Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred S PARSITY E NFORCING E DGE D ETECTION M ETHOD FOR BLURRED AND NOISY F OURIER D ATA Rosemary Renaut This is joint work with Wolfgang Stefan, Rice University, Aditya Viswanathan, Cal Tech, and Anne Gelb, Arizona State University T EMPLE U NIVERSITY 2011 April 27, 2011 National Science Foundation: Division of Computational Mathematics 1 / 41
Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred Outline Problem Statement and Test Examples 1 Background 2 Jump detection using the Concentration method 3 The Matching Waveform 4 l 1 minimization to detect edges in blurred signals 5 Extension for Edge Detection from Non-harmonic Coefficients 6 Conclusions 7 National Science Foundation: Division of Computational Mathematics 2 / 41
Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred Imaging from Spectral Data Spectral methods enjoy great popularity in a vast array of applications. Their superior numerical properties, however, break down when the underlying function is piecewise-analytic. Processing of such functions from spectral data requires special attention. Goal: leverage the interplay between local features, primarily jump discontinuities, and Fourier coefficients, to enable accurate and efficient processing of such data. National Science Foundation: Division of Computational Mathematics 3 / 41
Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred Motivating Application – Magnetic Resonance Imaging I We acquire spectral data. Data may be acquired along non-Cartesian sampling trajectories. resistance to motion artifacts ease in generating field gradients Data can be degraded by blur. Data can be noisy National Science Foundation: Division of Computational Mathematics 4 / 41
Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred Motivating Application – Magnetic Resonance Imaging II (a) Acquired Fourier Sam- (b) Spiral Sampling Trajec- (c) Basic Recon- ples tory structed Shepp Logan Figure: MR Imaging: Example without blur or noise, but non Cartesian Data 1 1 Sampling pattern courtesy Dr. Jim Pipe, Barrow Neurological Institute, Phoenix, Arizona National Science Foundation: Division of Computational Mathematics 5 / 41
Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred Brief Overview Objective Estimate the edges in a piecewise smooth function from blurred and noisy Fourier data on non-equispaced grids. Assume a finite number of Fourier Coefficients available for a piecewise function. These may be noisy The function may be blurred Data may be collected at non cartesian grid points Desire accurate and robust detection of jump discontinuities. Aim improve signal reconstructions, restorations and classification. Illustrate by counting true classifications of edges in data. National Science Foundation: Division of Computational Mathematics 6 / 41
Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred Background The function f is represented by a finite number of spectral coefficients f is 2 π -periodic and piecewise-smooth in [ − π, π ) . It has Fourier series coefficients � π f k = 1 ˆ f ( x ) e − ikx dx , k ∈ [ − N, N ] 2 π − π f is a global representation; i.e., ˆ ˆ f k are obtained using values of f over the entire domain [ − π, π ) . Assume f is piecewise smooth Its jump function is defined by [ f ]( x ) := f ( x + ) − f ( x − ) A jump discontinuity is a local feature; i.e., the jump function at any point x only depends on the values of f at x + and x − . National Science Foundation: Division of Computational Mathematics 7 / 41
Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred Concentration Factor Edge Detection Method (Gelb, Tadmor) Concentrating the edges using convolution with C σ N ( x ) Approximate [ f ]( x ) using generalized conjugate partial Fourier sum N � | k | � e ikx = ( f ∗ C σ ˆ S σ � N [ f ]( x ) = i f k sgn ( k ) σ N )( x ) (1) N k = − N (1) converges when concentration factors σ k,N ( η ) = σ ( | k | N ) satisfy admissibility properties: � 1 N σ ( η ) ( ii ) σ ( η ) � ∈ C 2 (0 , 1) ( i ) σ k,N sin( kx ) ( iii ) → − π, η η ǫ k =1 ǫ = ǫ ( N ) > 0 is small is odd Convergence S σ N [ f ]( x ) = [ f ]( x ) + O ( ǫ ) , depends on σ and the distance between x and a discontinuity of f . National Science Foundation: Division of Computational Mathematics 8 / 41
Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred Classical Concentration Factors Concentration Factors Factor Expression 4 Trigonometric σ T ( η ) = π sin( α η ) Polynomial 3.5 Trigonometric Exponential Si ( α ) Z α 3 sin( x ) Si ( α ) = dx 2.5 x 0 σ P ( η ) = − p π η p Polynomial 2 p is the order of the factor 1.5 „ 1 « σ exp ( η ) = C η exp Exponential 1 α η ( η − 1) 0.5 C - normalizing constant; α > 0 - order π 0 C = −80 −60 −40 −20 0 20 40 60 80 R 1 − 1 k “ ” 1 N exp dτ 1 α τ ( τ − 1) N Figure: Envelopes of the Table: Examples of concentration factors Concentration Factors in Fourier Space National Science Foundation: Division of Computational Mathematics 9 / 41
Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred Example for 2 D : No Noise, No Blur, Cartesian Grid Apply concentration to each dimension 2 „ | l | 1.5 N « S σ X 1 N [ f ]( x (¯ y )) = i sgn ( l ) σ N 0.5 l = − N f(x) 0 N −0.5 X ˆ f k,l e i ( kx + l ¯ y ) −1 · f −1.5 S N [f] k = − N −2 −3 −2 −1 0 1 2 3 x overbar represents constant dimension. (a) Trigonometric Factor: Jump Response (c) Edge Map (b) Shepp Logan phantom National Science Foundation: Division of Computational Mathematics 10 / 41
Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred Illustration of Edge Detection N = 64 . Black line is the jump function Example Case: No Noise, no blur, Cartesian Grid 8 − 3 π 4 ≤ x < − π 3 / 2 for 2 > > − π 4 ≤ x < π 7 / 4 − x/ 2 + sin(7 x − 1 / 4) for < 8 f ( x ) = (2) 3 π 8 ≤ x < 3 π x 11 / 4 − 5 for 4 > > 0 otherwise . : (d) Polynomial p=1 ( σ 1 ) (e) Polynomial p=2 ( σ 2 ) (f) Exponential ( σ exp ) National Science Foundation: Division of Computational Mathematics 11 / 41
Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred Observations Polynomial CFs only filter low frequencies Exponential also filters some high frequencies Fast convergence away from a jump leads to more oscillations around the jump Many false positive and false negatives with regard to classifying jumps. National Science Foundation: Division of Computational Mathematics 12 / 41
Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred Improving jump detection The minmod to improve the approximation (Gelb and Tadmor (2006)) Use the minmod function over different concentration functions � s min( | a 1 | , | a 2 | , . . . , | a n | ) s := sgn ( a i ) , ∀ i minmod { a 1 , . . . , a n } := , 0 otherwise (3) yielding the approximation obtained by finding the jump approximation with multiple σ S MM [ f ]( x ) = minmod { S σ 1 N [ f ]( x ) , S σ 2 N [ f ]( x ) , . . . , S σ n N [ f ]( x ) } . (4) N National Science Foundation: Division of Computational Mathematics 13 / 41
Problem Statement and Test Examples Background Concentration Method The Matching Waveform l 1 minimization to detect edges in blurred Minmod CF edge detection for noisy and blurred functions: 2% threshold (g) Under sampling (h) Blurring by a Gaussian (i) Noise contamination Figure: False positives & negatives. (g) 10% missing Fourier Coefficients. (h) Gaussian blur of variance τ = 0 . 05 , for point spread function coefficients h k = e − k 2 τ 2 ˆ 2 . (i) Noise of variance . 015 applied to Fourier Coefficients. For blurred functions the edges may be missed, for noisy functions or with missing data too many edges are determined. National Science Foundation: Division of Computational Mathematics 14 / 41
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