1. TOMOGRAPHIC IMAGE RECONSTRUCTION FOR PARTIALLY-KNOWN SYSTEMS AND IMAGE SEQUENCES M.S. Thesis Defense :Jovan Brankov

2. Project 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

3. Single Photon Emission Tomography (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

4. Positron Emission Tomography (PET) Radiotracers are positron emitters Isotopes 11 C, 13 N, 18 F Electronic collimation NaI(T1) Scintillator Photo Multiplier Tubes (PMT) Drawback: Requires a cyclotron Advantage: High sensitivity

5. Image sequence Gated study Synchronized with a periodic process in the body Like stroboscopy Dynamic study Not synchronized

6. Partially-known systems: System 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

7. Partially-known systems: Imaging model Idealized discrete model Discrete model based on distance-independent blur (not suitable for SPECT) Discrete model with PSF uncertainty dataimage System matrix

8. PWLS cost function: where S = circulant matrix composed of sinogram elements  2  a = PSF error variance,  2 n = additive noise variance, = regularization parameter Q = circulant Laplacian high-pass operator Partially-known systems: Cost functional

9. Partially-known systems: Cost Functional in DFT domain In discrete Fourier transform (DFT) domain: 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

10. Partially-known systems : Conjugate gradient minimization Conjugate gradient minimization modified conjugate gradient method application to non-convex cost functional quadratic interpolation for the line-search procedure nonnegativity constraint

11. Partially-known systems: Functional gradient in DFT domain Functional gradient : P n (i) is the i th coefficient of the DFT of the n th column of the projection matrix P.

12. Partially-known systems: Experiment o-True PSF +-Assumed PSF 0 0.1 0.2 0.3 0.4 Source ImagePoint spread functions Forward problem: Degrade the sinogram using the true PSF Inverse problem: Reconstruct using the incorrect (assumed) PSF 0 1 2 3 4 5 6 7 8 9 10

13. Partially-known system: Evaluation criteria Spatial mean squared error (MSE) MSE 2 of Region of interest (ROI) estimates true imagereconstructed image true valueestimated value

14. Partially-known systems: Evaluation criteria cont. 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

15. Partially-known systems: Evaluation criteria cont. MSE 2 for hot spots Vs. 0 50 100 150 200 250 300 350 400 00.0050.010.0150.02                             MSE 2 for cold spots Vs. 0 50 100 150 200 00.0050.010.0150.02                             MSE 2 : Hot spots MSE 2 : cold spots

16. Partially-known systems: Image results 051015202530 -2 0 2 4 6 8 10 12 14 051015202530 -2 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 16 Figure 3. Image reconstructed without modeling PSF uncertainty using: =0.013 and  2 n =100. MSE 1 =3642.81 0 2 4 6 8 10 12 051015202530 -2 0 2 4 6 8 10 12 14 051015202530 -2 0 2 4 6 8 10 12 14 Figure 4. Image reconstructed with model of PSF uncertainty using: =0.013  2  A =1.3e-5 and  2 n =100. MSE 1 =1187.23.

17. Partially-known systems: Partially-known systems: Point response Figure 5. Original image 0 1 2 3 4 5 6 7 8 9 10 05 15202530 -2 0 2 4 6 8 10 12 14 051015202530 -2 0 2 4 6 8 10 12 14

18. 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

19. Spatially adaptive temporal smoothing: Spatially adaptive temporal smoothing: Karhunen-Loève transformation (KL) 1 st 2 nd 9 th 16 th 23 rd Original sinograms KL transformed sinograms The Maximum noise fraction transform noise in all frame are equal - KL/PCA Green et al. 1988

20. 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

21. Spatially adaptive temporal smoothing: Spatially adaptive temporal smoothing: k-mean algorithm (Generalized Lloyd Algorithm) Step 3. Given y i recalculate cluster assignment according to : Step 2. Given C i calculate Centroids y i according to: Step 1. Initialization (random cluster assignment - C i, i=1..k) Step 4. Repeat steps 2 and 3 until no reassignment occurs

22. 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

23. Spatially adaptive temporal smoothing: Brain phantom Realistic MRI voxel-based numerical brain phantom developed by Zubal et al.

24. Spatially adaptive temporal smoothing: Compartment Kinetic Model 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

25. Spatially adaptive temporal smoothing: Kinetic brain model [ 11 C] Carfentanil Study JJ Frost et al.1990 Brain phantom Time activation curves

26. Source image Cluster map in sinogram domain Filterback projection of each cluster position separately Spatially adaptive temporal smoothing: Spatially adaptive temporal smoothing: Cluster map

27. 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

28. Spatially adaptive temporal smoothing: Image results Third frame from dynamic brain study reconstructed with different presmoothing techniques Reconstructed images Differences images

29. Spatially adaptive temporal smoothing: Lesion detection in dynamic PET Yu et al. 1997 Time activation curves for different region in lesion dynamic study Phantom

30. 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

31. Spatially adaptive temporal smoothing: Image results Some frames from dynamic lesion study reconstructed with different presmoothing techniques

32. Spatially adaptive temporal smoothing: Torso phantom The 4D gated mathematical cardiac-torso gMCAT (D1.01 version- fixed anatomy, dynamic (beating heart)) phantom. University of Massachusetts Medical School, Worcester, MA

33. 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

34. Spatially adaptive temporal smoothing: Spatially adaptive temporal smoothing: Time activation curves TAC of the ROI without presmootingTAC of the ROI with KL presmooting TAC of the ROI with KL/Clustering presmooting

35. 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