Edge Detection from Spectral Data With Applications to PSF - PowerPoint PPT Presentation

Edge Detection from Spectral Data With Applications to PSF Estimation and Fourier Reconstruction Aditya Viswanathan School of Electrical, Computer and Energy Engineering, Arizona State University aditya.v @ asu.edu Oct 22 2009 Aditya Viswanathan (Arizona State University) Edge Detection from Spectral Data Oct 22 2009 1 / 41

Prof. Anne Gelb is with the School of Prof. Douglas Cochran is with the School of Mathematical and Statistical Sciences, Electrical, Computer and Energy Arizona State University Engineering, Arizona State University Prof. Rosemary Renaut is with the School of Dr. Wolfgang Stefan is with the Department Mathematical and Statistical Sciences, of Computational and Applied Mathematics, Arizona State University Rice University Research supported in part by National Science Foundation grants DMS 0510813 and DMS 0652833 (FRG). Aditya Viswanathan (Arizona State University) Edge Detection from Spectral Data Oct 22 2009 2 / 41

Introduction Motivation Why do we need edge data? True Solution PDE Solution 4 t f = 0 Solve PDE’s with shocks more t f = 50, Exp. Filtering, p=2 2 t f = 50, Exp. Filtering, p=8 Un(x,t) 0 accurately. −2 −4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x (x pi) Jump Function Approximation 5 0 Identify tissue boundaries in [f](x) t f = 0 −5 t f = 50, Exp. Filtering, p=2 t f = 50, Exp. Filtering, p=8 medical images and segment them. −10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x (a) PDE solution with (b) MRI tissue bound- shock discontinuity ary detection Reconstruct piecewise-analytic Function Reconstruction Function Reconstruction functions from Fourier and other 2.5 2.5 True True 2 Fourier reconstruction 2 Alt. reconstruction 1.5 1.5 spectral expansion coefficients with 1 1 0.5 0.5 uniform and exponential g ( x ) g ( x ) 0 0 −0.5 −0.5 convergence properties. −1 −1 −1.5 −1.5 −2 −2 −2.5 −2.5 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 x x (c) Fourier reconstruc- (d) Alt. reconstruction tion showing Gibbs Aditya Viswanathan (Arizona State University) Edge Detection from Spectral Data Oct 22 2009 3 / 41

Introduction Motivation Application – Magnetic Resonance Imaging Fourier Coefficients Spiral scan trajectory 0.5 0.4 1 0.3 0.8 0.2 f ( ω k x , ω k y ) | 0.6 0.1 k y 0 0.4 | ˆ −0.1 0.2 −0.2 0 −0.3 100 120 100 −0.4 80 50 60 40 20 −0.5 ω ky 0 0 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 ω kx k x (e) Acquired Fourier Samples (f) Sampling Trajectory Reconstructed phantom (g) Reconstructed Image Figure: MR Imaging a a Sampling pattern courtesy Dr. Jim Pipe, Barrow Neurological Institute, Phoenix, Arizona Aditya Viswanathan (Arizona State University) Edge Detection from Spectral Data Oct 22 2009 4 / 41

Introduction Problem Statement Problem Statement Objective: To recover location, magnitude and sign of jump discontinuities from a finite number of spectral coefficients Assumptions: f is 2 π -periodic and piecewise-smooth function in [ − π, π ) . Its jump function is defined as [ f ]( x ) := f ( x + ) − f ( x − ) It has Fourier series coefficients Z π f ( k ) = 1 ˆ f ( x ) e − ikx dx , k ∈ [ − N, N ] 2 π − π 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 − . However, ˆ f is a global representation; i.e., ˆ f ( k ) are obtained using values of f over the entire domain [ − π, π ) . Aditya Viswanathan (Arizona State University) Edge Detection from Spectral Data Oct 22 2009 5 / 41

Concentration Method Outline 1 Introduction Motivation Problem Statement 2 Jump detection using the Concentration method The Concentration Method Concentration Factors Statistical Analysis of the Concentration Method Iterative Formulations 3 Applications PSF Estimation in Blurring Problems Applications to Fourier Reconstruction Aditya Viswanathan (Arizona State University) Edge Detection from Spectral Data Oct 22 2009 6 / 41

Concentration Method The Concentration Method Getting Jump Data from Fourier Coefficients Let f contain a single jump at x = ζ . Z π 1 ˆ f ( x ) e − ikx dx f ( k ) = 2 π − π Z ζ − Z π ! 1 f ( x ) e − ikx dx + ζ + f ( x ) e − ikx dx = 2 π − π Z ζ − ζ − f ( x ) e − ikx ˛ f ′ ( x ) e − ikx 1 ˛ = − − ik dx ˛ 2 π − ik ˛ − π − π Z π π + f ( x ) e − ikx ˛ ζ + f ′ ( x ) e − ikx « ˛ − − ik dx ˛ − ik ˛ ζ + f ( ζ − ) e − ikζ − f ( − π ) e ikπ Z ζ − f ′ ( x ) e − ikx 1 = − − ik dx − ik 2 π − π + f ( π ) e − ikπ − f ( ζ + ) e − ikζ Z π ζ + f ′ ( x ) e − ikx « − − ik dx − ik Aditya Viswanathan (Arizona State University) Edge Detection from Spectral Data Oct 22 2009 7 / 41

Concentration Method The Concentration Method Getting Jump Data from Fourier Coefficients Let f contain a single jump at x = ζ . Z π 1 ˆ f ( x ) e − ikx dx f ( k ) = 2 π − π Z ζ − Z π ! 1 f ( x ) e − ikx dx + ζ + f ( x ) e − ikx dx = 2 π − π Z ζ − ζ − f ( x ) e − ikx ˛ f ′ ( x ) e − ikx 1 ˛ = − − ik dx ˛ 2 π − ik ˛ − π − π Z π π + f ( x ) e − ikx ˛ ζ + f ′ ( x ) e − ikx « ˛ − − ik dx ˛ − ik ˛ ζ + f ( ζ − ) e − ikζ − f ( − π ) e ikπ Z ζ − f ′ ( x ) e − ikx 1 = − − ik dx − ik 2 π − π + f ( π ) e − ikπ − f ( ζ + ) e − ikζ Z π ζ + f ′ ( x ) e − ikx « − − ik dx − ik Aditya Viswanathan (Arizona State University) Edge Detection from Spectral Data Oct 22 2009 7 / 41

Concentration Method The Concentration Method Getting Jump Data from Fourier Coefficients Let f contain a single jump at x = ζ . Z π 1 ˆ f ( x ) e − ikx dx f ( k ) = 2 π − π Z ζ − Z π ! 1 f ( x ) e − ikx dx + ζ + f ( x ) e − ikx dx = 2 π − π Z ζ − ζ − f ( x ) e − ikx ˛ f ′ ( x ) e − ikx 1 ˛ = − − ik dx ˛ 2 π − ik ˛ − π − π Z π π + f ( x ) e − ikx ˛ ζ + f ′ ( x ) e − ikx « ˛ − − ik dx ˛ − ik ˛ ζ + f ( ζ − ) e − ikζ − f ( − π ) e ikπ Z ζ − f ′ ( x ) e − ikx 1 = − − ik dx − ik 2 π − π + f ( π ) e − ikπ − f ( ζ + ) e − ikζ Z π ζ + f ′ ( x ) e − ikx « − − ik dx − ik Aditya Viswanathan (Arizona State University) Edge Detection from Spectral Data Oct 22 2009 7 / 41

Concentration Method The Concentration Method Getting Jump Data from Fourier Coefficients f ( ζ − ) e − ikζ − f ( − π ) e ikπ Z ζ − f ′ ( x ) e − ikx 1 ˆ f ( k ) = − − ik dx 2 π − ik − π + f ( π ) e − ikπ − f ( ζ + ) e − ikζ Z π ζ + f ′ ( x ) e − ikx « − − ik dx − ik „ 1 2 πik + f ( − π ) e ikπ − f ( π ) e − ikπ ´ e − ikζ « f ( ζ + ) − f ( ζ − ) = ` + O k 2 2 πik Since f is periodic, f ( − π ) = f ( π ) and the second term vanishes. „ 1 f ( k ) = [ f ]( ζ ) e − ikζ « ˆ 2 πik + O k 2 Aditya Viswanathan (Arizona State University) Edge Detection from Spectral Data Oct 22 2009 8 / 41

Concentration Method The Concentration Method Getting Jump Data from Fourier Coefficients f ( ζ − ) e − ikζ − f ( − π ) e ikπ Z ζ − f ′ ( x ) e − ikx 1 ˆ f ( k ) = − − ik dx 2 π − ik − π + f ( π ) e − ikπ − f ( ζ + ) e − ikζ Z π ζ + f ′ ( x ) e − ikx « − − ik dx − ik „ 1 2 πik + f ( − π ) e ikπ − f ( π ) e − ikπ ´ e − ikζ « f ( ζ + ) − f ( ζ − ) = ` + O k 2 2 πik Since f is periodic, f ( − π ) = f ( π ) and the second term vanishes. „ 1 f ( k ) = [ f ]( ζ ) e − ikζ « ˆ 2 πik + O k 2 Aditya Viswanathan (Arizona State University) Edge Detection from Spectral Data Oct 22 2009 8 / 41

Concentration Method The Concentration Method Getting Jump Data from Fourier Coefficients f ( ζ − ) e − ikζ − f ( − π ) e ikπ Z ζ − f ′ ( x ) e − ikx 1 ˆ f ( k ) = − − ik dx 2 π − ik − π + f ( π ) e − ikπ − f ( ζ + ) e − ikζ Z π ζ + f ′ ( x ) e − ikx « − − ik dx − ik „ 1 2 πik + f ( − π ) e ikπ − f ( π ) e − ikπ ´ e − ikζ « f ( ζ + ) − f ( ζ − ) = ` + O k 2 2 πik Since f is periodic, f ( − π ) = f ( π ) and the second term vanishes. „ 1 f ( k ) = [ f ]( ζ ) e − ikζ « ˆ 2 πik + O k 2 Aditya Viswanathan (Arizona State University) Edge Detection from Spectral Data Oct 22 2009 8 / 41

Concentration Method The Concentration Method Extracting Jump Information Let us compute a ‘filtered’ partial Fourier sum of the form N „ iπk « ˆ X f ( k ) e ikx S N [ f ]( x ) = N k = − N N „ iπk « X ˆ f ( k ) e ikx S N [ f ]( x ) = N k = − N „ 1 N [ f ]( ζ ) e − ikζ „ iπk « » «– X e ikx = 2 πik + O k 2 N k = − N „ 1 N N « = [ f ]( ζ ) 1 e ik ( x − ζ ) + X X e ik ( x − ζ ) O 2 N k k = − N k = − N First term is a scaled (by the jump value) Dirac delta localized at x = ζ (the jump location) The second term is a manifestation of the global nature of Fourier data Aditya Viswanathan (Arizona State University) Edge Detection from Spectral Data Oct 22 2009 9 / 41