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Sparse signal processing, recovery of sparse signal via L1 minimization. Applications including face recognition, coupled dictionary learning for image super-resolution.

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- 1. Image Analysis & Retrieval CS/EE 5590 Special Topics (Class Ids: 44873, 44874) Fall 2016, M/W 4-5:15pm@Bloch 0012 Lec 17 Sparse Signal Processing & Applications Zhu Li Dept of CSEE, UMKC Office: FH560E, Email: lizhu@umkc.edu, Ph: x 2346. http://l.web.umkc.edu/lizhu p.1Z. Li, Image Analysis & Retrv, 2016 Fall
- 2. Outline Recap: Piece-wise Linear Models via Query Driven Solution Subspace Indexing on Grassmann Manifold Optimization of Subspace on Grassmann Manifold Sparse Signal Processing Sparse Representation and Robust PCA Sparse Signal Processing L1 norm and L1 Magic Solution Application in occluded face recognition Summary p.2Z. Li, Image Analysis & Retrv, 2016 Fall
- 3. Piece-wise Linear : Query Driven • Query-Driven Piece-wise Linear Model – No pre-determined structure on the training data – Local neighborhood data patch identified from query point q, – Local model built with local data, A(X, q) p.3Z. Li, Image Analysis & Retrv, 2016 Fall
- 4. DPC – Discriminant Power Coefficient The tradeoffs in local data support size p.4Z. Li, Image Analysis & Retrv, 2016 Fall
- 5. Face Recognition On ATT Data set: 40 subjects, 400 images: Extra credit: 10pts Develop a query driven local Laplacianface model for HW-3 p.5Z. Li, Image Analysis & Retrv, 2016 Fall
- 6. Subspace Indexing on Grassmann Manifold • Subspace Clustering by Grassmann Metric: – It is a VQ like process. – Start with a data partition kd-tree, their leaf nodes and associated subspaces {Ak}, k=1..2h – Repeat » Find Ai and Aj, if darc(Ai, Aj) is the smallest among all, and the associated data patch are adjacent in the data space. » Delete Ai and Aj, replace with merged new subspace, and update associated data patch leaf nodes set. » Compute the empirical identification accuracy for the merged subspace » Add parent pointer to the merged new subspace for Ai and Aj . » Stop if only 1 subspace left. p.6Z. Li, Image Analysis & Retrv, 2016 Fall
- 7. Simulation • Face data set – Mixed data set of 242 individuals, and 4840 face images – Performance compared with PCA, LDA and LPP modeling p.7Z. Li, Image Analysis & Retrv, 2016 Fall
- 8. Newtonian Method in Optimization Recall that in optimizing a functional over vector variables f(X), X in Rn, p.8 Credit: Kerstin Johnsson, Lund Univ Z. Li, Image Analysis & Retrv, 2016 Fall
- 9. Gradient & Hessian on Grassmann Manifold Gradient on Grassmann manifold: p.9Z. Li, Image Analysis & Retrv, 2016 Fall
- 10. Hessian on Grassmann Manifold Hessian: FY = nxp 1st order differentiation FYY= 2nd order differentiation along Y p.10Z. Li, Image Analysis & Retrv, 2016 Fall
- 11. Newton’s Method on Grassmann Manifold Overall framework Prof. A. Edelman’s matlab package: https://umkc.box.com/s/g2oyqvsb2lx2v9wzf0ju60wnspts4t9g p.11Z. Li, Image Analysis & Retrv, 2016 Fall
- 12. Outline Recap: Piece-wise Linear Models via Query Driven Solution Subspace Indexing on Grassmann Manifold Optimization of Subspace on Grassmann Manifold Sparse Signal Processing Sparse Representation and Robust PCA Sparse Signal Processing L1 norm and L1 Magic Solution Application in occluded face recognition Summary p.12Z. Li, Image Analysis & Retrv, 2016 Fall
- 13. Sparse representation • Signals/Images are sparse if it can have very few non-zero coefficients representation in certain subspace: – E.g. cameraman image X represented as 2-D DCT in Y: • How is this related to classification problem ? – Intuitively, sparse is good for classification, because it is to separate samples from different classes – Only when data points are dense and intertwined , classification is hard – How to characterize this mathematically ? x y=dct2(x) Eigen face p.13Z. Li, Image Analysis & Retrv, 2016 Fall
- 14. Sparsity in Human Visual System p.14Z. Li, Image Analysis & Retrv, 2016 Fall
- 15. Sparse Signal Recovery If x is sparse, i.e |x|0 is small, we can recovery x by a random projection measurement, y=Ax Basis pursuit de-noising: LASSO: p.15Z. Li, Image Analysis & Retrv, 2016 Fall
- 16. Sparse Face Model Consider a face recognition system We have k=1,2,…,K subjects, each subject has nk training samples {[v1,1, .., v1,n1], [v2,1, .., v2,n2], …, [vK,1, .., vK,nK]}, each is a thumbnail image with d=wxh pixels. Let us stack all training samples as a collection of column vectors, A, of d N, N=n1 + n2 + … + nK. The problem is, for a given thumbnail image, y, with unknown class label, how to solve for its label ?K p.16Z. Li, Image Analysis & Retrv, 2016 Fall
- 17. Assume y is belonging to class i, then, Or, Where only a small number of coefficients in x has non-zero entry, thus sparse. Sparsity p.17Z. Li, Image Analysis & Retrv, 2016 Fall
- 18. Assume y is belonging to class 1, then, Most co-efficients related to other classes are zero, only a small number of non-zero coefficients in alpha 1 Illustration of Sparsity p.18Z. Li, Image Analysis & Retrv, 2016 Fall
- 19. • So the problem is rather straight forward – Give y = Ax, where • y is the unknown face image in Rd, • A is the d x N training data matrix, or dictionary, with N large • x is the coefficients of y as linear combination of training samples that is sparse, out of total N coefficients, only a small number of them are non-zero – Mathematically, we are looking for : • Where |x|0 is L0 norm, which counts number of non-zero coefficients in x. Mathematical formulation 𝑥0 = arg min 𝑥 𝑥 0, 𝑠. 𝑡. , 𝐴𝑥 = 𝑦 p.19Z. Li, Image Analysis & Retrv, 2016 Fall
- 20. • The L0 minimization problem is basically a combinatorial optimization problem • Not much structure to exploit fast algorithm • Dumbest solution: – Assuming that x has at most 3 non-zero coefficients, then search total – Possible coefficients combinations and find the one gives the best match – It is an kNN search in effect ! L0 minimization is NP hard 𝑁 1 + 𝑁 2 + 𝑁 3 p.20Z. Li, Image Analysis & Retrv, 2016 Fall
- 21. L0 and L1 norm Lk norm (recall minkowski distance) p.21Z. Li, Image Analysis & Retrv, 2016 Fall
- 22. L1 solution L-2 ball p.22Z. Li, Image Analysis & Retrv, 2016 Fall
- 23. L1 based recognition p.23Z. Li, Image Analysis & Retrv, 2016 Fall
- 24. L1 solution for invalid input images • For non-face images: – Non sparse coefficients in x • Can threshold on residual to return not found result p.24Z. Li, Image Analysis & Retrv, 2016 Fall
- 25. Occlusion and Disguise • A big problem in biometrics is disguise and occlusion • The magic of sparsity and L1 minimization can deal with that effectively ! • Consider a face image with a small fraction p of its pixels corrupted: p.25Z. Li, Image Analysis & Retrv, 2016 Fall
- 26. • Let the occluded face images be y = Ax + e • Then re-state the constraint as, • then solve for P1 with y=Bw. Notice that sparsity in w is achieved thru sparsity in both x and e. Sparsity criteria takes care of occlusion p.26Z. Li, Image Analysis & Retrv, 2016 Fall
- 27. • Occlusion example – Large L2 errors, not recoverable by Eigenface/Fisherface: • Accuracy for sunglasses and scarves effects: Occluded face recognition p.27Z. Li, Image Analysis & Retrv, 2016 Fall
- 28. L1 vs L2 minimization • A natural question is why not solve y=Ax with L2 minimization ? – Typically, number of training samples is smaller than number of pixels in the training images, so why not do a pseudo-inverse like: – Which looks for a Maximum Likelihood estimation of true x, if noises are Gaussian with covariance sI. – However, the noises are non-gaussian and can be unbounded. The resulting L2 solution pretty bad p.28Z. Li, Image Analysis & Retrv, 2016 Fall
- 29. L2 solution for Occlusion • Example with occlusion: – (a): Occluded face – (b): x solved from L2 minimization, not sparse at all – (c ): error – (d ): reconstruction from x p.29Z. Li, Image Analysis & Retrv, 2016 Fall
- 30. L1 vs L2 minimization L1 vs L2 in 2D space: y=Ax p.30Z. Li, Image Analysis & Retrv, 2016 Fall
- 31. Sparsity is bad news for L2 • Given training set A, the unknown image y is under- determined in A: – R(A): a set of y that satisfies y=Ax: p.31Z. Li, Image Analysis & Retrv, 2016 Fall
- 32. Numeric solution for L1 minimization Candes (of CalTech)’s group has this L1 magic matlab toolbox Check out manual on course webpage Stephen Boyd: Boyd’s nice book on Optimization can be downloaded from his webpage at Stanford. Excellent book, with slides, homework and solutions. https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf p.32Z. Li, Image Analysis & Retrv, 2016 Fall
- 33. Numerical Tool from L1 Magic L1 Magic Toolbox: p.33 % signal length N = 512; % number of spikes in the signal, must be sparse w.r.t N T = 20; % number of observations to make K = 120; % random +/- 1 signal x = zeros(N,1); q = randperm(N); x(q(1:T)) = sign(randn(T,1)); subplot(3,1,1); plot(x); title('x(t)'); axis([1 500 -1.2 1.2]); % measurement matrix: random measuring fprintf('n Creating random measurement matrix...'); A = randn(K,N); % othorgonalize A = orth(A')'; % observations y = A*x; % initial guess = min energy x0 = A'*y; subplot(3,1,2); plot(x0); title('x_0(t)'); axis([1 500 -1.2 1.2]); % solve with primal-dual method xp = l1eq_pd(x0, A, [], y, 1e-3); subplot(3,1,3); plot(xp); title('x(t) recovered by L1 magic'); axis([1 500 -1.2 1.2]); % test l1magic end Z. Li, Image Analysis & Retrv, 2016 Fall
- 34. L1 Magic demo 50 100 150 200 250 300 350 400 450 500 -1 0 1 x(t) 50 100 150 200 250 300 350 400 450 500 -1 0 1 x0(t) 50 100 150 200 250 300 350 400 450 500 -1 0 1 x(t) recovered by l1 magic Original sparse signal L1 magic recovered sparse signal Pseudo-inverse: L2 recovery p.34Z. Li, Image Analysis & Retrv, 2016 Fall
- 35. Sparse face Recover face as sparse signal p.35 % create our measure matrix A: face + nonface icons A=zeros(1600, w*h); A(1:400, :) = faces; A(401:1600, :) = nfaces; [N, dim]=size(A); % in col vec form A = A'; % pick a face: offs in 1-400 figure(3); colormap('gray'); offs = 20; y = faces(offs, :)'; subplot(2,2,1); axis off; imagesc(reshape(y, h,w)); title('fontsize{11}original'); % solve for xp = min |x|, s.t. y=Ax % initial guess = min energy x0 = A'*y; % solve with primal-dual method xp = l1eq_pd(x0, A, [], y, 1e-3); % normalize x0 = x0./norm(x0); xp = xp./norm(xp); % reconstructed face yp = A*xp; subplot(2,2,2); axis off; imagesc(reshape(yp, h,w)); title('fontsize{11}sparse reconstruction'); Z. Li, Image Analysis & Retrv, 2016 Fall
- 36. L1Magic for Face Recognition p.36Z. Li, Image Analysis & Retrv, 2016 Fall
- 37. Super Resolution Super-Resolution Super-resolves a lower resolution patch, say k x k, to 3k x 3k. Mathematically, learn a function: p.37 𝑓 𝑥 → 𝑌, 𝑥 ∈ 𝑅 𝑑, 𝑌 ∈ 𝑅 𝐷 Z. Li, Image Analysis & Retrv, 2016 Fall
- 38. Basic Framework Super-resolve is the inverse of down scaling: Low res patch y is the blurred and scaled high res patch x: Assume the high res image is sparse on some dictionary (true, say DCT): p.38 Output OriginalInput Training patches ≈ 𝑦 = 𝑆𝐻𝑥 Z. Li, Image Analysis & Retrv, 2016 Fall
- 39. Coupled Dictionary Learning Pre-train a common set of coupled low and high resolution dictionary Super-resolve by solving L1 minimization on lower resolution patch, and use the same coeffiients to superresolve the higher resolution patch p.39Z. Li, Image Analysis & Retrv, 2016 Fall
- 40. Coupled Dictionary Learning Learn two sets of Dictionaries, Dh, Dl, that have common sparse coefficients for low and high resolution image patches, y and x: Reconstruction of low res patch with sparse coefficients: Furthermore, introduce a linear projection, F, to enforce perceptual metrics Then the high res patch x, can be constructed as p.40 min 𝛼 0 , 𝑠. 𝑡. , 𝐷𝑙 𝛼 − 𝑦 2 ≤ 𝜖 min 𝛼 0 , 𝑠. 𝑡. , 𝐹𝐷𝑙 𝛼 − 𝐹𝑦 2 ≤ 𝜖 𝑥 = 𝐷ℎ 𝛼 Yang, J Wright, TS Huang, Y Ma, Image super-resolution via sparse representation, IEEE Trans. Image Processing, vol.19 (11), 2861-2873 Z. Li, Image Analysis & Retrv, 2016 Fall
- 41. Coupled Dictionary Learning Put together, super resolve is to solve: Sparse reconstruction of lower resolution y Enforce local consistence with high res patches, extract adjacent overlapping stripes, via P, to be in agreement, w is the previously reconstructed patch pixels: Solution via Lagrangian relaxation: p.41Z. Li, Image Analysis & Retrv, 2016 Fall
- 42. Overall Algorithm Patch level super-resolution, complete with global image gradient search p.42Z. Li, Image Analysis & Retrv, 2016 Fall
- 43. Dictionary Training Training data: low and high resolution image patches Yl={yk}, Xh={xk}: Enforce the common sparse coefficients p.43Z. Li, Image Analysis & Retrv, 2016 Fall
- 44. Results Dictionary Training From flowers and animals data set, covering a variety of texture Training dictionary from more than 100,000 samples p.44 𝐷ℎ 𝐷𝑙 Z. Li, Image Analysis & Retrv, 2016 Fall
- 45. Results 3x super-resolution p.45 Bicubic Neighbor embedding [Chang CVPR ‘04] Low-resolution input OriginalCoupled Dictionary Z. Li, Image Analysis & Retrv, 2016 Fall
- 46. Related Work Potential paper review project p.46Z. Li, Image Analysis & Retrv, 2016 Fall
- 47. Summary Sparse Signal Processing If signal is sparse in some (unknown) domain, then from a random measurement, we can reliably recover the signal via L1 minimization Applications: Robust PCA and Face Recognition with Occlusion Face images are sparse linear combination from a face dictionary Recovery from solving L1 problem ~ caveat: only additive noises can be delt. Applications: Coupled Dictionary for Image Super Resolution Coupled dictionary: high and low res image patches sharing the same coefficients. p.47 min 𝑥 𝑥 1, 𝑠. 𝑡. 𝑦 = 𝐴𝑥 Z. Li, Image Analysis & Retrv, 2016 Fall

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