Reconstruction Algorithmsin Computerized Tomography

Nargol Rezvaninrezvani@cs.toronto.edu

CAIMS 2009UWO, London, Ontario.

University of Toronto

Tomography: Slice Imaging

Computerized Tomography (CT)A technique for imaging the cross sections of an object using aseries of x-ray measurements taken from different angles around

the object.

I First CT scanner invented by Godfrey Hounsfield in 1972.

I Allan M. Cormack invented a similar process in 1972.

I They shared the 1979 Nobel Prize in Medicine.I Since then researchers have tried to make

I Scanning faster,I Dosage used optimal,I Reconstruction algorithms more efficient.

I First CT scanner invented by Godfrey Hounsfield in 1972.

I Allan M. Cormack invented a similar process in 1972.

I They shared the 1979 Nobel Prize in Medicine.I Since then researchers have tried to make

I Scanning faster,I Dosage used optimal,I Reconstruction algorithms more efficient.

I First CT scanner invented by Godfrey Hounsfield in 1972.

I Allan M. Cormack invented a similar process in 1972.

I They shared the 1979 Nobel Prize in Medicine.

I Since then researchers have tried to makeI Scanning faster,I Dosage used optimal,I Reconstruction algorithms more efficient.

I First CT scanner invented by Godfrey Hounsfield in 1972.

I Allan M. Cormack invented a similar process in 1972.

I They shared the 1979 Nobel Prize in Medicine.I Since then researchers have tried to make

I Scanning faster,I Dosage used optimal,I Reconstruction algorithms more efficient.

Reconstruction Algorithms

I Direct, e.g. Filtered back-projection (FBP).

I Iterative

I Algebraic, e.g. algebraic reconstruction techniques (ART).

I Statistical image reconstruction techniques (SIRT):(Weighted) Least squares or Likelihood (Poisson), e.g.maximum log-likelihood.

FBP- Modeling Assumptions

I Straight lines (no refraction/diffraction)

I Monochromatic source (X-ray photons of the samewave-length/energy)

I Beer’s Law: X-ray intensity attenuates linearly in matter

dI

ds= −µI ⇒

∫lµ ds = − ln

(I

I0

)

I x-ray intensityµ attenuation coefficients arc length

FBP Mathematical BackgroundRadon Transform

In medical imaging, the Radon transform of a function provides amathematical model for the measured attenuation.

[Rµ(x , y)](t, ω) :=

∫l(t,ω)

µ(x , y) ds ,

- l(t,ω) = {tω + sω̂ : s ∈ R}- ω̂ unit vector perpendicular to unit vector ω .

ω(θ) = (cos θ, sin θ) , with orientation det(ωω̂) > 0 .

- t affine parameter.

Fourier TransformThe Fourier Transform of µ defined on R is

µ̂(t) =

+∞∫−∞

µ(x)e−ixt dx .

Fourier Inversion Formula

µ(x) =1

2π

+∞∫−∞

µ̂(t)e ixt dt .

These formulas can be extended to higher dimensions.

Central Slice Theorem

Key to efficient tomographic imagingRelates the measured projection data (Radon transform) to thetwo-dimensional Fourier transform of the object cross sections.

TheoremLet µ be an absolutely integrable function in the natural domain ofRadon transform. For any real number r and unit vector ω, wehave the identity

+∞∫−∞

[Rµ(x , y)](t, ω)e−itr dt = µ̂(rω) .

Filtered Back-Projection

Central slice theorem and fourier inversion formula give aninversion formula for the Radon transform,

a.k.a. filtered back-projection (FBP) formula.

FBP in two steps

I Radial integral: Filtering

I Angular integral: Back-Projection (BP)

FBP is a good starting point for the development of practicalalgorithms.

Different Geometries

Feldkamp-Davis-Kress Algorithm (1984)

FDK is the first practical algorithm for CBCT acquired from acircular source trajectory, assuming a planar detector.

I OSCaR- Open Source Cone-beam CT Reconstruction Tool forImaging Research

http://www.cs.toronto.edu/∼nrezvani/OSCaR.html

Feldkamp-Davis-Kress Algorithm (1984)

FDK is the first practical algorithm for CBCT acquired from acircular source trajectory, assuming a planar detector.

I OSCaR- Open Source Cone-beam CT Reconstruction Tool forImaging Research

http://www.cs.toronto.edu/∼nrezvani/OSCaR.html

FBP Deficiencies

I Difficulty in motion correction.

I Lack of flexibility.

I Not being able to model the noise and therefore moreradiation.

I FBP assumes monochromatic source, but in reality we havepolychromatic X-rays.

I When polychromatic X-ray beam passes through matter, lowenergy photons are absorbed.

I The beam gradually becomes harder, i.e. X-rays in ranges thatare more penetrating are referred to as hard, opposed to softX-rays that are more easily attenuated.

I If not corrected, this may cause artifacts in CT images, as aresult of inaccurate measurements.

This phenomenon is called Beam Hardening.

Algebraic Reconstruction Technique (ART)

A different approach for CT reconstruction, first introduced in1970.

In ART,

I it is assumed the cross-sections consist of arrays of unknowns.

I reconstruction problem can be formulated as a system oflinear equations.

I choosing a finite collection of basis functions and a method tosolve the systems are the next steps.

Pixel Basis

In ART localized basis, bjJj=1, such as pixel basis are typically used.

I bKj (x , y) =

{1 if (x , y) ∈ jth-square,0 otherwise.

I An approximation to µ is µ̄K =∑J

j=1 xjbKj .

I Radon transform is linear: Rµ̄K =∑J

j=1 xjRbKj .

I This stays true for many other bases.

Reconstruction Model

I Assume {bj} basis and Rµ sampled at

{(t1, ω1), . . . , (tI , ωI )} .

I For i = 1, . . . , I and j = 1, . . . , J , usually I 6= J , definemeasurement matrix as the line integrals: rij = Rbj(ti , ωi ) .

I Vector of measurements will be pi = Rµ(ti , ωi ) .

I Reconstruction problem:

J∑j=1

rijxj = pi .

Iterative methods are used for solving rx = p .

The size of these systems of equations is usually large making ARTmethods computationally expensive.

An Example

(Introduction to Mathematics of Medical Imaging– Epstein, 2003)

If the square is divided into J = 128× 128 ' 16, 000subsquares, then, using the pixel basis, there are 16,000unknowns. A reasonable number of measurements is 150samples of the Radon transform at each of 128 equallyspaced angles, so that I ' 19, 000. That gives a19, 000× 16, 000 system of equations.

Kaczmarz Method (Stefan Kaczmarz, 1937)

The principle method used in ART.

I If rx = p has a solution, method converges.

I Underdetermined systems, method converges (Tanabe 1971).

I More than one solution, initial vector 0 gives least squaressolution (Epstein).

I In medical applications a few complete iterations used.

I To reduce noise and accelerate the process, relaxation, isfrequently incorporated into ART type algorithm.

I For overdetermined linear systems, a randomized version ofKaczmarz has been proven to converge with expectedexponential rate (Strohmer and Vershynin).

I Many other results by Censor, Elfving, Herman, Gordon,Byrne, and others.

Kaczmarz Method (Stefan Kaczmarz, 1937)

The principle method used in ART.

I If rx = p has a solution, method converges.

I Underdetermined systems, method converges (Tanabe 1971).

I More than one solution, initial vector 0 gives least squaressolution (Epstein).

I In medical applications a few complete iterations used.

I To reduce noise and accelerate the process, relaxation, isfrequently incorporated into ART type algorithm.

I For overdetermined linear systems, a randomized version ofKaczmarz has been proven to converge with expectedexponential rate (Strohmer and Vershynin).

I Many other results by Censor, Elfving, Herman, Gordon,Byrne, and others.

Kaczmarz Method (Stefan Kaczmarz, 1937)

The principle method used in ART.

I If rx = p has a solution, method converges.

I Underdetermined systems, method converges (Tanabe 1971).

I More than one solution, initial vector 0 gives least squaressolution (Epstein).

I In medical applications a few complete iterations used.

I To reduce noise and accelerate the process, relaxation, isfrequently incorporated into ART type algorithm.

I For overdetermined linear systems, a randomized version ofKaczmarz has been proven to converge with expectedexponential rate (Strohmer and Vershynin).

I Many other results by Censor, Elfving, Herman, Gordon,Byrne, and others.

Kaczmarz Method (Stefan Kaczmarz, 1937)

The principle method used in ART.

I If rx = p has a solution, method converges.

I Underdetermined systems, method converges (Tanabe 1971).

I More than one solution, initial vector 0 gives least squaressolution (Epstein).

I In medical applications a few complete iterations used.

I To reduce noise and accelerate the process, relaxation, isfrequently incorporated into ART type algorithm.

I For overdetermined linear systems, a randomized version ofKaczmarz has been proven to converge with expectedexponential rate (Strohmer and Vershynin).

I Many other results by Censor, Elfving, Herman, Gordon,Byrne, and others.

Kaczmarz Method (Stefan Kaczmarz, 1937)

The principle method used in ART.

I If rx = p has a solution, method converges.

I Underdetermined systems, method converges (Tanabe 1971).

I More than one solution, initial vector 0 gives least squaressolution (Epstein).

I In medical applications a few complete iterations used.

I To reduce noise and accelerate the process, relaxation, isfrequently incorporated into ART type algorithm.

I For overdetermined linear systems, a randomized version ofKaczmarz has been proven to converge with expectedexponential rate (Strohmer and Vershynin).

I Many other results by Censor, Elfving, Herman, Gordon,Byrne, and others.

Kaczmarz Method (Stefan Kaczmarz, 1937)

The principle method used in ART.

I If rx = p has a solution, method converges.

I Underdetermined systems, method converges (Tanabe 1971).

I More than one solution, initial vector 0 gives least squaressolution (Epstein).

I In medical applications a few complete iterations used.

I To reduce noise and accelerate the process, relaxation, isfrequently incorporated into ART type algorithm.

I For overdetermined linear systems, a randomized version ofKaczmarz has been proven to converge with expectedexponential rate (Strohmer and Vershynin).

I Many other results by Censor, Elfving, Herman, Gordon,Byrne, and others.

Kaczmarz Method (Stefan Kaczmarz, 1937)

The principle method used in ART.

I If rx = p has a solution, method converges.

I Underdetermined systems, method converges (Tanabe 1971).

I More than one solution, initial vector 0 gives least squaressolution (Epstein).

I In medical applications a few complete iterations used.

I To reduce noise and accelerate the process, relaxation, isfrequently incorporated into ART type algorithm.

I For overdetermined linear systems, a randomized version ofKaczmarz has been proven to converge with expectedexponential rate (Strohmer and Vershynin).

I Many other results by Censor, Elfving, Herman, Gordon,Byrne, and others.

The k-th step of the algorithm

1:

x (k) → x (k) − x (k) · r1 − p1

r1r1· r1 = x(k,1) ,

2:

x (k,1) → x (k,1) − x (k,1) · r2 − p2

r2r2· r2 = x(k,2) ,

3:

x (k,I−1) → x (k,I−1) − x (k,I−1) · rI − pI

rI rI· rI = x(k+1) .

Reminder- Beam hardening is a phenomenon that apolychromatic X-ray beam becomes more penetrating, or harder,

as it traverses through matter.

Solutions for Beam Hardening

I Pre-filtering: Limiting beam hardening by physicallypre-filtering X-ray beams.

I Post-reconstruction: Correct measurements based on materialassumptions, e.g. Joseph and Spital method in (1978).

I By incorporating a polychromatic acquisition model.

Statistical image reconstruction techniques (SIRT) for X-rayCT can be developed based on a physical model that accounts

for polyenergetic X-ray source.

SIRT for Polyenergetic X-ray CT,(Elbakri and Fessler, 2002-2003)

SIR method for X-ray CT based on a physical model that accountsfor the polyenergetic X-ray source spectrum and the measurement

nonlinearities caused by energy-dependent attenuation.

I Object comprised of known non-overlapping tissue types.

I Attenuation coefficient in jth voxel is modeled:

µj(ε) =K∑

k=1

mk(ε)ρj fkj ,

I m(ε): energy-dependent mass attenuation, known for severaltissue types.

I ρ: unknown energy-independent density of the tissueI f k

j = 1 if jth voxel belongs to kth material, otherwise 0.

Basic Idea: Given measurements {yi}I1, find the distribution oflinear attenuation coefficients {µj}J1.

I The Poisson model for the measurements yi ’s is assumed.

I For estimating the attenuation coefficient, maximumlikelihood (ML) for the polyenergetic model, L(ρ) , isdeveloped.

I Data is noisy and ML gives noisy reconstruction, hence weneed regularization by adding a penalty term, R(ρ) to thelikelihood.

I Now our goal would be optimizing

Φ(ρ) = −L(ρ) + βR(ρ) ,

where β is a scalar that controls the tradeoff between thedata-fit and the penalty terms.

I An iterative method for estimating unknown densities in eachvoxel is developed,

I e.g. penalized weighted least squares with ordered subsets(PWLS-OS).

Other Approaches

I IMPACT (De Man et al, 2001)I Models object as a linear combination of the spectral

properties of two base substances, usually water and bone.I Knowledge of the X-ray spectrum is necessary, but a

pre-segmented image is not required.

I Simplified SIR for Polyenergetic X-ray CT(Srivastava and Fessler, 2005)

I Hybrid method between FBP-JS method and SIRT.I SIRT require more knowledge of the X-ray spectrum than is

needed in post-reconstruction methods.I Uses the same calibration data and tuning parameters in JS,

thereby facilitates its practical use.

I SR Algorithm for Polyenergetic X-ray CT (Chueh et al, 2008)I Assumes energy spectrum comprised of several sub-energy

spectrums.I Needs less a priori information in terms of tissue type and mass

attenuation coefficients.

Other Approaches

I IMPACT (De Man et al, 2001)I Models object as a linear combination of the spectral

properties of two base substances, usually water and bone.I Knowledge of the X-ray spectrum is necessary, but a

pre-segmented image is not required.

I Simplified SIR for Polyenergetic X-ray CT(Srivastava and Fessler, 2005)

I Hybrid method between FBP-JS method and SIRT.I SIRT require more knowledge of the X-ray spectrum than is

needed in post-reconstruction methods.I Uses the same calibration data and tuning parameters in JS,

thereby facilitates its practical use.

I SR Algorithm for Polyenergetic X-ray CT (Chueh et al, 2008)I Assumes energy spectrum comprised of several sub-energy

spectrums.I Needs less a priori information in terms of tissue type and mass

attenuation coefficients.

Other Approaches

I IMPACT (De Man et al, 2001)I Models object as a linear combination of the spectral

properties of two base substances, usually water and bone.I Knowledge of the X-ray spectrum is necessary, but a

pre-segmented image is not required.

I Simplified SIR for Polyenergetic X-ray CT(Srivastava and Fessler, 2005)

I Hybrid method between FBP-JS method and SIRT.I SIRT require more knowledge of the X-ray spectrum than is

needed in post-reconstruction methods.I Uses the same calibration data and tuning parameters in JS,

thereby facilitates its practical use.

I SR Algorithm for Polyenergetic X-ray CT (Chueh et al, 2008)I Assumes energy spectrum comprised of several sub-energy

spectrums.I Needs less a priori information in terms of tissue type and mass

attenuation coefficients.

Comparison of Different Reconstruction Algorithms

Compared to FBP,I ART is

I conceptually simpler.I more adaptable.I computationally more expensive and empirically speaking lacks

the accuracy and speed of implementation enjoyed by FBP.

I SIRTI incorporates the system geometry, detector response, object

constraints and any prior knowledge more easily.

I handles/models the noise more effectively, and thereforedosage used is more optimal.

I uses a polychromatic acquisition model, since it is nonlinear.

I has longer computation times, that hinder their use.

Comparison of Different Reconstruction Algorithms

Compared to FBP,I ART is

I conceptually simpler.I more adaptable.I computationally more expensive and empirically speaking lacks

the accuracy and speed of implementation enjoyed by FBP.

I SIRTI incorporates the system geometry, detector response, object

constraints and any prior knowledge more easily.

I handles/models the noise more effectively, and thereforedosage used is more optimal.

I uses a polychromatic acquisition model, since it is nonlinear.

I has longer computation times, that hinder their use.

Open Problems and Possible Future Work

I How to handle attenuation coefficient and energy levels inSIRT, e.g. decomposition of the energy-dependentattenuation coefficient.

I Reducing the number of coefficients, better optimizationmethods, more efficient and faster iterative techniques.

I Better estimation methods in SIRT.

I Hybrid method between ART and SIRT.

I Investigating more sophisticated variants to generate themeasurement matrix in ART.

I Possibly incorporating compressed sensing into SIRT.

I Using ART and SIRT in “4DCT”– fourth dimension is time.

Thank you for your attention!