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Image Reconstructionfrom Projections
J. Anthony Parker, MD PhD
Beth Israel Deaconess Medical Center
Boston, Massachusetts
Caveat Lector
ProjectionSingle Slice
Axial
Single Axial Slice: 3600
collimator
Ignoring attenuation, SPECT data are projections
Attenuation: 180o = 360o
keV
150
100
80
60
50
x
Tc-99m
htl(140 keV) ≈ 4 cm
Cardiac Perfusion Data Collection Special Case - 180o
AxialCoronal / Sagittal
Multiple simultaneous axial slices
Dual-Head General-Purpose Gamma Camera: 900 “Cardiac” Position
2 heads: 900 rotation = 1800 data
1
2
Inconsistent projections“motion corrected”
Original data
0
0
0
00
0
0
0
Single Axial Slice: 3600
Sinogram: ProjectionsSingle Axial Slice
0 60
060
pro
ject
ion
an
gle
x
x
x
Uniformity & Motion on Sinogram
1 h
ea
d2
4 m
in
2 h
ea
ds
12
min
12
min
Reconstruction by Backprojection
Backprojection tails
Backprojection2 projections2 objects
projection tailsmerge resulting
in blurring
Projection -> Backprojection of a Point
(1/r)
backprojectionlines add atthe point
tails spread point out
Projection -> Backprojection
Projection->Backprojection Smooths
Smooths or “blurs” the image
(Low pass filter)
((Convolution with 1/r))
Nuclear Medicine physics
Square law detector adds pixels
-> always blurs
Different from MRI (phase)
(Projection-Slice Theorem)“k-space (k,)”
detail
lowfrequency
spatial frequency domainspatial domain2D Fouriertransform
Spatial Frequency Basis Functionsf(u,v) ≠ 0, single u,0f(u,v) ≠ 0, single 0,v
f(u,v) ≠ 0, single u = v
Projection -> Backprojection: k-space
1/k
(Density ofslices is 1/k)
(Fourier Transform of 1/r <-> 1/k)
one projectionmultiple projections
Image Reconstruction: Ramp Filter
Projection -> Backprojection
blurs with 1/r in object space
k-space 1/k ( 1/r<-> 1/k)
Ramp filter
sharpen with k
(windowed at Nyquist frequency)k
k
Low Pass Times Ramp Filter
Low pass,Butterworth– noise
Ramp –reconstruct
What’s Good about FPB
Ramp filter exactly reconstructs projection
Efficient
(Linear shift invariant)
(FFT is order of n log(n)
n = number of pixels)
“Easily” understood
New Cardiac Cameras
Solid state - CZT: $$$, energy resolution
scatter rejection, dual isotope
Pixelated detector: count rate &
potential high resolution
poorer uniformity
Non-uniform sampling: sensitivity
potential for artifacts
Special purpose design
closer to patient: system resolution
upright: ameliorates diaphragmatic attenuation
Collimator Resolution*
Single photon imaging (i.e. not PET)
Collimators: image formation
Sensitivity / resolution trade-off
Resolution recovery hype
“Low resolution, high sensitivity ->
image processing = high resolution”
Reality - ameliorates low resolution
Steve Moore: “Resolution: data = target object”
Can do quick, low resolution image
* not resolution from reduced distance due to design
Dual Head: Non-Uniform Sampling
Activity Measurement: Attenuation
keV
150
100
80
60
50htl(140 keV) ≈ 4 cm
Attenuation Correction: Simultaneous Emission (90%) and Transmission (10%)
Gd-153 rods T1/2 240 d e.c. 100% 97 keV 29% 103 keV 21%
2 heads: 900 rotation = 1800 data
Semi-erect: Ameliorates Attenuation
Leaning Forward, < 500 Pounds
Digirad: Patient RotatesX-ray Attenuation Correction
CT: Polychromatic Beam -> Dose
keV
150
100
80
60
50
X-ray Tube Spectra
bremsstrahlung
characteristic X-rays
e- interaction:- ionization- deflection
X-ray tube: electrons on Tungsten or Molybdenum
Digirad X-ray Source: X-rays on Lead
74W
82Pb
X-rays interaction- ionization- no 10 bremsstrahlung
Digirad X-ray Spectrum
New Cardiac Cameras
D-SPECT CardiArc Digirad GE
Detector CZT* NaI(Tl) CsI(Tl) CZT*
Electronics SS* PMT PD*? SS*
Pixelated Y N Y Y
Collimation holes slits*? holes pinholes
Non-uniform Y* Y* ~N Y*
Limited angle Y Y N ~N
Closer to pt Y Y Y ~N
AC N CT? CT* CT
Position ~semi semi erect supine
Soft Tissue Attenuation: Supine
breast
lung
Soft Tissue Attenuation: Prone
breast
Soft Tissue Attenuation: Digirad Erect
breast
post
Sequential Tidal-Breathing Emission and Average-Transmission Alignment
Sensitivity / Resolution Trade-Off
Non-uniform sampling -> sensitivity
Special purpose design -> resolution
Advantages
Throughput at same noise
Patient motion - Hx: 1 head -> 2 head
Cost
Non-uniform sampling -> artifacts
History: 7-pinhole - failed
180o sampling - success
Sequential emission transmission
What’s Wrong with FilteredBackprojection, FBP, for SPECT
Can’t model:
Attenuation
Scatter
Depth dependant resolution
New imaging geometries
(Linear shift invariant model)
Solution
Iterative reconstruction
Uses:
Simultaneous linear equations
Matrix algebra
Can model image physics
(Linear model)
Projections as Simultaneous Equations(Linear Model)
But, exact solution for a largenumber of equations isn’t practical
Iterative Backprojection Reconstruction
Af
n
p
fn-1^ pn-1
^ en-1^
fn^
+
- x
+
f0^
r
H
H
A
object data
projection backprojection
estimate
model
error
estimate
estimateddata
estimate +backprojected
error
Reconstruction, H, can be Approximate
Af
n
p
fn-1^ pn-1
^ en-1^
fn^
+
- x
+
f0^
r
H
H
A
Accuracy of Model, A, is Key
Af
n
p
fn-1^ pn-1
^ en-1^
fn^
+
- x
+
f0^
r
H
H
A
^
Model, A, is Well-known PhysicsProblem: Model of the Body
^
Tc-99m half-tissue layer: 4 cm
Attenuation Map Gd-153 Transmission
Map adds noise to reconstructionand can introduce artifacts
Iterative ReconstructionNoise is “Blobby”
What’s Good About Iterative Reconstruction
Able to model:
Data collection, including new geometries
Attenuation
Scatter
Depth dependant resolution
Fairly efficient given current computers
(Iterative solution, e.g. EM, reasonable)
(OSEM is even better)
((OSEM has about 1/nsubsets of EM iterations))
What’s Wrong with Iterative Reconstruction
(Complicated by ill conditioned model)
((Estimating projections not object))
Noise character bad for oncology
To model attenuation & scatter
- need to measure attenuation
- adds noise
Conclusions
Filtered backprojection, FBP
Efficient
(Models noise)
“Easy” to understand
Iterative reconstruction, OSEM
Moderately efficient
Models noise, attenuation, scatter,
depth dependant resolution,
and new cameras
Applause
