach for a system with shift-invariant
blur having frequency response B(n) would be to use a Wiener
filter as the apodizing filter A(n) in [45] as follows:
A nð Þ ¼ B nð ÞjB nð Þj2 þ RðnÞ
where the ‘regularizer’ R(n) is a model for the ratio of the power
spectral density of the noise over that of p’(r) under the (ques-
tionable) assumption that both are wide-sense stationary ran-
dom processes.
Nonlinear sinogram preprocessing methods, including
classical methods based on view-adaptive Wiener filters (Tsui
and Budinger, 1979) and contemporary approaches like
wavelet-based denoising (Nowak and Baraniuk, 1999), have
the potential to reduce noise more than linear methods with
less degradation of spatial resolution. However, when a non-
linear sinogram filtering method is combined with the linear
FBP reconstruction method, the resulting spatial resolution
properties can be quite unusual. Modern methods typically
apply nonlinear processing in the image domain, for example,
by nonquadratic edge-preserving regularization, instead of in
the sinogram domain.
2.11.7 Sampling Considerations
In practice, one can acquire only finite radial and angular
samples, due to constraints such as cost and time. This section
describes considerations in choosing the radial and angular
sampling.
280 Fundamentals of CT Reconstruction in 2D and 3D
2.11.7.1 Radial Sampling
The radial sample spacing, DR, should be determined by the
spatial resolution (in the radial direction) of the tomographic
scanning instrument. The FWHM of the system radial resolution
is a function of the detector width, the source size in x-ray imag-
ing, etc. The radial detector response (e.g., a rectangular function
for square detector elements) generally is not exactly band-
limited, so Nyquist sampling theory can provide only general
guidance. A practical rule of thumb is to choose (if possible)
DR¼FWHM/2. Then, the number of radial samples should be
determined to cover the desired field of view (FOV) by choosing
nR¼FOV/DR.
2.11.7.2 Angular Sampling
For a given FOV and radial sampling, Nyquist sampling theory
can help determine the angular sampling r’. If we have nRradial samples spaced by DR, then in the Fourier domain (of
the 2D DFT), the corresponding spatial frequencies are spaced
by Dn¼1/(nRDR). It is natural to choose the angular sampling
so as to ensure that all samples in the 2D Fourier domain are
separated by no more than this amount.
Considering Figure 17, the appropriate angular spacing is
D’ ¼ Dn1= 2DRð Þ ¼ 2=nR. So, the total number of angles over 180�
should be
In practice, often, somewhat fewer angular samples are used
but usually n’nR. The reason for using fewer than pnR/2angular samples is that often, in real systems, there is blur in
the radial direction so the spatial resolution is somewhat lower
than that implied by just the radial sampling, that is, the radial
sampling may be a little finer than necessary from a strict
Nyquist perspective. However, we rarely use Nyquist (sinc)
reconstruction, but rather only linear interpolation, so some
‘oversampling’ is reasonable. Inadequate angular sampling can
lead to significant aliasing artifacts. On the other hand, for the
FBP reconstruction method, the computation time is directly
proportional to the number of angles. Figure 18 illustrates the
effects of angular undersampling.
2.11.8 Linogram Reconstruction
For tomographic imaging systems with certain geometries
involving flat detectors, it can be convenient to use an alterna-
tive coordinate system for the sinogram of the form
1
Dj
2DRDn
u
v
Figure 17 Angular sampling considerations.
pEW s; bð Þ≜ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ b2
p p arctan bð Þsffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ b2p
!
pNS s; bð Þ≜ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ b2
p pp=2þ arctan bð Þsffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ b2p
! [63]
for |b|�1. This is called a linogram (Axel et al., 1990; Edholm
and Herman, 1987; Edholm et al., 1988; Magnusson, 1993;
Potts and Steidl, 2001), because in this coordinate system, the
projection of a point source is a straight line.
Taking the 1D FT of pEW(s, b) and pNS(s, b) along s and
applying the Fourier-slice theorem yield the Fourier
relationships
PEW’ nð Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ b2
pP arctan bð Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ b2n
p �¼F n,bnð Þ
PNS’ nð Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ b2
pPp=2þ arctan bð Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ b2n
p �¼F �bn, nð Þ
[64]
So, the 1D FT of linogram data corresponds to the samples
of the object spectrum F(u, v) along lines with slope b or 1/b. Inparticular, for a projection at slope b, if the linogram data have
NS equally spaced samples along s with spacing DS, then the
corresponding samples of the object spectrum F(u, v) in 2D
Fourier space along the line at slope b are spaced byffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ b2
p= NSDSð Þ, corresponding to the pseudopolar grid
shown in Figure 19. Using this sampling pattern, one can
develop direct Fourier reconstruction methods for linogram
data akin to Section 2.11.4.1 (e.g., Averbuch et al., 2008a,b).
The linogram concept has been generalized to higher-dimen-
sion data, called planogram reconstruction (Brasse et al., 2004;
Kazantsev et al., 2004, 2006; Kinahan et al., 1999).
By equating ’ with arctan(b) or pþ/2þarctan(b), one can
show thatffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ b2
p¼ 1
max cos ’j, jsin ’j jð Þ. Therefore, the radial sam-
plespacing in[63] is thesameasthatofMojettesamplingdescribed
in Brady (1993), Donohue and Saniie (1989), Guedon and
Normand (1997), Guedon et al. (2004), Guedon and Normand
(2005), Herman (1972), Katz (1978), Nuyts et al. (1994),
Oppenheim (1974), Schmidlin (1994), and Subirats et al.
(2004). The primary difference between linogram and Mojette
sampling is in the angular sampling.
2.11.9 2D Fan-Beam Tomography
The preceding sections have focused on case of 2D parallel-
beam projections. Although first-generation x-ray CT scanners
did correspond to that geometry, many contemporary tomo-
graphic imaging systems have fan-beam geometries, including
commercial x-ray CT scanners and some collimators for SPECT
systems. For hypothetical continuous measurements, one
could transform fan-beam projections into parallel-beam pro-
jections by a simple change of variables. For discrete, noisy
measurements, rebinning fan-beam measurements into
parallel-beam projections requires an interpolation operation
that could degrade spatial resolution. To avoid such rebinning,
one can derive analytic reconstruction methods directly in
terms of the fan-beam coordinates, as described in the succeed-
ing text.
Figure 20 illustrates the fan-beam geometry that will be
considered here. Because it can be challenging mechanically
to ensure that the line between the x-ray source and the
u
v
Figure 19 Illustration of pseudopolar grid in 2D Fourier spaceassociated with linogram sampling. The cross marks and circlescorrespond to the P’
EW(n) samples and the P’NS(n) samples, respectively.
Original 2 angles 4 angles
32 angles 64 angles 128 angles
Figure 18 Illustration of the effects of angular undersampling on image quality for FBP reconstruction. The image is 128�128, and the truevalues of the digital phantom are 1 in the background disk and 8 in the small disks. The grayscale display is windowed from 0.5 to 1.5 to enhancethe visibility of the artifacts.
Fundamentals of CT Reconstruction in 2D and 3D 281
midpoint of the detector passes through the exact center of
rotation, we allow an offset roff between that line and the center
(Gullberg et al., 1986). Let P denote the point along that line
that intersects the circle of radius roff centered at the rotation
isocenter. D0d denotes the distance from the point P to the
detector, Ds0 denotes the distance from the x-ray source to P,
and Dfs denotes the distance from the focal point of the
detector arc to the x-ray source. Define Dsd≜D0dþDs0 to be
the total distance from the x-ray source to the center of the
detector. This formulation allows the detector focal point to
differ from the x-ray source location to encompass a variety of
system configurations. For flat detectors, Dfs¼1. For third-
generation x-ray CT systems, Dfs¼0. For fourth-generation
x-ray CT systems, Dfs¼�Ds0.
In our notation, the distances D0d and Ds0 are constants,
rather than being functions of angle b. Generalizations exist toallow noncircular source trajectories (Besson, 1996).
Let s2 [�smax, smax] denote the (signed) arc length along
the detector, with s¼0 corresponding the detector center. Arc
length is a natural parameterization for detector elements that
are spaced equally along the detector. (For a flat detector
with Dfs¼1, the arc length s is simply the position along
the detector.) The various angles have the following
relationships:
a sð Þ ¼ s
Dfd, g sð Þ ¼ arctan
Dfd sin a sð ÞDfd cos a sð Þ �Dfs
� �[65]
where Dfd≜DfsþDsd. The two most important cases are
g sð Þ ¼ s=Dsd Dfs ¼ 0 equiangularð Þarctan s=Dsdð Þ Dfs ¼ 1 equidistantð Þ
[66]
The function g(s) is antisymmetric and increasing and the
(inverse) relationship between g and s is
s gð Þ ¼Dsd g, Dfs ¼ 0
Dfd g� arcsinDfs
Dfdsin g
� �� �, 0 � Dfs <1
Dfs tan g, Dfs ¼ 1
8><>: [67]
The ray corresponding to angle b and detector element s is
P
x-Ray source
a
g
b
b
jFocal point
Dfs
Ds0
D0d
s = 0
s
roff
r
Figure 20 Illustration of fan-beam geometry.
282 Fundamentals of CT Reconstruction in 2D and 3D
L s;bð Þ ¼ x; yð Þ : x cos ’ s;bð Þ þ y sin ’ s; bð Þ ¼ r sð Þf g [68]
where
’ s; bð Þ≜bþ g sð Þ [69]
r sð Þ≜Ds0 sin g sð Þ þ roff cos g sð Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2
s0 þ r2off
qsin g sð Þ þ boffð Þ [70]
where we define boff≜∠(Ds0, roff). When b¼0, the coordi-
nates of a point on the detector are
xd sð Þ, yd sð Þð Þ ¼roff þDsd sin
s
Dsd
� �,
Ds0 �Dsd coss
Dsd
� �0BB@
1CCA, Dfs ¼ 0 equiangularð Þ
roff þ s, �D0dð Þ, Dfs ¼ 1 equidistantð Þ
8>>>><>>>>:
[71]
Unlike in our analysis of parallel-beam tomography, here the
range of r is limited inherently by the position of the x-ray
source and the extent of the detector:
jr sð Þj � rmax≜Ds0 sin gmax [72]
where gmax≜g (smax) and smax is half of the total arc length of
the detector. The radius rmax defines the circular field of view
of the imaging system: the subset of the plane that is measured
completely. (We assume that D0d� rmax, because otherwise,
even the detector center would limit the field of view.) The
angle 2gmax is called the fan angle. A typical clinical CT scanner
has Dsd95 cm and smax45 cm, so gmaxp/6.Figure 21 illustrates the sampling [70] associated with the
fan-beam geometry, before and after the values of (r, ’) are
collapsed to the range described in [8].
The line-integral projection p(s, b) of f along L s;bð Þ is:
p s;bð Þ ¼ðL s;bð Þ
f x; yð Þd‘
¼ðð
f x; yð Þd x cos’ s; bð Þ þ y sin’ s;bð Þ � r sð Þð Þdx dy[73]
for |s|� smax and 0�b�bmax. We assume bmax�pþ2gmax to
ensure complete sampling. Practically speaking, the integral
should be restricted to the field of viewffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2
p � rmax, but
this restriction would complicate analysis by introducing a
shift variance into the problem, so we ignore it. By [70],
fan-beam projections satisfy the following general symmetry
property:
p g�1 �g sð Þ � 2boffð Þ, b� pþ 2g sð Þ þ 2boff � ¼ p s;bð Þ [74]
In particular, if Dfs¼0, then
p �s� 2Dsd boff , b� pþ 2s=Dsd þ 2boffð Þ ¼ p s; bð Þ [75]
Alternatively, if boff¼0, then
p �s, b� pþ 2g sð Þð Þ ¼ p s; bð Þ [76]
The reconstruction problem is to estimate f from the
fan-beam projections {p(s, b)}.
2.11.9.1 Fan-Parallel Rebinning Methods
In continuous space, to rebin from fan-beam to parallel-beam
coordinates requires a simple change of variables based on
[70]. In the usual case where roff¼0, we can express the
parallel-beam projections in terms of the fan-beam projections
using the following relationship:
p’ rð Þ ¼ p s;bð Þjs¼s rð Þ, b¼b r;fð Þ¼ p g�1 arcsin r=Ds0ð Þð Þ,f� arcsin r=Ds0ð Þ �
[77]
where g�1 is defined by [67]. For sampled measurements,
usually, one first performs 1D interpolation along the source
position using the relationship f¼bþg. Then, one performs
1D interpolation along the detector by relating r and s. When
the s samples include a quarter-detector offset, then the radial
sampling can be improved by a factor of two, at least for 360�
scans, as illustrated in Figure 21.
In the presence of motion, it can be preferable to extend the
data range slightly and then use transition weights similar to the
Parker weighting described in Section 2.11.9.3 to reduce arti-
facts frompossibly inconsistent views (Allmendiger et al., 2012).
2.11.9.2 The FBP Approach for 360� Scans
This section derives FBP methods for the fan-beam geometry
shown in Figure 20, including the case of displaced center of
rotation (Gullberg et al., 1986). We focus on the case of
−0.5 0 0.5
0
p p
r / Ds0
r / Ds0 r / Ds0
r / Ds0
j j
p p
j j
0β = 0
β = 4π/3
β = 0
β = 4π/3
−0.5 0 0.5
−0.5 0 0.5
0 0
−0.5 0 0.5
Figure 21 Left: (r, ’) coordinates for equiangular fan-beam samples based directly on [70]. The fan angle is 2gmax¼p/3. Right: after converting ’
to the range [0, p) using the periodicity property [17]. Top: for equally spaced samples in s symmetrical around s¼0. Bottom: for equally spacedsamples in s with quarter-detector offset. The samples for one particular value of b are circled for illustration.
Fundamentals of CT Reconstruction in 2D and 3D 283
360� rotation of the x-ray source and detector (Kak and Slaney,
1988, p. 77).
We start by rewriting the parallel-ray FBP formula [48] for
the case of 360� rotation:
f x; yð Þj j ¼ 1
2
ð2p0
ðp’ rð Þh∗ x cos’þ y sin’� rð Þdr d’ [78]
where h∗(�) denotes the ramp filter in [52] with 1D FT
H∗(n)¼ |n|.Now, change to fan-beam coordinates by making the trans-
formation of variables r¼ r(s), ’¼’(s, b), defined in [70]. The
Jacobian matrix is
@
@sr sð Þ @
@br sð Þ
@
@s’ s;bð Þ @
@b’ s;bð Þ
2664
3775 ¼ Ds0 cos g sð Þ � roff sin g sð Þ½ � _g sð Þ 0
_g sð Þ 1
� �
[79]
the determinant of which is
J sð Þ≜ Ds0 cos g sð Þ � roff sin g sð Þj jj _g sð Þj [80]
The reconstruction formula becomes
f x; yð Þj j ¼ 1
2
ð2p0
ðp s; bð Þh∗ x cos’ s; bð Þ þ y sin’ s; bð Þ � r sð Þð Þ J sð Þdsdb
¼ 1
2
ð2p0
ðp s; bð ÞJ sð Þh∗ x cos bþ g sð Þð Þ þ y sin bþ g sð Þð Þð
�Ds0sin g sð Þ � roff cos g sð ÞÞdsdb
Although this expression is a fan-beam reconstruction
formula, it is inconvenient for practical use; the challenge
is to manipulate it so that it has a FBP form to facilitate
implementation.
Using trigonometric identities, one can simplify the argu-
ment of h∗ in the preceding text as follows:
x cos bþ gð Þ þ y sin bþ gð Þ �Ds0sin g� roff cosg¼ Lb x; yð Þ sin gb x; yð Þ � g
�[81]
wherexb≜x cos bþ y sin byb≜� x sin bþ y cosb
Lb x; yð Þ≜ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDs0 � yb �2 þ xb � roff
�2qyb x; yð Þ≜ arctan
xb � roffDs0 � yb
! [82]
Define sb(x, y), or s0 for short, by gb (x, y)¼g (s0) via [67], that
is,
sb x; yð Þ ¼ Dsdgb x; yð Þ, Dfs ¼ 0Dsd tan gb x; yð Þ, Dfs ¼ 1
¼Dsd arctan
xb � roffDs0 � yb
!, Dfs ¼ 0
Dsdxb � roffDs0 � yb
, Dfs ¼ 1
8>>><>>>:
[83]
Using [81] and applying the scaling property of the ideal
ramp filter (using this property leads to nonuniform image
noise variance in practice (Zeng, 2004), and an alternative
approach based on Hilbert transforms has been proposed
(You and Zeng, 2007)) [53] lead to the following form for
the fan-beam reconstruction formula:
284 Fundamentals of CT Reconstruction in 2D and 3D
f x; yð Þ ¼ 1
2
ð2p0
ðp s;bð ÞJ sð Þh∗ Lb x; yð Þ sin gb x; yð Þ � g sð Þ � �
dsdb
¼ 1
2
ð2p0
ðp s;bð Þ J sð Þ sb x; yð Þ � s
Lb x; yð Þ sin gb x; yð Þ � g sð Þ � � !2
h∗ sb x; yð Þ � s �
dsdb
For certain special fan-beam geometries, we can express the
sin term in the preceding text as follows (Besson, 1996):
sin gb x; yð Þ � g sð Þ � ¼W2 x; y;bð ÞW1 sð ÞW0 sb x; yð Þ � s �
[84]
for some geometry-dependent weighting functions W0, W1,
and W2. For such geometries, the fan-beam reconstruction
formula simplifies as follows:
f x; yð Þ ¼ 1
2
ð2p0
1
W22 x; y; bð ÞL2b x; yð Þ
ðp s;bð Þ J sð Þ
W21 sð Þ g∗ sb x; yð Þ � s
�ds
� �db
where the inner integral is convolution with the following
modified ramp filter:
g∗ sð Þ≜ s
W0 sð Þ� �2
h∗ sð Þ [85]
Thus, the fan-beam FBP method uses the following three
steps:
• Step 1. Compute weighted projections for each b :
~p s;bð Þ≜p s;bð Þ J sð ÞW2
1 sð Þ [86]
• Step 2. Filter those weighted projections (along s) for each busing the modified ramp filter:
�p s;bð Þ≜~p s;bð Þg∗ sð Þ, 8b
• Step 3. Perform a weighted back projection of those filtered
projections:
f x; yð Þ ¼ 1
2
ð2p0
1
W22 x; y; bð ÞL2b x; yð Þ�p sb x; yð Þ, b �
db [87]
This three-step procedure is quite practical and has been
used routinely in commercial x-ray CT systems.
The existence of this efficient FBP approach hinges on
whether a factorization of the form [84] exists. There are exactly
four fan-beam geometries for which this is possible (Besson,
1996). Fortunately, two of the four cases are the important
ones described in [66]. Somewhat unfortunately, the fourth-
generation CT scanner geometry is not one of the four cases,
although most commercial CT scanners now are third
generation.
2.11.9.2.1 Equiangular caseThe case where Dfs¼0 corresponds to third-generation x-ray
CT systems using an arc detector is called the equiangular case
(Herman and Naparstek, 1977). In this case, g(s)¼ s /Dsd, so we
factor the sin term in [84] as follows:
sin gb x; yð Þ � g sð Þ � ¼W2W0 sb x; yð Þ � s �
whereW2 ¼ 1Dsd
,W0 sð Þ ¼Dsd sins
Dsd
�, sb x; yð Þ ¼Dsd gb x; yð Þ:
with W1¼1. Thus, the modified ramp filter [85] is
g∗ sð Þ ¼ s=Dsd
sin s=Dsdð Þ� �2
h∗ sð Þ [88]
Because _g sð Þ ¼ 1=Dsd, the projection weighting in [86] sim-
plifies to
J sð ÞW2
1
¼ J sð Þ ¼ 1
Dsd
����Ds0 coss
Dsd� roff sin
s
Dsd
���� Ds0
Dsdcos
s
Dsd[89]
and the back projection weighting in [87] is
1
W22L
2b x; yð Þ ¼
D2sd
L2b x; yð Þ ¼D2
sd
Ds0 � yb �2þ xb � roff
�2 [90]
Note that as Ds0!1, that is, as the rays become more
parallel, the modified ramp filter approaches the usual ramp
filter, and the projection and back projection weightings
approach unity, as expected for the parallel-ray case.
For practical implementation, the filter must be band-
limited and sampled. Combining [88] with [51] for ha(r), the
necessary filter samples are
h n½ � ¼
1
4D2S
, n ¼ 0
0, neven�1pDsd sin nDS=Dsdð Þ½ �2 , nodd
8>>><>>>:
[91]
where DS denotes the detector element spacing. (Note that
there is an error in Kak and Slaney (1988, eqn [96]).)
There is a subtle issue in the aforementioned derivation
(You and Zeng, 2007; Zeng, 2004). We first applied the scaling
property of the ramp filter, continued the derivation, and then
eventually considered a band-limited ramp filter [91]. If we
had considered the band-limiting effect first, which might be
more appropriate for sampled data, then the scaling property
would not hold exactly. It has been stated that this practice
‘results in a nonstationary cutoff frequency in the image’ (You
and Zeng, 2007; Zeng, 2004).
2.11.9.2.2 Equidistant caseThe case where Dfs¼1 corresponds to a flat detector and is
called the equidistant case (Lakshminarayanan, 1975). In this
case, g(s)¼arctan(s/Dsd) and (with some trigonometric identi-
ties) we factor the sin term in [84] as follows:
sin g0 � g
�¼ sin g
0cos g� sin g cos g
0
¼ cos g0cos g tan g
0 � tan g �
¼ cos g0
� DsdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2
sd þ s2q s
0 � s
Dsd
where sb(x, y)¼Dsd tan gb(x, y); or equivalently,
W2 ¼ cos gb x; yð ÞDsd
, W1 ¼ DsdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2
sd þ s2q , W0 sð Þ ¼ s
Because W0(s)¼ s, we can use the usual ramp filter [51]
without modification. Because here,
_g sð Þ ¼ cos2g sð Þ �=Dsd ¼W2
1 sð Þ=Dsd
the projection weighting in [86] becomes
Fundamentals of CT Reconstruction in 2D and 3D 285
J sð ÞW2
1
¼ 1
DsdDs0 cos g sð Þ � roff sin g sð Þj j Ds0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
D2sd þ s2
q [92]
and the back projection weighting in [87] is
1
W22L
2b x; yð Þ ¼
D2sd
cos gb x; yð ÞLb x; yð Þ �2 D2sd
Ds0 � yb �2 [93]
Note that as Ds0!1, the projection and back projection
weightings again approach unity.
The preceding derivation started with the parallel-ray FBP
formula. Other formulations are possible (e.g., Chen, 2003).
2.11.9.3 FBP for Short Scans
The preceding analyses have assumed a full 360� rotation. For a360� rotation, every ray is sampled exactly twice, leading to the
1/2 factor in [78]. There are also fan-beam reconstruction
methods that use a short scan with less than a full rotation.
A standard short scan uses bmax¼pþ2gmax, so that every point
in the equivalent parallel-beam sinogram is sampled at least
once. As illustrated in Figure 22, some areas of the sinogram
are sampled twice, so these methods require appropriate
weighting, sometimes called Parker weighting (Parker, 1982).
In particular, by the symmetry property [76], fan-beam sino-
gram data where 0<b<2(gmax�g(s)) correspond to the same
rays as p�2g(s)<b<p�2gmax.
One can compensate for this sampling by applying an
additional weighting w(s, b) to the ramp-filtered fan-beam
sinogram �p s;bð Þ prior to back projection, such as the following
weighting:
w s; b½ � ¼q
b2 gmax � gð Þ� �
, 0 � b � 2 gmax � gð Þ1, 2 gmax � gð Þ < b < p� 2g
qpþ 2gmax � b2 gmax þ gð Þ
� �, p� 2g � b � pþ 2gmax
8>>>>><>>>>>:
[94]
where q xð Þ ¼ sin2 p2 x �
and g¼g(s). Note that eqn [12] of
Parker (1982) has a typo.
Amazingly, one can also reconstruct an image from less
than a short scan (Noo et al., 2002). Of course, one can
apply iterative reconstruction methods to arbitrary geometries.
The relative performance of iterative methods and analytic
methods for the ‘less than a short scan’ cases is an interesting
open problem.
3gmax
g
gmax
p
j
p- gmax
b [°
]
0
50
100
150
200
Figure 22 For short scan with bmax¼pþ2gmax, the left figure in the precedinare sampled twice. The other areas are sampled once. The right figure shows
2.11.9.4 The BPF Approach
Although the FBP approach is used most widely for fan-beam
reconstruction, it is also possible to use BPF methods for fan-
beam tomography (Gullberg and Zeng, 1995).
Given fan-beam projections p(s, b) as defined in [73], for
b2 [0, bmax], and given a weighting function w(s, b), the
weighted back projection is given by
b x; yð Þ ¼ðsmax
�smax
ðbmax
0
d x cos’ s;bð Þ þ y sin’ s;bð Þ � r sð Þð Þ
p s; bð Þw s;bð ÞdsdbWe now analyze the impulse response of the sequence of
linear operations f(x, y)!p(s, b)!b(x, y), by considering an
object f(x, y) that is a Dirac impulse at (x0, y0), for which the
fan-beam projections are given by p(s, b)¼d(x0 cos’(s, b)þy0sin’(s, b)� r(s)). The overall impulse response of the weighted
projection/back projection operation is thus
h x; y; x0; y0ð Þ ¼ðbmax
0
ðsmax
�smax
d x cos’ s; bð Þ þ y sin’ s; bð Þ � r sð Þð Þd x0 cos’ s; bð Þ þ y0 sin’ s;bð Þ � r sð Þð Þw s;bð Þdsdb
Now, make the change of variables defined in [70], the
Jacobian determinant of which is given in [80]. Assume that
the weighting term w(s, b) consists of two terms:
• A term that compensates for the Jacobian determinant in
[80] (Gullberg and Zeng, 1995).
• A term that ensures that for any (r, ’) pair that gets contri-
butions frommore than one (s, b) pair, these (s, b) pairs areappropriately weighted (Kak and Slaney, 1988, p. 98; Pan,
1999) (no such weighting is needed for a full rotation
where bmax¼2p).
Assume furthermore that bmax�pþ2gmax, so that there is
complete sampling. Then, after making the aforementioned
change of variables, the impulse response expression simplifies to
h x; y; x0; y0ð Þ ¼ðp0
ðrmax
�rmax
d x cos’þ y sin’� rð Þd x0 cos’þ y0 sin’� rð Þdrd’
¼ðp0d x� x0ð Þ cos’þ y� y0ð Þ sin’ð Þ
1 jx0 cos’þ y0 sin’ j � rmaxf gd’
¼ðp0d r� x� x0, y� y0ð Þ cos ’�∠p x� x0, y� y0ð Þð Þð Þ
Parker weighting
g [°]-25 -20 -15 -10 -5 0 5 10 15 20 25
0
1
g text illustrates (by shading) the areas of a sinogram (in ’, g space) thatthe fan-beam sinogram Parker weighting [94].
286 Fundamentals of CT Reconstruction in 2D and 3D
1 jx0 cos’þ y0 sin’ j � rmaxf gd’
¼ 1
r� x� x0, y� y0ð Þj j1 jx0 cos’þ y0 sin’ j � rmaxf gj’¼∠p x�x0, y�y0ð Þ�p=2
where rmax was defined in [72]. Thus, within the field of view,
the composition of projection with appropriately weighted back
projection is linear and shift-invariant, and similarly to
Theorem 11, in polar coordinates, the impulse response is
(Gullberg, 1979) h rð Þ ¼ 1rj j. Using this ‘restricted’ shift invari-
ance, one can formulate an approximate BPF reconstruction
method (Gullberg and Zeng, 1995). However, the practical
problems described in Section 2.11.4.2 are exacerbated here
by the inherently finite field of view of the fan-beam geometry.
2.11.10 3D Cone-Beam Reconstruction
A particularly challenging problem for analytic reconstruction
methods is that of reconstructing a 3D object from cone-beam
projection views. As x-ray detectors have evolved from 1D to
2D arrays, the practical importance of this problem has
increased. The literature on this problem is extensive and
includes both exact and approximate methods, for various
types of x-ray source trajectories and detector configurations.
For the case of a circular x-ray source trajectory (axial scan),
the most popular method is the Feldkamp cone-beam algo-
rithm or FDK approach (Feldkamp et al., 1984). This approach
extends the fan-beam reconstruction formula to the cone-beam
case; it matches the fan-beam formula in the plane of the
circular source trajectory but is inexact (due to missing data)
away from that plane. Exact 3D reconstruction for a circular
trajectory (axial scan) is likely impossible because such a scan
does not satisfy Tuy’s conditions (Tuy, 1983).
Here, we summarize the classical FDK method for a circular
trajectory. The basic idea of the FDK method is to filter
(a weighted version of ) the measured data along each row of
the detector as if it were part of a 2D fan-beam acquisition and
then to perform 3D back projection. With this approximation,
the central slice is reconstructed exactly as it would be by
conventional 2D fan-beam methods, whereas the noncentral
slices have artifacts that increase with distance from the central
slice. We focus on the case roff¼0 for simplicity.
2.11.10.1 Equidistant Case (Flat Detector)
Figure 23 illustrates the flat-detector cone-beam geometry,
where the rotated coordinates (xb, yb) were defined in [82].
As in Figure 20, let b denote the angle of the source point
z
Ds0 D0d
ys
t
xSource
Figure 23 Cone-beam flat-detector geometry.
counterclockwise from the y-axis. The cone-beam line-integral
projections are given by
p s; t;bð Þ ¼ðL s;t;bð Þ
f x, y, zð Þd‘
where the locations of the source and a point on the detector
and the line between them are
p!0 ¼ p
!0 bð Þ ¼ �Ds0 sin b,Ds0 cos b, 0ð Þ [95]
p!1 ¼ p
!1 s; t; bð Þ
¼ s cos bþD0d sin b, s sin b�D0d cos b, tð Þ [96]
L s; t;bð Þ ¼ p!0 þ a
p!1 � p
!0
p!1 � p
!0
��� ��� : a 2 R
8<:
9=; [97]
Note that p!1 � p
!0 ¼ s cos bþDsd sin b, s sin b�Dsd cos b, tð Þ
and p!1 � p
!0
��� ��� ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2
sd þ s2 þ t2q
.
Consider one row of themeasured projection at angle b and
vertical position t. That row forms a tilted fan with the source
point, with distances ‘inflated’ by the factor
p!1 s; t; bð Þ � p
!0
��� ���p!1 s;0;bð Þ � p
!0
��� ��� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2
sd þ s2 þ t2q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2
sd þ s2q
Following Section 2.11.9.2.2, we apply the projection
weighting in [92], compensating for the aforementioned infla-
tion factor, that is,
w1 s; tð Þ ¼ Ds0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDsd þ s2p
p!1 s;0; bð Þ � p
!0
��� ���p!1 s; t;bð Þ � p
!0
��� ���¼ Ds0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
D2sd þ s2 þ t2
q [98]
We apply the usual ramp filter [51] to the (weighted) mea-
surements from each row of the detector. Then, we perform 3D
cone-beam back projection using the image-domain weighting
given in [93]. No modifications of [93] are needed. To sum-
marize, the FDK algorithm for the equidistant case consists of
the following steps:
• Step 1. Compute weighted projections (Kak and Slaney,
1988, p. 106, eqn [175]):
~p s; t;bð Þ ¼ w1 s; tð Þp s; t; bð Þ, w1 s; tð Þ Ds0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2
sd þ s2 þ t2q [99]
• Step 2. Filter each row of those projections using the ordi-
nary ramp filter [51]:
�p s; t;bð Þ ¼ ~p s; t; bð Þh∗ sð Þ [100]
where h∗(s) denotes the ramp filter given in [50] or [51].
• Step 3. Perform weighted cone-beam back projection of
those filtered projections:
Fundamentals of CT Reconstruction in 2D and 3D 287
f x; y; zð Þ ¼ 1
2
ð2p0
w2 x; y; bð Þ�p sb x; yð Þ, tb x; y; zð Þ;b �db,
D2
w2 x; y;bð Þ sd
Ds0 � ybð Þ2 [101]
where from [83],
sb x; yð Þ ¼ Dsd
Ds0 � ybxb, tb x; y; zð Þ ¼ Dsd
Ds0 � ybz
(The factor Dsd
Ds0�yb is a magnification factor.) For view b, theray from the source through the point (x, y, z) intersects
the detector at point (sb(x, y), tb(x, y, z)).
Although the aforementioned algorithm is inexact in gen-
eral, if the object is ‘cylinder-like,’ that is, if f(x, y, z)¼ f(x, y, 0),
then one can verify that [101] is exact (Grass et al., 2000).
2.11.10.2 Equiangular Case (Third-GenerationMultislice CT)
For a third-generation multislice CT scanner, the FDK algo-
rithm has the same general steps as in the preceding text.
Using [71], here, the distances in the tilted fans are ‘inflated’
by the factor
p!1 s; t; bð Þ � p
!0
��� ���p!1 s;0;bð Þ � p
!0
��� ��� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2
sd þ t2q
Dsd
Thus, we replace the 1D weighting in [99] by
w1 s; tð Þ ¼ Ds0
Dsdcos
s
Dsd
� � p!1 s;0;bð Þ � p
!0
��� ���p!1 s; t;bð Þ � p
!0
��� ���¼ Ds0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
D2sd þ t2
q coss
Dsd
� �[102]
The filtering in [100] for each projection view row uses
the modified ramp filter [88] in its sampled form [91]. The
2D weighting in [101] is replaced by the equiangular counter-
part in [90]. Finally, the sb(x, y) term in [101] is replaced by
[83]. (The tb(x, y, z) term is unchanged.)
Again, if the object is ‘cylinder-like,’ then one can verify that
[101] is exact (Grass et al., 2000).
2.11.10.3 Extensions (Data Truncation and Helical Scans)
2.11.10.3.1 RebinningBy analogy with the fan-beam case discussed in
Section 2.11.9.1, there are cone-parallel rebinning methods
for cone-beam CT that can have simplified computation and
reduce noise (Grass et al., 2000; Hsieh and Tang, 2006).
2.11.10.3.2 Offset detectorsIf the detector is too narrow (transaxially) to record the entire
projection of the object, sometimes, its position is offset. This
is called the offset detector geometry. Reconstruction from
such data requires appropriate weighting functions (Maass
et al., 2008).
2.11.10.3.3 Long object problemIn practice, the object axial extent usually exceeds the axial
range of the detector, which is called the long object problem.
Extrapolation techniques have been proposed for reconstruct-
ing somewhat more of the axial extent of the object from such
axially truncated data (Grimmer et al., 2009).
2.11.10.3.4 Helical scansFor helical source trajectories, Katsevich (2002) developed a
‘theoretically exact’ FBP type of reconstruction method for
cone-beam reconstruction, a significant breakthrough in that
field. As a result, this topic is evolving rapidly (e.g., Pan et al.,
2004). Versions of the Fourier-slice theorem have even been
proposed for fan-beam and cone-beam geometries (Chen
et al., 2005). In practice, usually, ‘approximate’ algorithms
are often used rather than the exact methods to save computa-
tion. A typical approximate algorithm involves some rebin-
ning, ramp filtering, and weighted back projection steps
(e.g., Stierstorfer et al., 2004). A particularly simple method
is single-slice rebinning (Noo et al., 1999).
2.11.10.3.5 Region of interest reconstructionInterestingly, even though the topics in this chapter have been
studied for decades, advances continue to be made. For exam-
ple, recently, the subject of reconstructing a region of interest
(ROI) (from truncated sinogram data) has had breakthroughs
(Clackdoyle and Noo, 2004; Defrise et al., 2006; Noo et al.,
2004; Xia et al., 2006). The ROI reconstruction problem is
closely related to interior tomography (Courdurier et al.,
2008; Kudo et al., 2008; Louis and Rieder, 1989; Wang et al.,
2009; Xu et al., 2011; Yang et al., 2010; Yu and Wang, 2009).
2.11.10.3.6 Motion compensationTomographic imaging of moving objects, such as the beating
heart, is particularly challenging. Methods for estimating
motion and compensating for that motion in the image recon-
struction process are an active research area (Blondel, 2004;
Chun and Fessler, 2013; Grangeat et al., 2002; Hinkle et al.,
2012; Isola et al., 2010; van Stevendaal et al., 2008).
2.11.10.3.7 Local tomographyMethods for local tomography aim to reveal the singularities
(e.g., edges) in the object f rather than reconstructing the object
itself (Faridani and Ritman, 2000; Faridani et al., 1992, 1997;
Gelfand and Graev, 1991; Katsevich, 1997, 2006, 2008;
Katsevich et al., 2011; Ramm and Katsevich, 1996; Smith and
Keinert, 1985; Spyra et al., 1990; Vainberg et al., 1981). Like
interior tomography approaches, these methods also can work
from transaxially truncated sinogram data. Some local tomog-
raphy methods have been based on wavelet decompositions
(Olson and DeStefano, 1994; Rashid-Farrokhi et al., 1997).
2.11.11 Iterative Image Reconstruction
The preceding sections focused on analytic image reconstruc-
tion methods that are noniterative. The analytic methods are
derived for idealized models with a continuum of measure-
ments, and the data sampling is an afterthought as discussed
288 Fundamentals of CT Reconstruction in 2D and 3D
in Section 2.11.5. In contrast, iterative methods account for
the fact that real sinograms are discretely sampled, as described
in [57]. Practical CT scanners record only a finite set of mea-
surements yi ¼ y1; . . . ; yndð Þ, where nd denotes the total number
of measured rays, for example, nd¼n’nR for the 2D parallel-
beam geometry [57]. Because y has a finite dimension but the
unknown object f(x, y) lies in an infinite dimensional space,
the problem of reconstructing the object f(x, y) from the mea-
sured sinogram y is ill posed.
2.11.11.1 Object Model
To make the problem tractable, the first step is to parameterize
the object using a finite-series expansion (Censor, 1983):
f x; yð Þ ¼Xnpj¼1
xjbj x; yð Þ [103]
where {bj(x, y)} are user-selected basis functions that should
be linearly independent. With such a parameterization,
the image reconstruction problem is to estimate the coefficients
x ¼ x1; . . . ; xnp �
from the sinogram y. Square pixels are themost
commonly used basis function bj x; yð Þ ¼ rectx�~xjDX
�rect
y�~yjDY
�,
where ~xj; ~yj
�denotes the coordinates of the jth pixel and (DX,
DY) denotes the pixel size (usually square). Another popular
choice of basis function is Kaiser–Bessel window functions
(also known as blobs) (Lewitt, 1990). These are smooth func-
tions with finite support that are approximately band-limited.
2.11.11.2 Measurement Model
Substituting the finite-series representation [103] into the
measurement models [57] and [6] yields
yi¼ðL ri;’ið Þ
f x; yð Þd‘ ¼ðL ri ;’ið Þ
Xnpj¼1
xjbj x; yð Þd‘ ¼Xnpj¼1
aijxj [104]
where ri, ’i are the radial and angular coordinates of the ith ray
and the nd�nd system matrix A has elements
aij¼ðL ri ;’ið Þ
bj x; yð Þd‘ [105]
that correspond to the length of intersection of the ith ray
through the jth pixel. The data are also corrupted by noise eso we rewrite [104] in matrix–vector form as
y5Axþe [106]
The length of intersection model [105] does not account
the finite size of x-ray detectors and the x-ray focal spot and can
lead to aliasing artifacts (De Man and Basu, 2004). In practice,
usually, more accurate models for the system matrix A are used
that account for the detector response, such as the distance-
driven (De Man and Basu, 2004) or separable footprint (Long
et al., 2010) methods. Any such models are approximations
because the finite detector effects take place in the intensity
domain, which is nonlinear (Joseph and Spital, 1981). We
focus hereafter on reconstructing from the linear model
[106]; see Fessler (2000) for a survey of iterative methods for
reconstructing from intensity measurements.
2.11.11.3 Algebraic Methods
The algebraic algorithms for image reconstruction focus on
‘solving’ the linear system of equations y¼Ax in [106] itera-
tively. We begin with an initial guess of the reconstructed
image x(0) (typically either a uniform image or a FBP image)
and then compute recursively a sequence of estimates {x(n)}
that ideally should approach some desired limit.
The original algebraic reconstruction technique (ART) iter-
atively updated x(n) using one ray at a time (the ith row of A
and the measurement yi) (Gordon et al., 1970; Herman and
Lent, 1976, p. 280). Under certain conditions, ART converges
(Herman et al., 1978) to the minimum-norm solution of
Ax¼y, that is, it finds an image that is consistent with the
data. For low-dose CT, the measurements are very noisy, so a
solution consistent with the data is also likely to be very noisy.
Variations of ART include multiplicative ART (Gordon
et al., 1970); (Herman and Lent, 1976, p. 284) and simulta-
neous ART (Andersen and Kak, 1984). See Qu et al. (2009) for
necessary and sufficient convergence conditions.
2.11.11.4 Statistical Models
Algebraic methods are suboptimal for low-dose CT scans
because they do not consider the statistical properties of CT
measurements. If CT scanners had photon-counting detectors,
the measurements would have simple Poisson distributions,
provided that dead-time losses are modest (Yu and Fessler,
2000, 2002). However, the current clinical x-ray CT scanners
use current integrating detectors for which the measurement
statistics are considerably more complicated (Nuyts
et al., 2013).
Noise models for x-ray CT measurements are important
(Whiting, 2002; Whiting et al., 2006) for statistical image
reconstruction methods (particularly for low-dose scans), for
performing realistic simulations of CT scans, for adding noise
to CT measurements to ‘synthesize’ lower-dose scans, and
for designing sinogram data compression schemes (Bae and
Whiting, 2001).
There are numerous sources of variability that affect the
measurement statistics in CT, including x-ray tube current fluc-
tuations (Nien and Fessler, 2013; Thibault et al., 2007b), the
number of x-ray photons transmitted toward a given detector
element, the energy of each transmitted photon, the absorp-
tion and scattering of x-ray photons within the object, the
interactions of x-ray photons within a given detector element,
the light photons produced by scintillation, the conversion of
light photons into photoelectrons, and electronic noise in the
data acquisition system, including quantization in the analog-
to-digital converters.
Because of the complexity of the noise statistics, in practice,
approximate statistical models are used for iterative reconstruc-
tion. Perhaps, the simplest statistical model is to treat the
(intensity) measurements as if they have Poisson distributions
(Fessler, 2000). That model ignores electronic noise, so an
alternative is to consider a ‘Poisson–Gaussian’ model (Ma
et al., 2012; Snyder et al., 1993, 1995) that approximates the
Fundamentals of CT Reconstruction in 2D and 3D 289
combined effects of x-ray quanta and electronic noise. A model
that accounts for the polyenergetic x-ray spectrum is the
compound Poisson distribution (Elbakri and Fessler, 2003;
Whiting, 2002). An approximation to this model is the expo-
nential dispersion approach of Xu and Tsui (2007, 2009).
Currently, the most prevalent approach to handling the
complex statistics of x-ray CT measurements is to approximate
that log sinogram as having a Gaussian distribution with
ray-dependent variances (Fessler, 1994; Sauer and Bouman,
1993; Thibault et al., 2007a), leading to WLS criteria for
image reconstruction, where the weighting uses an appropriate
model for the noise variance: wi ¼ 1Var yif g. The weighting should
account for the recorded x-ray intensity and electronic noise
(Thibault et al., 2006).
2.11.11.5 Regularized Weighted Least Squares
A WLS approach to image reconstruction would form a recon-
structed image x by minimizing a cost function of the form
x ¼ arg minx��0
1
2y � Axk k2W [107]
where W¼diag{wi} is a diagonal weighting matrix that
accounts for the variance of each ray and kzkW2 ¼z 0Wz. Unfor-
tunately, even with statistical weighting W, in the presence of
noise, the best fit image x according to the WLS cost function
will be undesirably noisy because x-ray CT image reconstruc-
tion is an ill-conditioned problem for typical values of nd and
np. Indeed, often the WLS cost function has multiple solutions.
To reconstruct images with better noise properties, we find
an image that compromises between fitting the data and
matching our a priori expectations about desirable image char-
acteristics by minimizing a regularized WLS cost function:
x ¼ arg minx��0
C xð Þ, C xð Þ≜ 1
2y � Axk k2W þ bR xð Þ [108]
where R(x) is a regularizer that typically encourages the image
to be piecewise smooth and b is a regularization parameter that
controls the trade-off between spatial resolution and noise
(Fessler, 1996; Fessler and Rogers, 1996). A typical edge-
preserving regularizer penalized the differences between neigh-
boring voxels (Thibault et al., 2007a):
R xð Þ ¼Xnpj¼1
Xk2N j
cjk xj � xk �
[109]
where Nj denotes the set of indices of the neighbors of the jth
voxel (typically the nearest 6 or 26 neighbors for 3D CT) and
each c is a potential function that determines how strongly to
penalizes differences. Numerous potential functions have been
investigated, and their choice can strongly influence image
resolution, noise variance, and noise texture (Thibault et al.,
2007a). Unfortunately, there is no known theoretical criterion
for selecting from the many potential function options, so
empirical studies are needed. Quadratic potential functions,
that is, c(t)¼ t2, are the simplest to implement and analyze
but lead to undesirable resolution loss. To avoid blurring
edges, one must use a potential function that increases less
rapidly than a quadratic function, for example, a hyperbola:
c tð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2 þ t2
p� d [110]
When the difference between neighboring voxel values is
less than the parameter d, this potential function is approxi-
mately quadratic and smooths noise. For values greater than d,the potential function increases approximately linearly, rather
than quadratically, thereby penalizing less across edges and
better preserving edges between regions of sufficiently high
contrast.
2.11.11.6 Total Variation Regularization and Sparsity
For problems that are severely underdetermined, such as
limited-angle tomography or sparse-angle tomography, strong
forms of regularization are needed and many recent publica-
tions have used the absolute value function as the potential
function c(t)¼ |t|. This is equivalent to letting the parameter
d!0 in the hyperbola potential function [110]. For a 2D
digital image f[n, m], the discrete-space version of the edge-
preserving regularizer [109] for the absolute value potential
function becomesXn
Xn
f n;m½ � � f n� 1,m½ �j j þ f n;m½ � � f n,m� 1½ �j j
This regularizer is called the anisotropic total variation (TV) or
the bilateral TV of the image f[n,m]. The isotropic version of TV is
Xn
Xn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif n;m½ � � f n� 1,m½ �j j2 þ f n;m½ � � f n,m� 1½ �j j2
q[111]
(Strictly speaking, this version is isotropic only in continu-
ous space.) TV regularizers often yield images that are some-
what ‘cartoonlike,’ apparently in part due to the nonsmooth
nature of the absolute value potential function at zero. This
property may be acceptable in interventional applications
where one may be tracking positions of relatively high-contrast
objects from a limited number of projection views. But for
clinical diagnosis from low-dose CT scans that are angularly
well sampled, potential functions that are differentiable at
zero, like [110], seem to have preferable noise properties
(Thibault et al., 2007a).
TV regularization is best suited to objects that are piecewise
constant, that is, objects whose gradient is sparse (zero every-
where except at region boundaries). The field of compressed
sensing is built upon assuming that signals are sparse with
respect to some representation. The other most popular family
of sparsifying transforms is wavelets. Compressed sensing
methods seem particularly well suited to dynamic imaging
problems where the data are often unavoidably undersampled
due to object motion or contrast dynamics (Chen et al., 2008).
2.11.11.7 Optimization Algorithms
Image quality in iteratively reconstructed images is determined
by the choice of the object model, that is, the basis functions in
[103]; the systemmodel, that is, a systemmatrix A in [106]; the
statistical model, for example, a weighting matrix W in [107];
and the regularizer in [109]. After one chooses all those
290 Fundamentals of CT Reconstruction in 2D and 3D
components, what remains is the numerical problem of min-
imizing the cost function C(x) in [108]. This optimization
problem requires iterative algorithms, and finding algorithms
that converge quickly to the minimizer x is an area of active
research. Classical algorithms include coordinate descent
(Sauer and Bouman, 1993) and the conjugate-gradient
method (Fessler and Booth, 1999). Most clinical positron
emission tomography (PET) and SPECT systems today use a
maximum likelihood expectation maximization (ML-EM)
algorithm for emission tomography. This emission ML-EM
algorithm is unsuitable for transmission CT because the statis-
tics of transmission and emission tomography are entirely
different (Fessler, 2000). There does exist an EM algorithm
for transmission tomography (Lange and Carson, 1984), but
it converges too slowly for practical use. More recent algo-
rithms use optimization transfer principles to provide faster
convergence (Erdogan and Fessler, 1999a; Kim et al., 2013b).
These algorithms can be further accelerated by incorporating
momentum terms (Kim et al., 2013a) or ordered subsets
(Erdogan and Fessler, 1999b). Augmented Lagrangian
methods are also being investigated (McGaffin et al., 2012;
Ramani and Fessler, 2012).
2.11.11.8 Example
Figure 24 illustrates the sparse-view tomography problem of
reconstructing a 128�128 image from a noisy sinogram
with only 50 projection views. FBP with a ramp filter has
unacceptable streak artifacts due to the sparse angular
units: 1 cm-1
x1
1281 128
0
0.4
1
RMSE = 0.048 cm-1
FBP ramp1
1281 128
0
0.4
1
Figure 24 Reconstruction from sinogram with only 50 views by FBP with a rthe regularized weighted least squares (WLS) cost function [108] using the h
sampling and noise due to the ramp filter. FBP with a Hanning
window reduces the noise but significant streaks remain and
the image is visibly blurred. In contrast, reconstructing the
image using the regularized WLS cost function [108], mini-
mized by an iterative ordered-subset algorithm (Erdogan and
Fessler, 1999b), yields an image with reduced noise and
reduces streak artifacts without too much visible blur for this
piecewise constant phantom. Here, we used the hyperbola
potential function [110] with d¼0.01 cm�1, which is fairly
small, so the results are similar to using TV regularization
[111]. The drawback of iterative methods is their computation
time; the WLS image in Figure 24 used 20 iterations, each of
which used both a forward projection and a back projection
calculation, so it required about 40 times the computation
of the FBP image that needs only a single back projection.
Accelerating iterative algorithms is an active area of research.
2.11.12 Summary and Future Trends
This chapter first reviewed analytic methods for tomographic
image reconstruction. We have considered the parallel-beam
geometry, fan-beam geometries for both flat and curved detec-
tors, and both types of cone-beam geometries. For analytic
methods, each geometry requires its own derivation. In contrast,
for iterative image reconstruction, the basic linear model [106]
is the same for any geometry, just the system matrix A changes.
This flexibility of iterative methods is one of their advantages.
RMSE = 0.010 cm-1
PWLS1
281 128
0
0.4
RMSE = 0.029 cm-1
FBP hanning1
281 128
0
0.4
amp filter, FBP with a Hanning window, and iterative reconstruction usingyperbola potential function [110] with d¼0.01 cm�1.
Fundamentals of CT Reconstruction in 2D and 3D 291
Because of the shift invariance of (parallel-beam) projec-
tion and back projection, the primary tool for understanding
analytic image reconstruction methods is Fourier analysis. In
contrast, linear algebra, statistics, and optimization provide the
foundation for iterative methods.
Further acceleration of iterative algorithms for x-ray CT
image reconstruction will continue to be an active research
area because faster computation is essential for routine clinical
use. Moore’s law alone will not solve the computation problem
in iterative reconstruction because the problem sizes in CT
continue to grow due to higher angular sampling rates, the
use of multiple x-ray sources, and wider cone angles. More
accurate modeling of the x-ray physics, such as focal spot size
and detector fill fraction, will also increase computation
(La Riviere and Vargas, 2008). Model-based methods for
motion-compensated image reconstruction (MCIR) are also
computationally demanding (Fessler, 2010; Jacobson and
Fessler, 2003, 2006). MCIR is particularly challenging for non-
rigid motion (as expected in cardiac scans) and will remain an
important research area. Dynamic imaging problems (such as
perfusion CT) are also a significant challenge for iterative
reconstruction.
Dual-energy CT (DECT) has become available commer-
cially in recent years through various advances in technology.
There need to be more studies on iterative methods for DECT
image reconstruction to make full use of the energy informa-
tion with acceptable doses. The extension of DECT to spectral
CT measurements with more than two energy bins is also an
important future direction in image reconstruction research.
An important challenge in using iterative methods for
image reconstruction is that they typically are nonlinear,
which makes it difficult to analyze and predict the image
quality properties of the resulting images. There has been
some progress on analyzing the spatial resolution and noise
properties of nonquadratically regularized image reconstruc-
tion methods (Ahn and Leahy, 2008), but predicting perfor-
mance of detection tasks, particularly for unknown location
problems, remains a challenging problem (Zeng and
Myers, 2011).
For surveys of this active field, see De Man and Fessler
(2010), Fessler (2000), Hsieh et al. (2013), Nuyts et al.
(2013), and Peyrin and Douek (2013).
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Relevant Websites
http://en.wikipedia.org/wiki/List_of_open-source_healthcare_software.http://stir.sourceforge.net/.http://web.eecs.umich.edu/ fessler/code/.
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