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Image reconstruction in PET

Image reconstruction in PET. Luis Manuel Janeiro. Instituto de Biofísica e Eng. Biomédica - Fac. Ciências Univ. Lisboa – Lisboa Service Hospitalier Frédéric Joliot – C.E.A - Orsay. m26307@fc.ul.pt. Aims. To briefly show how data could be acquired and organized, in PET.

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Image reconstruction in PET

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  1. Image reconstruction in PET Luis Manuel Janeiro Instituto de Biofísica e Eng. Biomédica - Fac. Ciências Univ. Lisboa – Lisboa Service Hospitalier Frédéric Joliot – C.E.A - Orsay m26307@fc.ul.pt

  2. Aims • To briefly show how data could be acquired and organized, in • PET. • To overview the different approaches used for the image reconstruction, in PET. “PET” understood as the standard PET.

  3. z Data acquisition in PET LOR y Transaxial plane Line of response Oblique planes y’  y’ y x 3D aquisition x’ y’ Scanner axis x’   2 0 = 0 1  x’ x Values go to this line Values go to this line Values go to this line x’ x’ = xcos + ysin y’ = -xsin + ycos Sinogram organization

  4. Data acquisition and image reconstruction in PET Data Acquisition Septa used between axial planes No septa between axial planes 2D 3D Reconstruction 2D Rec. Rebinning 3D Rec. • Concerning the scanner sensitivity, a 3D acquisition is better than 2D. • The rebinning operation preserves the increased sensitivity of a 3D acquisition.

  5. Different approaches for image reconstruction (I) 3D data Analytical How to reconstruct? y’ y The total number of coincidences between any detector pair is, approximately, a line integral through the source distribution. g(x,y) x’  O x Reference frame Search for an exact solution for the equation: P(x’) (example for a transaxial plane) Radon transform

  6. Analytical reconstruction: FBP Inversion of the Radon transform: The objective... The tool... Central slice theorem Relates de 2D Fourier Transform of the object with the 1D FT of its projection, along a certain direction. The algorithm... Two steps Different filters are used in practice: Hanning, Hamming, Butterworth, etc... 1st) Filtering 2nd) Integration An integration along a sinusoid in the Radon domain. Backprojection operation

  7. z n FBP 2D FBP 3D y Image Estimate 3D Image x Complete projections Forward projection Virtually completed projections Analytical reconstruction: 3DRP 3D FBP Non truncated projections are needed... The objective... To recover g(x,y,z) from Finite size The tool... Central section theorem Truncated oblique proj. Relates each section of the 3D Fourier transform of the image through the origin with the 2D FT of its projection, along a certain direction. 3D RP Reconstruction is possible for projections satisfying Orlov’s sufficienty condition The algorithm... 1st) Filtering step Colsher Filter 2nd) Backprojection

  8. Disadvantages of analytical reconstruction The main disadvantages of an analytical reconstruction, are: • The limitation imposed by the approximations implicit in the line integral model onto which the formulae are based. It is not possible to model the detection and acquisition process. • Does not take in account the statistical variability inherent to the photon limited coincidence detection. The noise is controlled at the expense of resolution, varying the cut-off frequency of a filter applied to the sinograms. • The reconstruction problem is formulated on a continuous base. Discretization is a contingence.

  9. Advantages of analytical reconstruction The main advantages of an analytical reconstruction, are: • Well known properties • Easy to implement There are many groups using this type of reconstruction. Can be useful to compare results • Fastest reconstruction Differences are becoming less significant with the use of faster computers

  10. Different approaches for image reconstruction (II) 3D data Algebraic How to reconstruct? FBP (2D rec.) 3DRP (3D rec) Analytical Needed... • Image parameterisation:  = {j, j = 1, …, n}. • Model for the acquisition process: • Objective function – a distance to be minimized. • Iterative algorithm The object as a group of VOXELS(volume elements) Relates the estimated image - f(A,) - with the acquired data – y. System matrix Least-squares

  11. System matrix Used to model the aquisition and detection processes Factorized representation... A = Adet.sen.Adet.blur.Aattn.Ageom.Apositron Adet.sen: matrix that contains the detection efficiency of each detector pair. Adet.blur: local blurring function applied to the sinogram, that accounts for: the not exact co-linearity of photons, the scattering of photons from one crystal to another; and the fact that a photon may penetrate through one or more crystals before being stopped. Aattn: matrix with the attenuation terms. Ageom: contains the geometrical mapping between the source and data. The (i, j)th element is equal to the probability that a photon pair produced in voxel j reaches the front faces of the detector pair i Apositron: includes the effect of the distance travelled by the emitted positron.

  12. Different approaches for image reconstruction (III) 3D data How to reconstruct? FBP (2D rec.) 3DRP (3D rec) Algebraic Analytical Needed... Statistical model for the data? • Image parameterisation:  = {j, j = 1, …, n}. • Model for the acquisition process • Objective function – a distance that should be minimized • Iterative algorithm (to minimize the objective function) No Yes Non Statistical Reconstruction It is better to go back to the objective function !!! ART Algebraic Reconstruction Technique

  13. Objective function and likelihood A statistical measure which is maximized when the difference between estimated and measured projections is minimized. Likelihood: Obtain an estimate, , of  (source activity) which maximizes the probability p(y| ) of observing the actual detector count data, y, over all possible densities . Assuming a distribution for the data, the maximum of the likelihood function is the minimum of a distance. Least squares Gaussian data Likelihood function Regularization term Relates the image with the measured data. Relates the image with a prior.

  14. Different approaches for image reconstruction (IV) 3D data How to reconstruct? FBP (2D rec.) 3DRP (3D rec) Algebraic Analytical Needed... Statistical model for the data? • Image parameterisation:  = {j, j = 1, …, n}. • Model for the acquisition process Objective function – a distance that should be minimized • Iterative algorithm (to minimize the objective function) Yes No Objective function Non Statistical Reconstruction Non Bayesian rec. Iterative algorithms ART Bayesian reconstruction

  15. Bayesian reconstruction In the objective function it is includedaregularization term Inclusion of informationa priori Examples of Bayesian algorithms: GEM: Generalized Expectiation Maximization OSL: One-step-late SAGE: Space-alternating generalized EM Gradient ascent methods These algorithms could produce superior results than analytical or non-Bayesian methods for image reconstruction. Advantages: High computational cost. Disadvantages: The behavior of these nonlinear methods is not well understood “Statistical Approaches in Quantitative Positron Emission Tomography”, R. Leahy and J. Qi, Statistics and Computing, Vol. 10, April 2000, pp 147-165

  16. Non-Bayesian reconstruction: ML-EM ML-EM: Maximum-Likelihood Expectation-Maximization The iterative algorithm used to maximize the likelihood The objective function to maximize Discretized object. Model for the acquisition and detection process (system matrix). Poisson data. Assumptions: Measured data The algorithm includes two steps: • E-step: calculation of the expectation The estimated data, based on the model and the source distiribution estimated in the previous iteration. • M-step: maximization of the expectation

  17. Non-Bayesian reconstruction: OS-EM OS-EM: Ordered Subsets Expectation-Maximization In practice, understood as an accelerated version of ML-EM M subsets 1 iteration  (M EM iterations) 1 subset (only part of the acquired projections) 1 sub-iteration Steps involved... 1) Initialization of : , n = 0 2) To repeat in each iteration (update): 64 projections (64 samples in ) 2 projections / subset for subset s = 1, ..., M 32 subsets / iteration

  18. Advantages and disadvantages of OS-EM over analytical reconstruction Advantages: • The inclusion of a model for the aquisition/detection process. It is possible to account for the attenuation, scatter, detectors efficiency, and other effects. • Statistical properties of the data are considered (Poisson data) • Better noise properties Noise amplitude is lower in regions of low counts. Absence of striking artifacts. Disadvantages: • Less well known properties. • Computational cost. Something that has been improved with the use of OS-EM instead of ML-EM

  19. Different approaches for image reconstruction (V) 3D data How to reconstruct? FBP (2D rec.) 3DRP (3D rec) Algebraic Analytical Needed... Statistical model for the data? • Image parameterisation:  = {j, j = 1, …, n}. • Model for the acquisition process Objective function – a distance that should be minimized • Iterative algorithm (to minimize the objective function) Yes No Objective function Non Statistical Reconstruction Rebinning? Non Bayesian rec. ML-EM OS-EM ART Bayesian reconstruction GEM, OSL, SAGE, ...

  20. f s f s Rebining algorithm An algorithm which sorts the 3D-data into an ordinary 2D data set. FORE1 – Fourier Rebinning SSRB FORE B N2 Oblique Sinograms A 3D ACQUISITION FORE - 2Rec. REBINNING 3D Image 2N-1 Slices Fourier Space 2D Rec. 2N-1 Direct Sinograms (1) “Exact and Approximate Rebinning Algorithms for 3D PET Data”, Michel Defrise, P. E. Kinahan, D. Townsend, C. Michel, M. Sibomana, D,F Newport, IEEE TMI 16(2), 1997, pp 145-158

  21. Different approaches for image reconstruction (VI) 3D data How to reconstruct? FBP (2D rec.) 3DRP (3D rec) Algebraic Analytical Needed... Statistical model for the data? • Image parameterisation:  = {j, j = 1, …, n}. • Model for the acquisition process Objective function – a distance that should be minimized • Iterative algorithm (to minimize the objective function) Yes No Objective function Non Statistical Reconstruction Rebinning? Non Bayesian rec. ML-EM OS-EM ART Yes Bayesian reconstruction GEM, OSL, SAGE, ... 2D Reconstruction

  22. Rebinning + 2D reconstruction 3D data Consistent projections, corrected for randoms, attenuation, scatter, etc. Needs... FORE (“Fourier Rebinning”) Rebinning OS - EM (“Ordered Subsets Expectation Maximization”) 2D Reconstruction Assumes... No more POISSON DATA X POISSON DATA AW-OSEM NEC-OSEM Decorrect data and include the corrections in the matrix Aij

  23. Different approaches for image reconstruction (VII) 3D data How to reconstruct? FBP (2D rec.) 3DRP (3D rec) Algebraic Analytical Needed... Statistical model for the data? • Image parameterisation:  = {j, j = 1, …, n}. • Model for the acquisition process Objective function – a distance that should be minimized • Iterative algorithm (to minimize the objective function) Yes No Objective function Non Statistical Reconstruction Rebinning? Non Bayesian rec. ML-EM OS-EM ART Yes No Bayesian reconstruction GEM, OSL, SAGE, ... Fully 3D Reconstruction How to deal with corrections applied to the data? 2D Reconstruction

  24. CONCLUSIONS In most of the situations, a 3D aquisition is better than a 2D aquisition. The properties of analytical algorithms are well established and then these algorithms could be useful, for example, to test the performance of a new scanner. The assumption of the statistical properties of data and the inclusion of a model for the acquisition/detection process support the better results obtained with statistical algebraic algorithms. Among the statistical algorithms, OS-EM is becoming widespread and assuming an important role as an alternative to FBP. Due the computational cost (timing constraints), a solution to reconstruct 3D data is the use of a rebinning algorithm followed by a 2D reconstruction. There is no significant loss, but a fully 3D reconstruction, if affordable, would be preferable.

  25. S. Miguel - Azores

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