Conditioning of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples

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1 of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples Benjamin C. Lee Dissertation Oral Defense May 0, 00 University of Michigan Electrical Engineering: Systems Chair: Prof. Andrew Yagle

2 Introduction Statement Previous Wor My Contributions Overview Approach Approach Divide-and-Conquer Approach Conclusion Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples

3 -D Fourier Transform Introduction Statement Previous Wor My Contributions Images are -D functions of intensity. Image High Intensity (Dense) Low Intensity (Clear) Fourier Transform analyzes image in frequency domain. -D Fourier Transform High Frequency (Detail) Low Frequency (Smoothness) Head: Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 3

4 Sampling Continuous-Space Images Assume image is spatially bandlimited to wavenumber Sampled image Introduction Statement Previous Wor My Contributions Given continuous image x ( s, s C ) with -D Fourier Transform with wavenumbers, * X j( s s ) C (, ) xc ( s, s) e dsds with -D Discrete-Time Fourier Transform (DTFT) ** X ( e x( i, i) xc ( i, i j j j( i i ), e ) x( i, i) e i i is now -D periodic in wavenumber. * Also nown as the Continuous-Space Fourier Transform (CSFT). ** Also nown as the Discrete-Space Fourier Transform (DSFT). Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 4 )

5 Statement Introduction Statement Previous Wor My Contributions Given some N values of the -D Discrete-Time Fourier Transform X j j,, ( e, e ),,..., N reconstruct its M M image x( i, i) Why not use inverse -D Discrete Fourier Transform (DFT)? Only some -D DTFT values are nown on -D rectangular grid. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 5

6 Introduction Statement Previous Wor My Contributions Spotlight-Mode Synthetic Aperture Radar (SAR) Ground area is imaged from radar signals at many angles. -D DTFT is nown on arcs. SAR Side View: SAR Image: Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 6

7 Introduction Statement Previous Wor My Contributions Limited-Angle Tomography (LAT) Microscope transmits electrons through cells on slides. -D DTFT is nown on only some slices. Microscope: Slide: Sandberg, J Struct Bio, 44 (003) 6-7, Pneumonia Cell: www-db.embl.de/jss/emblgroupsorg/g_47.html Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 7

8 -D Filter Design Introduction Statement Previous Wor My Contributions Filter is designed with a prescribed frequency response. -D DTFT is nown on passband and stopband contours. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 8

9 Introduction Statement Previous Wor My Contributions Spiral Scan Magnetic Resonance Imaging (MRI) Body is imaged from induced oscillating magnetic fields. -D DTFT is nown on spiral trajectory. Machine: Coils: Image: Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 9

10 Computed Tomography (CT) Introduction Statement Previous Wor My Contributions Body is imaged from rotating fan-beam x-ray projections. After rebinning, -D DTFT is nown on slices over 360. Machine: XRay: Image: Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 0

11 Filtered Bac-Projection (FBP) Introduction Statement Previous Wor My Contributions Filtered bac-projection only approximates (but still good). Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples

12 Introduction Statement Previous Wor My Contributions Direct Fourier Inversion (Gridding) Interpolate frequency values over whole grid from slices, then tae inverse -D Fourier transform. Projection Slice Theorem Interpolation Inverse Fourier Transform But once again this is still an approximation. Is there an exact solution? Yes Head: Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples

13 System of Linear Equations Introduction Statement Previous Wor My Contributions Why not solve as large system of linear equations? Ax=b e e j( M )( j( M )(,, N System Matrix,, N ) ) x(0,0) x( M, M Solution ) X ( e X ( e j j,, N, e, e Data j j,, N ) ) (Frequency Locations) (Image) (Frequency Values) This is an exact solution, but uniqueness (how many sufficient frequencies), conditioning (stability), determined by frequency locations and algorithm choice, are unclear in -D. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 3

14 Introduction Statement Previous Wor My Contributions Uniqueness How many -D DTFT values are sufficient to ensure that a -D signal is uniquely determined? Just M (support size). How many -D DTFT values are sufficient to ensure that the M M image is uniquely determined? In some cases, more than M. j j,, X ( e, e ),,..., x( i, i) N Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 4

15 Introduction Statement Previous Wor My Contributions On what set of projection angles should data be collected? Projection Slice Theorem Solve by System of Linear Equations One that leads to the least amount of noise in the solution.? Head: Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 5

16 Introduction Statement Previous Wor My Contributions Sensitivity of the solution to perturbations in the data. Noise in the frequency data is amplified in reconstructed image. Condition number of system matrix determines noise amplification. depends on frequency locations in system matrix. A ( x ) b Freq. Data b Noise δ Freq. Config. A - Noisy Sol n x+ε Solution x Amp. Noise ε Head: Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 6

17 Reconstruction Method Introduction Statement Previous Wor My Contributions. Cost Function: What are we solving? Minimize: Cost function = Data-fit term + Regularization term * How many data samples? (uniqueness) Square or just-determined system Over-determined system How is noise handled? (conditioning) Closed-form formula Iterative method Non-iterative method Divide-and-conquer method Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 7 ( A H Axˆ A) xˆ H ( A A I) xˆ A Regularization ~ ~ ~ Well-conditioned frequency selection ( A A) xˆ A. Optimization Algorithm: How do we solve it? b A H H b b b H H ~ * Examples shown are linear least squares estimation solutions.

18 Introduction Statement Previous Wor My Contributions Goals To develop a simple procedure for evaluating the relative conditioning of various configurations of given -D DTFT samples without having to compute the condition number of the system matrix. To develop a fast algorithm to reconstruct an image by performing iterative computations offline once and then reapplying the method in one step to any measurement data with the same -D DFT configuration. To develop a partitioning procedure to brea up a large reconstruction problem in to many smaller ones to reduce computation time or memory usage, or when a partial reconstruction is sufficient. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 8

19 Introduction Statement Previous Wor My Contributions Previous Wor Uniqueness -D DTFT on lines [Zahor 90, Zahor 9] Unwrapping to -D [Bresler 96, Yagle 00] Reconstruction Projection onto Convex Sets [Youla 8] Solve band-limited lexicographic -D system [Strohmer 97] Non-uniform FFT interpolation [Fessler 03] Sparsifiable Image Model [Candes 06, Gilbert 06] Requires random frequency locations Requires linear programming or orthogonal matching pursuit Outside scope of problem considered in this thesis Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 9

20 Introduction Statement Previous Wor My Contributions My Contributions. Computed closed-form expression for condition number and its upper bound of a just-determined problem using the Lagrange interpolation formula.. Developed the variance measure as an accurate and fast estimate of conditioning not restricted to the just-determined case. 3. Implemented simulated annealing with the variance sensitivity measure as the objective function in order to find wellconditioned frequency configurations. 4. Developed 45 rotated image support Kronecer substitution reconstruction method with a unique -D to -D mapping for which the system matrix is smaller to solve. 5. Showed non-rectangular regular -D DTFT configuration that is perfectly conditioned when unwrapped to -D. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 0

21 Introduction Statement Previous Wor My Contributions My Contributions Algorithms 6. Introduced fast non-iterative DFT-based image reconstruction algorithm that deconvolves a precomputed filter from filtered data using just three -D DFTs. 7. Precomputed the filter using a finite-support constraint but which converges faster than POCS based on the dual of band-limited image interpolation. 8. Introduced a divide-and-conquer image reconstruction method using subband decomposition using Gabor filters. 9. Presented results of varying the amount of regularization or the over-determining factor for each subproblem separately with decreased error or computation time. 0. Applied presented reconstruction methods to actual CT sinogram data, provided by Adam M. Alessio of Univ. of Washington by the way of Jeffrey A. Fessler of Univ. of Michigan. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples

22 -D to -D Unwrapping Sensitivity Measure Frequency Selection Overview Approach Approach Divide-and-Conquer Approach Conclusion Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples

23 Unwrapping from -D to -D -D to -D Unwrapping Sensitivity Measure Frequency Selection Kronecer substitute x z ( M )/, y z ( M )/ in -D z-transform of 45 rotated support image -D DTFT Samples on Parallel Lines I( z) z I( x, y) a 7.5 a 0 0 x z 0 0 y 3 a x a z y a a -D DTFT samples constrained on -D periodic line M Y X M where they now map to -D j j( y Y X ) e e z x Solved -D signal is wrapped bac to -D with index i x z y M M i X i Y... a... a Rotated Support Image Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples x z y 4 a a 5 5 x z 0 y 5 3

24 -D to -D Unwrapping Sensitivity Measure Frequency Selection Variance Sensitivity Measure In -D, conditioning is easier to determine. Condition number: O(M 3 ), r=.0 ( A) F A F A F M M M M m l, l u z m z z l l, z e j Upper bound: O(M ), r=0.9 ( A) F M M M l, l cos l Variance measure *: O(M), r=0.8 ** M ( A) M M,... M M * Normalized variance of distances between adjacent frequency locations measures departure from a uniform distribution. ** Sample correlation coefficient (r) between the variance measure and the condition number in the over-determined case. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 4

25 Frequency Selection -D to -D Unwrapping Sensitivity Measure Frequency Selection Variance measure relatively determines if a frequency configuration leads to a well-conditioned system. -D -D How do we find such a configuration? Simulated annealing uses the variance measure as its cost function to find a global minimum, in the case of CT, a near-optimal angle configuration. -D DTFT Intersections Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 5

26 -D to -D Unwrapping Sensitivity Measure Frequency Selection Frequency Configuration Example M=3 band-limited diagonals intersecting with CT slices. The baseline uniform-angle configuration has 44 slices intersecting at 973 data samples and variance measure of 7.7. The near-optimal variable-angle configuration has 68 slices intersecting at 969 data samples and variance measure of.. Uniform-Angle Projections High-Variance DTFT Locations Variable-Angle Projections Low-Variance DTFT Locations Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 6

27 -D to -D Unwrapping Sensitivity Measure Frequency Selection Simulation Data Frequency Configuration Simulation data slices intersect 55 diagonal lines at 64,64 samples. Image has size M M =5353 and -D solution has support 3,3. Given -D DTFTs, minimize over-determined non-regularized costs: ~ ~ ~ ( x) Ax b ( x) Ax b Uniform-Angle Configuration (VarMeas = 6.7) Variable-Angle Configuration (VarMeas =.8) Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 7

28 -D to -D Unwrapping Sensitivity Measure Frequency Selection Simulation Data Reconstructed Images Each system is solved iteratively using preconditioned conjugate gradient (PCG) method with convergence tolerance of 0-6. H H ~ ~ ~ H ( A A) xˆ A b ( A A) xˆ A b H ~ Uniform-Angle Reconstruction (RMSE = 3.9) Variable-Angle Reconstruction (RMSE =.) Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 8

29 Simulation Data Performance -D to -D Unwrapping Sensitivity Measure Frequency Selection Relative residual error versus PCG iteration Uniform-angle configuration requires 03 iterations. Variable-angle configuration requires 34 iterations. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 9

30 -D to -D Unwrapping Sensitivity Measure Frequency Selection Actual CT Data Actual CT sinogram data* is rebinned From fan beam with 80 projections over 360 and 888 detectors To parallel beam with 40 projections over 80 and 889 detectors. A reference FBP reconstructed image is shown to the right. Fan Beam Sinogram Parallel Beam Sinogram FBP Image * Courtesy of Adam M. Alessio of the University of Washington by the way of Jeffrey A. Fessler of the University of Michigan. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 30

31 Actual CT Reconstructed Images -D to -D Unwrapping Sensitivity Measure Frequency Selection Actual CT data slices intersect M=55 band-limited diagonals. The baseline uniform-angle configuration has 355 slices intersecting at 65,37 data samples and variance measure of The near-optimal variable-angle configuration has 57 slices intersecting at 65,095 data samples and variance measure of 7.4. Image has size 5555 and -D solution has support 65,05. FBP Image* Uniform-Angle PCG Image Variable-Angle PCG Image * Reference FBP image reconstructed from 8 uniform slices with 509 bins each and a non-apodized ramp filter. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 3

32 Algorithm Filter Construction Overview Approach Approach Divide-and-Conquer Approach Conclusion Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 3

33 Why? Algorithm Filter Construction Iterative reconstructions algorithms are computationally expensive. Projection Onto Convex Sets (POCS) requires many iterations to converge. Preconditioned Conjugate Gradient (CG) method converges quicer but each iteration is slow. Solving normal equation has worse conditioning. The condition number of A H A is square of condition number of A, which amplifies noise and can require drastic regularization. CG iterations increases roughly with condition number even with help of preconditioning. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 33

34 Modified Statement Given some values of the N N -D Discrete Fourier Transform (DFT) X ( reconstruct its M M image where N >>M., ) N N i 0 i 0 x( i x( i, i) Algorithm Filter Construction, i ) e j ( i N i ) Why not use the inverse -D DFT this time? Not all -D DFT values are nown on the -D rectangular grid. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 34

35 Algorithm Algorithm Filter Construction DFT-based Deconvolution using a Data Masing Filter. Precompute a filter with zero magnitude response at unnown -D DFT sample locations.. Filter nown -D DFT samples in the frequency domain and set to zero elsewhere. Y(, H(, h( i, i H( ) ) 0, ) 0,, ) X ( 0, (, ( i, i ) (, (, ) ) KNOWN KNOWN 3. Deconvolve original image from filtered image y using three -D DFT operations where g is, ) ), y( i, i) F Y(, ) KNOWN SUPPORT M N P the inverse filter. x( i, i ) g( i, i )** y( i, ) ˆ i Operator ** denotes -D convolution. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 35

36 Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 36 Deconvolution Noisy system model in -D DFT domain is If no noise and H (, ) 0 then Tihonov regularization numerically stabilizes solution Regularized inverse filter is then the Wiener filter Deconvolution using -D DFTs of size L different from N * ), ( ), ( ), ( ~ ), ( ), ( H H G H G ), ( ), ( ), ( ), ( Y X H ), ( ), ( ), ( ), ( ), ( H Y Y G X arg min ˆ X X H Y X X Algorithm Filter Construction ), ( ), ( ), ( ˆ Y F F G F i i x N N L L L * Additive white Gaussian noise.

37 Algorithm Filter Construction Filter Design Filter Specification h (i,i ) = 0 for N -M+ i, i N H (, ) = 0 for (, ) not in the set of Ω KNOWN Filter Size h (i,i ) unnown of size P P = (N -M+) (N -M+) h (i,i ) nown as 0 at N P points H (, ) arbitrarily nown at N P points H (, ) nown as 0 at P points Filter is unique since P unnowns = P equations Filtered Image y (i,i ) = h (i,i )**x (i,i ) must be N N is over-determined by (N P )/M N /M M N N P Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 37

38 Filter Construction Algorithm Filter Construction Projection Onto Convex Sets (POCS) Alternatingly projects onto the spatial domain imposing finite support and on the DFT domain imposing the nown DFT values. Not much storage needed. Slow convergence. Finite-Support Regularized Conjugate Gradient (FSR) Finite-support constraint is implemented as regularization in the cost function minimally biases the solution. Minimizing cost function using the conjugate gradient algorithm has faster convergence. Adapted as the Fourier dual of the band-limited interpolation approach [Strohmer 97]. Although iteratively computed it is done offline and just once. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 38

39 Filter Construction using Finite-Support Regularization Objective Function Algorithm Filter Construction The objective function includes a data-fit term of only elements in set Ω C of missing -D DFT values and a regularization term including only elements in set Π C of non-support spatial values ( h) J C Fh H C J C h h C where J is an irregular downsampling matrix, F is the -D DFT matrix, H Ω c are unnown DFTs, and h Ω c are non-support spatial values of 0. Minimizing the objective function with respect to filter solution h leads to the regularized normal equation of the form ( F H I C F I where J H is an upsampling matrix and I Ω c=j Ω c H J Ω c is an indicator matrix. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 39 C ) hˆ F H J H C H C

40 Filter Construction using Finite-Support Regularization Uniqueness Typically, M M samples of an N N -D DFT are not sufficient to uniquely determine a M M image. However it can be shown our added constraints where non-support spatial values are nown combined with the sampled -D DFT values result in uniqueness. Preconditioning Preconditioning the system can improve the convergence rate P H ( F I C F I ˆ C One simple but effective preconditioner is to replace the masing diagonal matrix with a scaled identity matrix as such P F H Algorithm Filter Construction ) h ( I) F I I I where α = tr(i Ω c) / tr(i Π c) = K /N and K is the cardinality of Ω C Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 40 C P F H J H C C H C

41 Filter Construction using Finite-Support Regularization Optimization Algorithm Algorithm Filter Construction As the regularization parameter λ, the non-support region goes to 0 and we solve the transformed preconditioned normal equation ( I F H I C FI I C ) gˆ hˆ I I F gˆ H J H C H C where the P has been Cholesy decomposed. Conjugate Gradient (CG) is used to the solve the above, where each iteration only requires only two -D DFT operations per CG iteration in O(N log (N )). This final form gives the same answer as POCS, yet converges at the faster rate of the preconditioned CG method. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 4

42 Simulation Data Configuration Algorithm Filter Construction Simulation data on nearest neighbor analytical spiral with 44 revolutions and up to 5 angular samples per revolution on the DFT of size NxN=768x768, with conjugate symmetry. Image size MxM=8x8 is over-determined by.5. Filter size PxP=64x64 is over-determined by.3, no regularization, and no noise.* Mas of -D DFT Samples Log DFT of POCS Filter (Residual=3.3x0-3 ) Log DFT of FSR Filter (Residual=.8x0-3 ) * Both POCS and FSR filters were generated with 50 iterations. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 4

43 Simulation Data Reconstructed Images Deconvolution image size LxL=844x844. Approximate operations of POCS * is 5.x0 7 in 5 iterations, Deconvolution (POCS Filter) is.x0 7, and Deconvolution (FSR Filter) is.x0 7. Algorithm Filter Construction True DFT and Image RMSE=0.0 POCS Recon. DFT and Image RMSE=.9 Deconv. DFT and Image (POCS Filter) RMSE=7.4 Deconv. DFT and Image (FSR Filter) RMSE=3. * POCS was stopped when RMSE was approximately that of the FSR-filtered deconvolution method and did not run to convergence. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 43

44 Algorithm Filter Construction Simulation Data Reconstruction Performance RMSE versus approximate operations of reconstruction methods including deconvolution using over-determined POCS-filter and FSR-filter. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 44

45 Actual CT Medium Configuration Actual CT data with 369 slices fill the DFT size NxN=04x04, with zeros beyond circular band-limit. Image size MxM=56x56 is over-determined by 6.9. Filter size PxP=769x769 is over-determined by.0, regularization λ=x0 -, and no noise. * Mas of -D DFT Samples Log DFT of Filter H Algorithm Filter Construction Log DFT of Inverse Filter G Log DFT of Regularized Inverse Filter * Filter too 6 minutes to precompute offline. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 45

46 Algorithm Filter Construction Actual CT Medium Reconstructed Images Deconvolution image size LxL=8x8. Runtimes of POCS * is.4s in 3 iterations, Deconvolution is.s, and Regularized Deconvolution is.s. POCS DFT and Zoomed Image Deconv. DFT and Zoomed Image Reg. Deconv. DFT and Zoomed Image * POCS was stopped when computation time was approximately that of the deconvolution method and did not run to convergence. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 46

47 Algorithm Filter Construction Actual CT Noisy Reconstructed Images 385 slices; N=536; M=5; P=05; L=690; λ=x0 - ; SNR=.6dB * Filter over-determined by.5 **; Image over-determined by.9. Runtimes of FBP is 76s, POCS is 3.0s for 3 iterations, and Regularized Deconvolution is.3s. FBP DFT and Image POCS DFT and Image Reg. Deconv. DFT and Image * Noise is zero-mean additive white Gaussian. ** Filter too 3 minutes to precompute offline. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 47

48 Algorithm Filter Construction Actual CT Large Reconstructed Images 667 slices; N=33; M=888; P=445; L=465; λ=x0-3 ; no noise. Filter over-determined by 3.7; Image over-determined by.3. Runtimes of FBP is 33s, POCS is 3.6s for 3 iterations, and Regularized Deconvolution is 4.7s. FBP DFT and Image POCS DFT and Image Reg. Deconv. DFT and Image Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 48

49 Divide Step Conquer Step Overview Approach Approach Divide-and-Conquer Approach Conclusion Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 49

50 Why Divide and Conquer? Divide Step Conquer Step A large problem is replaced with similar smaller problems. image sized N=56 solved in O(N ) taes x 56 4 =4.3x0 9 ops. 64 subproblems each sized N=3 taes 64 x 3 4 =6.7x0 7 ops. Each subproblem can be regularized independently, depending on its conditioning. Poorly conditioned or underdetermined subproblems (not enough frequency samples in that subband) can be discarded altogether, regularizing the overall problem. An unaliased low-resolution image can be reconstructed using the lowest-frequency subband. This may be sufficient for recognition in some applications. statement is same as earlier where the image is reconstructed from some values of its -D DTFT. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 50

51 Divide Step: Gabor Logons Divide Step Conquer Step Formulation (Divide). Divide D frequency plane into ( K ) subbands.. Construct an over-complete set of modified D discrete-time Gabor logons truncated to ensure finite-support in time -D DTFT ( i,, i ;,, ) e K 0,,, ( i i overlap ) e j ( i i K K ) 3. Project image onto modified Gabor logons 4. Demodulate problem to baseband 5. Downsample by and in each dimension. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 5

52 Conquer Step: Subband Subproblems Reconstruction (Conquer) 6. Solve each self-contained subproblem separately and in parallel. 7. Deconvolve each subsolution from downsampled Gabor logon by point-wise division in the -D DFT domain. 8. Combine all ( K ) -D DFT subbands and inverse -D DFT to compute the original image. Regularization A Divide Step Conquer Step, y ( i, i ) b,, y, (, ), (, )** (, ;,, ) i i x i i i i Uniform regularization may be applied. Any badly conditioned subproblem can be regularized separately or even discarded if under-determined. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 5

53 Divide Step Conquer Step Simulation Data Uniform Regularization Simulation CT data with 8 slices each with 8 bins, divided in to K x K =8x8 (K=3) subbands, each of size 3x3 due to α overlap =. M=8. SNR is 5.7 db. Uniform regularization is λ=x0. FBP RMSE=7.4 at.6s. D&C RMSE=.9 at.6s / sub. Original Image and DFT FBP Image and DFT D&C Image and DFT Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 53

54 Simulation Data Variable Regularization Simulation CT data same as before. SNR is now 5.74 db. Variable regularization from λ=x0 at lowest to λ=5x0 at highest frequency subbands increasing log-linearly. FBP RMSE=.3 at.8s. Var.D&C RMSE=3. at.5s/sub. D&C RMSE=3.5 (λ=x0 ). D&C RMSE=6.3 (λ=5x0 ). Original Image and DFT Divide Step Conquer Step FBP Image and DFT Var.D&C Image and DFT Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 54

55 Actual CT Reconstructed Images Actual CT data with 05 slices each with 445 bins, divided in to K x K =3x3 (K=5) subbands, each size 3x3 with α overlap =. M=5. SNR is 50. db. Uniform λ=x0 0. Runtimes of FBP is 0s, D&C is 0.7s / subproblem solved by PCG and 78s total. FBP Image and DFT Divide Step Conquer Step D&C Image and DFT Lowest-Subband Image and DFT Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 55

56 Divide-and-Conquer Conclusion Contributions Evaluation Future Research The End Overview Approach Approach Divide-and-Conquer Approach Conclusion Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 56

57 Divide-and-Conquer Conclusion Contributions Evaluation Future Research The End Summary of Contributions Introduced a more efficient 45 rotated-support -D to -D unwrapping procedure which allows easier analysis of conditioning. Developed a quic O(M) and accurate (r=0.80 with κ(a) ) sensitivity measure of the variance of distances between -D DTFT locations, based on a closed-form formula for the upper-bound on the Frobenius condition number of a square Vandermonde matrix. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 57

58 Divide-and-Conquer Conclusion Contributions Evaluation Future Research The End Summary of Contributions Algorithms Developed a fast non-iterative algorithm that deconvolves a precomputed data masing filter from a filtered DFT of the data using just three -D DFTs. Reinterpreted band-limited image interpolation as a finite-support regularization method for computing the data masing filter much faster than POCS. Developed a procedure to sub-divide a reconstruction problem using Gabor logons and subband decomposition and applied varying regularization to each sub-problem. Numerical Applied each of the conditioning, non-iterative, and divide-andconquer methods to actual CT sinogram data with excellent reconstructions. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 58

59 Divide-and-Conquer Conclusion Contributions Evaluation Future Research The End Evaluation of The variance measure is the only estimate of sensitivity reliable for the over-determined case and reduces the amount of regularization. The Kronecer substitution based wrapping methods allow DTFT samples to be chosen more freely on parallel lines which was more restrictive in the Good-Thomas unwrapping. The helical scan FFT does not need a rotated support. Frequency selection is achieved using simulated annealing with the variance measure when physical system geometry restrictions exist. In applications lie magnetic resonance spectroscopic imaging (MRSI) or 3-D MRI where data at each x-y sample is expensive, - space trajectories maybe designed with a fixed number of samples with the best if not perfect conditioning without extra sampling. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 59

60 Divide-and-Conquer Conclusion Contributions Evaluation Future Research The End Evaluation of Algorithms The non-iterative approaches are recommended when frequency samples are already determined and fixed for multiple uses as we precompute the frequency mas filter once. The speed advantage comes from only requiring three -D FFTs. The divide-and-conquer approach is also recommended for when sample locations are determined or when the image problem is large. This subdivision allows customization of each subproblem in terms of regularization and the type of algorithm used to solve (closedform, iteratively or non-iteratively) based on its conditioning. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 60

61 Divide-and-Conquer Conclusion Contributions Evaluation Future Research The End Suggestions for Future Research The conditioning approaches may benefit from a stronger theoretical lin between the fast sensitivity measures and the condition number which explain the strong empirical evidence relating the two quantities, especially in the overdetermined case. The non-iterative method computes a very quic solution, which in cases when resources are available, can be further refined with iterative methods. As for the advantage of the divide-and-conquer method, a non-aliased low-resolution image is quicly computed which can then be updated with higher resolution subbands. Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 6

62 Divide-and-Conquer Conclusion Contributions Evaluation Future Research The End Than You! of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples Benjamin C. Lee Dissertation Oral Defense May 0, 00 University of Michigan Electrical Engineering: Systems Chair: Prof. Andrew Yagle Benjamin Lee of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples 6