TOMOGRAPHIC IMAGE RECONSTRUCTION TOMOGRAPHIC IMAGE RECONSTRUCTION FOR FOR

PARTIALLY-KNOWN SYSTEMS AND IMAGE PARTIALLY-KNOWN SYSTEMS AND IMAGE SEQUENCESSEQUENCES

M.S. Thesis Defense : Jovan Brankov

Project GoalsProject Goals

• New reconstruction algorithms

• Image reconstruction with Partially-Known system model• Applicable for PET

• Spatially adaptive temporal smoothing for image sequence reconstruction

• Applicable for dynamic PET and gated SPECT

Single Photon Emission Tomography Single Photon Emission Tomography

(SPECT(SPECT))

• Radiotracers are gamma emitters

• Isotopes Tc-99, I-123 and Ga-67

• Metal collimators

• NaI(T1) Scintillator

• Photo Multiplier Tubes (PMT)

• Drawback:

• Low sensitivity

• Advantage:

• Inexpensive

• Cyclotron not required

Positron Emission Tomography (PET)Positron Emission Tomography (PET)

• Radiotracers are positron emitters

• Isotopes 11C, 13N, 18F

• Electronic collimation

• NaI(T1) Scintillator

• Photo Multiplier Tubes (PMT)

• Drawback:

• Requires a cyclotron

• Advantage:

• High sensitivity

Image sequenceImage sequence

• Gated study

• Synchronized with a periodic process in the body

• Like stroboscopy

• Dynamic study

• Not synchronized

Partially-known systems:Partially-known systems:System modelingSystem modeling

• The behavior of the system is not exactly known• object dependent (scattering)

• errors in modeling PSFs

• errors in measurement of PSFs

• System is modeled as the sum of:• a known deterministic part

• an unknown random part

o- Actual PSF+- Assumed PSF

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Assumption

Added noise

Different variance

Laplasian

Partially-known systems:Partially-known systems:Imaging modelImaging model

• Idealized discrete model

• Discrete model based on distance-independent blur (not suitable for SPECT)

• Discrete model with PSF uncertainty

data imageSystemmatrix

E[ ]g Pf

E[ ]g APf

E[ ]g (A A)Pf

J g( ) ( ) ( )f g APf C g APf Qf T 1 2

PWLS cost function:

C SS Ig a 2 2T

nwhere

• S = circulant matrix composed of sinogram elements 2

a = PSF error variance,

2n = additive noise variance,

• = regularization parameter• Q = circulant Laplacian high-pass operator

Partially-known systems:Partially-known systems:Cost functionalCost functional

Partially-known systems:Partially-known systems:Cost Functional in DFT domainCost Functional in DFT domain

In discrete Fourier transform (DFT) domain:

Jn

f Qfa

bg

FHGG

IKJJ

1

2

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2

2 2N

G(i) A(i)S(i)

S(i)i

N-1

A(i), S(i) , and G(i) are DFT coefficients of the blurring kernel a, the computed sinogram s, and the observed sinogram g, respectively.

convolution

Partially-known systems :Partially-known systems : Conjugate gradient minimizationConjugate gradient minimization

Conjugate gradient minimizationConjugate gradient minimization

• modified conjugate gradient method

• application to non-convex cost functional • quadratic interpolation for the line-search procedure

• nonnegativity constraint

Partially-known systems:Partially-known systems:Functional gradient in DFT domainFunctional gradient in DFT domain

Functional gradient :

Pn (i) is the ith coefficient of the DFT of the nth column of the projection matrix P.

grf

fn

n a n

J

FHG

af a fc h22 2 2

0N

A (i)P (i) G(i) A(i)S(i)

S(i)n

i

N-1 Re

IKJJ

a

a n

2 2

2 2 2 2 2G(i) A(i)S(i) Re P (i)S (i)

S(i)

n

d i Q QfT

Partially-known systems:Partially-known systems:ExperimentExperiment

o-True PSF+-Assumed PSF

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Source Image Point spread functions

Forward problem:

Degrade the sinogram using the true PSF

Inverse problem:

Reconstruct using the incorrect (assumed) PSF

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Partially-known system:Partially-known system: Evaluation criteriaEvaluation criteria

Spatial mean squared error (MSE)

MSE = E1

N1 f fLNM

OQP

2

MSE = E2 LNMOQPe j2

MSE2 of Region of interest (ROI) estimates

true image reconstructed image

true value estimated value

Partially-known systems:Partially-known systems: Evaluation criteriaEvaluation criteria cont. cont.

MSE1 Vs.

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MSE as a function of for different values of the PSF noise variance 2

a assumed by the reconstruction algorithm.

Conclusions: 1. Accounting for PSF error helps. 2. Not very sensitive to variance estimate

Partially-known systems:Partially-known systems: Evaluation criteriaEvaluation criteria cont. cont.

MSE2 for hot spots Vs.

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MSE2 for cold spots Vs.

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MSE2 : Hot spots

MSE2 : cold spots

Partially-known systems:Partially-known systems: Image resultsImage results

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Figure 1. Original image

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Figure 3. Image reconstructed without modeling PSF uncertainty using: =0.013 and 2

n=100.MSE1=3642.81

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Figure 4. Image reconstructed with model of PSF uncertainty using: =0.013 2

A =1.3e-5 and 2n=100.

MSE1 =1187.23.

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Figure 2. Image reconstructed from blurred noisy sinogram using filtered back-projection.

MSE1 =5428.68

Partially-known systems:Partially-known systems: Point response

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Figure 6. Image reconstructed from blurred noisy sinogram using filtered back-projection

MSE1 =5428.68

Figure 5. Original image

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Figure 7. Image reconstructed without modeling PSF uncertainty using: =0.013 and 2

n=100. MSE1 =3642.81

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Figure 8. Image reconstructed with model of PSF uncertainty using: =0.013 2

A=1.3e-5 and 2n=100.

MSE1 =1187.23.

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Partially-known systems:Partially-known systems:Conclusion and future work

Future work Increase the rate of computation speed and reduce required

memory Use more realistic model Evaluate with different types of uncertainties Develop automatic estimation of algorithm parameters

Conclusion Improvements in the reconstructed image

• visually

• quantitatively

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:Karhunen-Loève transformation (KL)

1st 2nd 9th 16th 23rd

Original

sinograms

KL transformed

sinograms

• The Maximum noise fraction transform

• noise in all frame are equal - KL/PCAGreen et al. 1988

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:

Karhunen-Loève transformation (KLT)

Steps:

1. Karhunen-Loève transformation

2. Discard components

3. Inverse KLT

4. Reconstruct

Kao et al. IEEE TMI, 1998

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing: k-mean algorithm (Generalized Lloyd Algorithm)

y x, y xy

i i iE d C* arg min b g

C d d j iin

i j x R x, y x, y: ( ),b gn s

Step 3. Given yi recalculate cluster assignment according to :

Step 2. Given Ci calculate Centroids yi according to:

Step 1. Initialization

(random cluster assignment - Ci, i=1..k)

Step 4. Repeat steps 2 and 3 until no reassignment occurs

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:

Compare results of three reconstruction procedures:

• no sinogram preprocessing

• sinogram presmoothing by using KL transform (KL)

• sinogram presmoothing by KL transform taking into account different statistics of pixels (KL/Clustering)

• all three reconstructed with Expectation Maximization algorithm (EM).

Tested for three possible applications:• Kinetic study of the brain

• Lesion detection in dynamic PET

• Gated SPECT

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:Brain phantomBrain phantom

Realistic MRI voxel-based numerical brain phantom developed by Zubal et al.

Spatially adaptive temporal smoothing: Spatially adaptive temporal smoothing: Compartment Kinetic ModelCompartment Kinetic Model

dC t

dtk C t k C t k C t k k k C tf

p b n f

( )( ) ( ) + ( ) - ( ) ( ) 1 4 6 2 3 5

dC t

dtk C t k C tb

f b

( )( ) - ( ) 3 4

dC t

dtk C t k C tn

f n

( )( ) ( ) 5 6

C t C t C t C tb n f( ) ( ) ( ) ( )

C t e L e C tdec

t

t

iR t

pi

ni( ) ( )

( )

12

1

2

1

ln

C tL e t t

c e t tp

pmt

iD t

i

i( )

( ) ( )

( )

RS|T|

1 0 0

00

3

Equilibrium solution:

Blood curve model:

The blood sample values obtained in a PET study conducted by the Department of Radiology at the University of Chicago.

Four Compartment Kinetic Model

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing: Kinetic brain modelKinetic brain model

Brain region Tree compartment Four compartment

Thalamus - 1Caudate - 0.87

Front Cortex - 0.805Ant. Temporal +Sup. Temporal

Cortex

- 0.805

White matter - 0.1Occipital Cortex 1 -

[11C] Carfentanil Study JJ Frost et al.1990

Brain phantom

Time activation curves

Source image Cluster map in sinogram domain

Filterback projection of each cluster position separately

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing: Cluster map

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:Time activity curves (TAC)

Estimated TAC’s for thalamus and occipital cortex

Difference between the original and estimated TAC’s

thalamus and occipital cortex

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:Image resultsImage results

Third frame from dynamic brain study reconstructed with different presmoothing techniques

Reconstructed images

Differences images

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing: Lesion detection in dynamic PETLesion detection in dynamic PET

Yu et al. 1997

Time activation curves for different region in lesion dynamic studyPhantom

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing: Time activity curves

Estimated TAC’s for small lesion

Difference between the original and estimated TAC’s for small lesion

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:Image resultsImage results

Some frames from dynamic lesion study reconstructed with different presmoothing techniques

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:Torso phantomTorso phantom

The 4D gated mathematical cardiac-torso gMCAT (D1.01 version- fixed anatomy, dynamic (beating heart)) phantom.

University of Massachusetts Medical School, Worcester, MA

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:

Gated SPECT

• The goal is to preserve heart motion

• Difficult to evaluate quantitatively

• ROI on the heart wall

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing: Time activation curves

TAC of the ROI without presmooting TAC of the ROI with KL presmooting

TAC of the ROI with KL/Clustering presmooting

Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing: Conclusion and future work

Future work Apply filtering before KL coefficients estimation

(Manoj et al. 1998) Evaluate on real SPECT/PET data Evaluate for clinical use

Conclusion Possible improvement in estimation of time activation

curves for ROI’s which leads to better:• kinetic model parameters estimation

• delectability of the lesion

• observation of the heart motion and abnormalities