IMAGE RECONSTRUCTION FROM UNDERSAMPLED FOURIER DATA USING THE POLYNOMIAL ANNIHILATION TRANSFORM Anne Gelb, Rodrigo B. Platte, Rosie Renaut Development and Analysis of Non-Classical Numerical Approximation Tools AFOSR955-12-1-393 (212-215) Developing Fast, Accurate, and Robust Numerical Algorithms for Extracting Actionable Information from Acquired Sensing Data AFOSR955-15-1-152 (215-218) AFOSR Computational Mathematics Program Review Meeting (August 215)
PROJECT: DEVELOPMENT AND ANALYSIS OF NON-CLASSICAL NUMERICAL APPROXIMATION TOOLS THEME: Sampling (direct and indirect) and function (or feature) reconstruction. Consider {(T N [f (x)]) j : j = 1,..., N} is given. Design P M, such that P M (T N [f ])(x) g(f (x)). RELATED QUESTIONS: Restraints on the collected data, {(T N [f ](x)) j : j = 1,..., N}? Convergence rates and stability? What regularization techniques can be applied to ensure numerical accuracy and stability?
APPLICATIONS RADAR IMAGING Data from synthetic aperture radar can be seen as incomplete Fourier data.
APPLICATIONS RADAR IMAGING Data from synthetic aperture radar can be seen as incomplete Fourier data. Medical imaging: MRI and tomography
APPLICATIONS RADAR IMAGING Data from synthetic aperture radar can be seen as incomplete Fourier data. Medical imaging: MRI and tomography Solution of PDEs on irregular domains
Function Reconstruction from Undersampled Fourier Data Using the Polynomial Annihilation Transform In collaboration with Richard Archibald (Oak Ridge)
THE FOURIER RECONSTRUCTION PROBLEM Let Given 1 ˆf (ω) = f (x)e iπω x dx. 2 R d ˆf (ωj ), j =,..., N, We want to recover f (x j ), j =,... M. DIFFICULITES: 1 If f is not smooth (Gibbs phenomenon). 2 ˆf is not sampled on a structured grid (no FFT). 3 The data may be undersampled (N < M). Remark: NUFFT addresses 2 to some extent, but not 1 and 3. Regularization is needed to address lack of uniqueness.
1D EXAMPLES EXAMPLE f a (x) = { 1 x if 1 x < 1 x otherwise ; f b(x) = cos πx 2 if 1 x < 1 2 cos 3πx 2 if 1 2 x < 1 2 cos 7πx 2 if 1 2 x 1 Test Function 1 Uniform Grid.8.6.4.2.2.4.6.8 1 1.5.5 1 1.8.6.4.2.2.4.6.8 1 Test Function Uniform Grid 1.5.5 1 128 data points are shown
FOURIER DATA, 1D EXAMPLE 1 Target function.5 f(x) -.5-1 -1 -.5.5 1 x.3 Fourier data.15 R(ˆf(ω)).2.1 -.1 I(ˆf(ω)).1.5 -.5 -.1 -.2-5 5 ω -.15-5 5 ω Wave numbers ω j were drawn from a normal distribution with standard deviation 2N/6, where 2N + 1 is the number of recovered function values.
FOURIER RECONSTRUCTION FROM PARTIAL SUM, 1D EXAMPLE 1 Test Function Uniform Grid 1 Fourier Reconstruction Test Function 1 Filtered Fourier Reconstruction Test Function.8.8.8.6.6.6.4.4.4.2.2.2.2.2.2.4.4.4.6.6.6.8.8.8 1 1 1 1.5.5 1 1.5.5 1 1.5.5 1 f a (x) S N f a (x) S σ N f a(x) 1 1 1.8.8.8.6.6.6.4.4.4.2.2.2.2.2.2.4.4.4.6.6.6.8.8.8 1 Test Function 1 Fourier Reconstruction 1 Filtered Fourier Reconstruction Uniform Grid Test Function Test Function 1.5.5 1 1.5.5 1 1.5.5 1 f b (x) S N f b (x) S σ N f b(x) S N f (x) = N ˆf (ωj )e iπωjx and SNf σ (x) = j= N σ jˆf (ωj )e iπωjx j=
l 1 REGULARIZATION [Candès, Tao, Romberg, Donoho, Lustig, Goldstein, Osher,...] min f J(f) such that Ff ˆf 2 =, ˆf consists of samples of the Fourier transform of the target function/image, f. F Is the Fourier transform operator (matrix). J is an appropriate l 1 regularization term, (e.g. Lf 1 ). Typically for measured data the related (TV) denoising problem, min f J(f) such that Ff ˆf 2 < σ,
THE POLYNOMIAL ANNIHILATION TRANSFORM [Archibald, Gelb, Yoon, 25] Let [f ](x) = f (x + ) f (x ) be the jump function in 1D. Edge map: f [f ]. POLYNOMIAL ANNIHILATION TRANSFORM L m f (x) = 1 c j (x)f (x j ), q m (x) x j S x S Ij = {x j m,, x j+ m }, S I 2 2 j = {x j m+1,, x 2 j+ m 1 } 2 for m even and odd respectively (centered case). c j (x)p l (x j ) = p (m) l (x), j = 1,..., m + 1, x j S x q m (x) = x j S + x c j (x), where S + x is the set of points x j S x such that x j x.
THE POLYNOMIAL ANNIHILATION TRANSFORM [Archibald, Gelb, Yoon, 25] Let [f ](x) = f (x + ) f (x ) be the jump function in 1D. Edge map: f [f ]. POLYNOMIAL ANNIHILATION TRANSFORM For example, not assuming periodicity, for m = 4, L m = 1 3 3 12 18 12 3 1 4 6 4 1 1 4 6 4 1 1 4 6 4 1. 1 4 6 4 1 1 4 6 4 1 1 4 6 4 1 3 12 18 12 3
FOURIER RECONSTRUCTION USING THE PA TRANSFORM IN 1D min f L m f 1 such that Ff ˆf 2 =, 1 PA m=1 Test Function 1 PA m=2 Test Function 1 PA m=3 Test Function.8.8.8.6.6.6.4.4.4.2.2.2.2.2.2.4.4.4.6.6.6.8.8.8 1 1 1 1.5.5 1 1.5.5 1 1.5.5 1 PA (m = 1) for f a PA (m = 2) for f a PA (m = 3) for f a 1 1 1.8.8.8.6.6.6.4.4.4.2.2.2.2.2.2.4.4.4.6.6.6.8.8.8 1 PA m=1 1 PA m=2 1 PA m=3 Test Function Test Function Test Function 1.5.5 1 1.5.5 1 1.5.5 1 PA (m = 1) for f b PA (m = 2) for f b PA (m = 3) for f b
PHASE TRANSITION DIAGRAMS Recovery of piecewise constant functions m = 1 (TV) m = 2 m = 3 The colormap shows the fraction of successful recoveries in 2 trials. A trial is deemed successful when the relative l 2 error is below 1 2.
PHASE TRANSITION DIAGRAMS Recovery of piecewise linear functions m = 1 (TV) m = 2 m = 3 The colormap shows the fraction of successful recoveries in 2 trials. A trial is deemed successful when the relative l 2 error is below 1 2.
PHASE TRANSITION DIAGRAMS Recovery of piecewise quadratic polynomials m = 1 (TV) m = 2 m = 3 The colormap shows the fraction of successful recoveries in 2 trials. A trial is deemed successful when the relative l 2 error is below 1 2.
THE SPLIT BREGMAN ALGORITHM [Goldstein and Osher, 29] min f L m f 1 such that Ff ˆf 2 < σ, ALGORITHM While Ff ˆf 2 > σ 1 f k+1 = arg min Ff ˆf 2 + λ d k L m f b k 2 2 f 2 d k+1 = arg min d 1 + λ d L m f k+1 b k 2 2 d 3 b k+1 = b k + L m f k+1 d k+1 Remarks: (1) is a linear least-squares problem (2) is solved using the shrink operator (3) is a feedback term (the residual is added back into the regularization)
2D EXAMPLES Masks used for sampling in Fourier space Gaussian Sampling Tomographic Sampling
PA (m = 2) PA (m = 2) 2D EXAMPLE - NO NOISE, GAUSSIAN SAMPLE (5%) TV TGVSH
2D EXAMPLE: CROSS SECTION ERROR COMPARISON 1 2 3 4 5 Fourier TV 6 TGVSHCS Spa m=2 Spa m=3 7 1.5.5 1
Noisy Fourier data (1 db SNR) and using tomographic sampling (5%) TV TGVSH PA (m = 2) PA (m = 3)
Noisy Fourier data (5 db SNR) and using Gaussian sampling (5%) TV TGVSH PA (m = 2) PA (m = 3)
SAR image: Noisy Fourier data (5 db SNR) undersampled by 5% Original SAR Image TV TGVSH Rel. l error =.14 Rel. l2 error =.13 2 PA m = 2, Rel. l2 error =.96 PA m = 3, Rel. l2 error =.9
REMARKS The Polynomial Annihilation transform improves accuracy away from discontinuities. The Split Bregman Algorithm can be adapted to use the PA transform. The method is especially effective when data are undersampled and noisy. The computational cost is comparable to TV regularized recovery.
A frame theoretic approach to the nonuniform fast Fourier transform (Gelb and Song, 214) Frame operator Frame approximation S(f )(x) = j ˆf (ωj )e iωjx f (x) = j ˆf (ωj )S 1 e iωjx Change of Frames S 1 e iωjx l k b l,j e ijx New frame approximation A frm (f ) = ˆf (ωj )b l,j e ijx Convolutional gridding A cg (f )(x) = j n l m j n l ω j q α jˆf (ωj )ĝ(l ω j) e ilx g(x)
A frame theoretic approach to the nonuniform fast Fourier transform (Gelb and Song, 214)
Fourier reconstruction of univariate piecewise-smooth functions from non-uniform spectral data with exponential convergence rates (Platte, Gutierrez and Gelb, 215)
y 3 A windowed Fourier method for approximation of non-periodic functions on equispaced nodes (Platte 215) u 2.5 2 1.5 uw 1 w.5 3 1 1 3 x
y 3 A windowed Fourier method for approximation of non-periodic functions on equispaced nodes (Platte 215) u 2.5 2 1.5 uw 1 w.5 3 1 1 3 x
STARTING GRANT (215-218) Developing Fast, Accurate, and Robust Numerical Algorithms for Extracting Actionable Information from Acquired Sensing Data AFOSR955-15-1-152 Supported by: Dr. Arje Nachman Sensing, Surveillance and Navigation Program and Dr. Jean-Luc Cambier Computational Mathematics Program
OBJECTIVES Construct fast, accurate, and robust numerical algorithms that extract actionable information from acquired sensing data in collaboration with Air Force laboratories. Study synthetic aperture radar (SAR) as the prototype sensing system, with the focus on the detection, classification, tracking of targets, and the extraction of important image information. Thoroughly analyze current localized feature detection and image reconstruction methods so that it is possible to develop new algorithms that are effective under non-idyllic circumstances (flight path variations, clutter, jamming). Expand current localized feature detection and imaging techniques, when possible, to a framework amenable to other relevant representations (such as wavelets and short time Fourier transforms). Develop high-order algorithms for solving partial differential equations (PDEs) where non-standard algorithms may be more suitable for computations.
WINDOWED FOURIER METHODS FOR PDES Combining the benefits of Fourier spectral methods with domain decomposition (partition of unit) schemes. 1.5 1 1 WF Fourier WFadapt RBF ε =1 RBF ε =2 RBF ε =1 y.5.5 error (log scale) 1 5 1 1 1 1.5 1 2 3 4 5 6 x 1 15 2 4 6 8 1 12 degrees of freedom
WINDOWED FOURIER METHODS FOR PDES Combining the benefits of Fourier spectral methods with domain decomposition (partition of unit) schemes.
PUBLICATIONS D. Denker, R. Archibald and A. Gelb, (215) An Adaptive Fourier Filter for Relaxing Time Stepping Constraints for Explicit Solvers, Lecture Notes in Computational Science and Engineering (ICOSAHOM Proceedings), Springer, to appear. R. Archibald, A. Gelb, and R.B. Platte (215) Image Reconstruction from Undersampled Fourier Data Using the Polynomial Annihilation Transform, J.Sci. Comput., in revision. R.B. Platte, A. Gutierrez, and A. Gelb, (215) Edge informed Fourier reconstruction from non-uniform spectral data with exponential convergence rates, Appl. Comput. Harmon. Anal., DOI:1.116/j.acha.214.1.2. G. Wasserman, R. Archibald and A. Gelb,(214) Image Reconstruction from Fourier Data Using Sparsity of Edges, J. Sci. Comput., DOI 1.17/s1915-14-9973-3. A. Gelb and G. Song, (214) A Frame Theoretic Approach to the Non-Uniform Fast Fourier Transform, SIAM J.Numer. Anal, 52:3 1222 1242. A. Martinez, A. Gelb and A. Gutierrez, (214) Edge Detection from Non-Uniform Fourier Data Using the Convolutional Gridding Algorithm, J. Sci. Comput, 61:3 49 512. G. Song and A. Gelb, (213) Approximating the Inverse Frame Operator from Localized Frames, Appl. Comput. Harmon. Anal., 35:1, 94-11. Q. Huang, R. Eubank and R. A. Renaut, (215), Functional Partial Canonical Correlation, Bernoulli, 21:2, 147-166. S. Vatankhah, V. E Ardestani and R. A Renaut, (214) Automatic estimation of the regularization parameter in 2-D focusing gravity inversion: an application to the Safo manganese mine in northwest of Iran, J. Geophysics and Engineering, 11:4, 44 51.
PUBLICATIONS, CONT. S. Vatankhah, R. A. Renaut and V. E. Ardestani, (214), Regularization Parameter Estimation for Underdetermined problems by the χ 2 principle with application to 2D focusing gravity inversion, Inverse Problems,3:8 85 862. J. Hansen, J. Hogue, G. Sander, R. A. Renaut, and S. C. Popat (214), Non-negatively constrained least squares and parameter choice by the residual periodogram for the inversion of electrochemical impedance spectroscopy, submitted. R.A. Renaut, R. Baker, M. Horst, C. Johnson and D. Nasir (213) Stability and error analysis of the polarization estimation inverse problem for microbial fuel cells, Inverse Problems 29:4 4 522. R.B. Platte (215) C compactly supported and positive definite radial kernels, SIAM J. Sci. Comput. (to appear). R.B. Platte (215) A windowed Fourier method for approximation of non-periodic functions on equispaced nodes, Lecture Notes in Computational Science and Engineering (ICOSAHOM Proceedings), Springer (to appear). A.A. Mitrano, R.B. Platte (215) A numerical study of divergence-free kernel approximations, Appl. Numer. Math., 96, 97 17. R.B. Platte, A.J. Gutierrez, A. Gelb (214) Fourier reconstruction of univariate piecewise-smooth functions from non-uniform spectral data with exponential convergence rates, Appl. Comput. Harmon. Anal., DOI 1.116/j.acha.214.1.2. J.M. Martel, R.B. Platte (215) Stability of radial basis function methods for convection problems on the circle and sphere, J. Sci. Comput. (submitted). B. Adcock, R.B. Platte (215) A mapped polynomial method for high-accuracy approximations on arbitrary grids, SIAM J. Numer. Anal. (submitted). Thank you!