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Lecture 22 Exemplary Inverse Problems including Filter Design

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Syllabus Lecture 01Describing Inverse Problems Lecture 02Probability and Measurement Error, Part 1 Lecture 03Probability and Measurement Error, Part 2 Lecture 04The L 2 Norm and Simple Least Squares Lecture 05A Priori Information and Weighted Least Squared Lecture 06Resolution and Generalized Inverses Lecture 07Backus-Gilbert Inverse and the Trade Off of Resolution and Variance Lecture 08The Principle of Maximum Likelihood Lecture 09Inexact Theories Lecture 10Nonuniqueness and Localized Averages Lecture 11Vector Spaces and Singular Value Decomposition Lecture 12Equality and Inequality Constraints Lecture 13L 1, L ∞ Norm Problems and Linear Programming Lecture 14Nonlinear Problems: Grid and Monte Carlo Searches Lecture 15Nonlinear Problems: Newton’s Method Lecture 16Nonlinear Problems: Simulated Annealing and Bootstrap Confidence Intervals Lecture 17Factor Analysis Lecture 18Varimax Factors, Empircal Orthogonal Functions Lecture 19Backus-Gilbert Theory for Continuous Problems; Radon’s Problem Lecture 20Linear Operators and Their Adjoints Lecture 21Fréchet Derivatives Lecture 22 Exemplary Inverse Problems, incl. Filter Design Lecture 23 Exemplary Inverse Problems, incl. Earthquake Location Lecture 24 Exemplary Inverse Problems, incl. Vibrational Problems

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Purpose of the Lecture solve a few exemplary inverse problems image deblurring deconvolution filters minimization of cross-over errors

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Part 1 image deblurring

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three point blur (applied to each row of pixels)

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null vectors are highly oscillatory

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solve with minimum length

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note that GG T can deduced analytically and is Toeplitz might lead to a computational advantage

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Solution Possibilities 1.Use sparse matrix for G together with mest=G’*((G*G’)\d) (maybe damp a little, too) 2. Use analytic version of GG T together with mest=G’*(GGT\d) (maybe damp a little, too) 3. Use sparse matrix for G together with bicg() to solve GG T λ=d (maybe with a little damping, too) and then use m est = G T λ

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Solution Possibilities 1.Use sparse matrix for G together with mest=G’*((G*G’)\d) (maybe damp a little, too) 2. Use analytic version of GG T together with mest=G’*(GGT\d) (maybe damp a little, too) 3. Use sparse matrix for G together with bicg() to solve GG T λ=d (maybe with a little damping, too) and then use m est = G T λ we used the simplest, which worked fine

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image blurred due to camera motion (100 point blur)

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(A)(B)(C) (D)(E)(F) pixel number

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[G -g ] 728 R 728 row number (A) (B) sidelobes

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Part 2 deconvolution filter

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Convolution general relationship for linear systems with translational invariance

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Convolution general relationship for linear systems with translational invariance model m(t) and data d(t) related by linear operator

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Convolution general relationship for linear systems with translational invariance only relative time matters

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underlying principle linear superposition

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time, t, after impulse d(t)= g (t) 0 time, t, after impulse m(t)=δ(t) 0 If the input of a spike m(t)= δ (t) spike causes the output of d(t)= g (t)

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m(t 0 )g(t t 0 ) m(t) time, t t0t0 d(t) time, t t0t0 spike of amplitude, m(t 0 ) Then the general input m(t) causes the general output d(t)=m(t)*g(t)

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convolution d=m*g

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discrete convolution d=m*g standard matrix from d=Gm

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seismic reflection sounding

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want airgun pulse to be as spiky as possible p(t) = g(t) * r(t) pressure = airgun pulse * sea floor response so as to be able to detect pulses in sea floor response p(t) ≈ r(t)

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time, t g(t) actual airgun pulse is ringy

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so construct a deconvolution filter m(t) so that g(t) *m(t) = δ(t) and apply it to the data p(t)*m(t) = g(t)*m(t)*r(t) = r(t) p(t) =g(t) * r(t)

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g(t) *m(t) = δ(t) and apply it to the data p(t)*m(t) = g(t)*m(t)*r(t) = r(t) p(t) =g(t) r(t) this is the equation we need to solve so construct a deconvolution filter m(t) so that

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g(t) *m(t) = δ(t) Gm = d discrete approximation of delta function m = use discrete approximation of convolution...

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solve with damped least squares m est = [G T G + ε 2 I] -1 G T d with d = [1, 0, 0,..., 0] T (or something similar) matrices G T G and G T d can be calculated analytically

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approximately Toeplitz with elements

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autocorrelation of g

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cross-correlation of g and d

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Solution Possibilities 1.Use sparse matrix for G together with mest=(G’*G)\(G’*d) (maybe damping a little, too) 2. Use analytic versions of G T G and G T d together with mest=GTG\GTd (maybe damp a little, too) 3. Never form G, just work with its columns, g use bicg() to solve G T G m = G T d but use conv() to compute G T (Gv) 4. Same as 3 but add a priori information of smoothness

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Solution Possibilities 1.Use sparse matrix for G together with mest=(G’*G)\(G’*d) (maybe damping a little, too) 2. Use analytic versions of G T G and G T d together with mest=GTG\GTd (maybe damp a little, too) 3. Never form G, just work with its columns, g use bicg() to solve G T G m = G T d but use conv() to compute G T (Gv) 4. Same as 3 but add a priori information of smoothness we used this complicated but very fast method

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time, t g(t) m(t) m(t)*g(t) (A) (B) (C)

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(A) Original (B) After deconvolution d(t) d(t)*m(t)

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Part 3 minimization of cross-over errors

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longitude latitude true estimated gravity anomaly, mgal longitude note streaks

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general idea data s is measured along tracks data along each track is off by an additive constant theory s j obs (track i) = s j true (track i) + m (track i) goal is to estimate the constants by minimizing the error at track intersections

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cross-over points

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i th intersection has ascending track A i and descending track D i s Ai obs = s Ai true + m Ai s Di obs = s Di true + m Di subtract s Ai obs -s Di obs = m Ai - m Di has form d=Gm

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the matrix G is very sparse every row is all zeros, except for a single +1 and a single -1

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note that this problem has an inherent non-uniqueness m is determined only to an overall additive constant one possibility is to use damped least squares, to choose the smallest m (you can always add a constant later)

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the matrices G T G and G T d can be calculated semi-analytically

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recipe starting with zeroed G T G and G T d

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Solution Possibilities 1.Use sparse matrix for G together with damped least squares mest=(G’*G+e2*speye(M,M))\(G’*d) 2. Use analytic versions of G T G and G T d add damping directly to the diagonal of G T G then use mest=GTGpe2I\GTd 3. Use sparse matrix for G together with bicg() version of damped least squares 4. Methods 1 or 2, but use hard constraint instead of damping to implement Σ i m i = 0

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Solution Possibilities 1.Use sparse matrix for G together with damped least squares mest=(G’*G*e2*speye(M,M))\(G’*d) 2. Use analytic versions of G T G and G T d add damping directly to the diagonal of G T G then use mest=GTG\GTd 3. Use sparse matrix for G together with bicg() version of damped least squares 4. Methods 1 or 2, but use hard constraint instead of damping our choice

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longitude latitude (A) (B) (D) (C) gravity anomaly, mgal longitude

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