Pixel Recovery via Minimization in the Wavelet Domain

1. Pixel Recovery via Minimization in the Wavelet Domain Ivan W. Selesnick, Richard Van Slyke, and Onur G. Guleryuz * * # presenting author *: Polytechnic University, Brooklyn, NY #: DoCoMo Communications Laboratories USA, Inc., San Jose, CA

2. Overview • Problem statement: Estimation/Recovery of missing data. • Formulation as a linear expansion over overcomplete basis. • Expansions that minimize the norm. • Why do this? • Connections to adaptive linear estimators and sparsity. • Connections to recent results and statistics • Simulation results and comparisons to our earlier work. • Why not to do this: Analysis of what is going on. • Conclusion and ways of modifying the solutions for better results. ( Presentation is much more detailed than the paper.) ( Some software available, please check the paper.)

3. available pixels lost pixels Image (assume zero mean) 2. Lost Block 3. Derive predicted Problem Statement 1. Original

4. Formulation available data projection 1. Take NxM matrix of overcomplete basis, 2. Write y in terms of the basis 3. Find the expansion coefficients (two ways)

5. subject to Find the expansion coefficients to minimize the norm norm of expansion coefficients Regularization Available data constraint

6. subject to Bogus reason norm Why minimize the norm? “Under i.i.d. Laplacian model for coefficient probabilities, min Real reason: sparse decompositions.

7. 1. Any such decomposition builds an adaptive linear estimate. 2. In fact “any” estimate can be written in this form. Onur G. Guleryuz, ``Nonlinear Approximation Based Image Recovery Using Adaptive Sparse Reconstructions and Iterated Denoising: Part I - Theory,‘’ IEEE Tr. on IP, in review. http://eeweb.poly.edu/~onur (google: onur guleryuz). What does sparsity have to do with estimation/recovery?

8. The recovered signal must be sparse 3. The recovered becomes null space of dimension y has to be sparse Onur G. Guleryuz, ``Nonlinear Approximation Based Image Recovery Using Adaptive Sparse Reconstructions and Iterated Denoising: Part I - Theory,‘’ IEEE Tr. on IP, in review. http://eeweb.poly.edu/~onur (google: onur guleryuz).

9. Who cares about y, what about the original x? If successful prediction is possible x also has to be ~sparse i.e., if small, then x ~ sparse 1. Predictable sparse 2. Sparsity of x is not a bad leap of faith to make in estimation If not sparse, cannot estimate well anyway. (caveat: the data may be sparse, but not in the given basis) Onur G. Guleryuz, ``Nonlinear Approximation Based Image Recovery Using Adaptive Sparse Reconstructions and Iterated Denoising: Part I - Theory,‘’ IEEE Tr. on IP, in review. http://eeweb.poly.edu/~onur (google: onur guleryuz).

10. subject to Why minimize the norm? Under certain conditions the problem gives the solution to the problem: subject to Find the “most predictable”/sparsest expansion that agrees with the data. (solving convex, not combinatorial) D. Donoho, M. Elad, and V. Temlyakov, ``Stable Recovery of Sparse Overcomplete Representations in the Presence of Noise‘’. http://www-stat.stanford.edu/~donoho/reports.html

11. Why minimize the norm? Experience from statistics literature. The “lasso” is known to generate sparse expansions. subject to R. Tibshirani, ``Regression shrinkage and selection via the lasso’’. J. Royal. Statist. Soc B., Vol. 58, No. 1, pp. 267-288.

12. subject to H: Two times expansive M=2N, real, dual-tree, DWT. Real part of: N. G. Kingsbury, ``Complex wavelets for shift invariant analysis and filtering of signals,‘’ Appl. Comput. Harmon. Anal., 10(3):234-253, May 2002. Simulation Results vs. Iterated Denosing (ID) with no layering and no selective thresholding: Onur G. Guleryuz, ``Nonlinear Approximation Based Image Recovery Using Adaptive Sparse Reconstructions and Iterated Denoising: Part II –Adaptive Algorithms,‘’ IEEE Tr. on IP, in review. http://eeweb.poly.edu/~onur (google: onur guleryuz).

13. Simulation Results

14. available pixel constraint x Sparse Modeling Generates Non-Convex Problems x missing pixel x available pixel Pixel coordinates for a “two pixel” image Transform coordinates

15. “Sparse=non-convex”, who cares. What about reality, natural images? = +

16. ball x x Case 1 Case 2 Case 3 Not sparse Bogus reason Geometry x

17. overwhelming noise: error due to missing data modeling error Why not to minimize the norm What about all the optimality/sparsest results? Results such as: D. Donoho et. al. ``Stable Recovery of Sparse Overcomplete Representations in the Presence of Noise‘’. are very impressive, but they are tied closely to H providing the sparsest decomposition for x.

18. subject to (problem due to ) “nice” basis, “decoherent” “not nice” basis (due to cropping), may become very “coherent” Why not to minimize the norm

19. unnormalized coherency= normalized coherency= 1 (worst possible) 1. Optimal solution sometimes tries to make coefficients of scaling functions zero. 2. solution never sees the actual problem. Examples orthonormal, coherency=0

20. Decomposes the big problem into many progressions. • Arrives at the final complex problem by solving much simpler problems. • is conceptually a single step, greedy version of ID. Progression 1: Progression 2: ... What does ID do? Uses correctly modeled components to reduce the overwhelming errors/”noise”

21. ID is all about robustly selecting sparsity • Tries to be sparse, not the sparsest. • Robust to model failures. • Other constraints easy to incorporate

22. Conclusion 1. Have to be more agnostic than smoothest, sharpest, smallest, sparsest, *est. minimum mse not necessarily = sparsest 2. Have to be more robust to modeling errors. When a convex approximation is possible to the underlying non-convex problem, great. But have to make sure assumptions are not violated. For lasso/ fans: 3. Is it still possible to use , but with ID principles? Yes

23. subject to subject to available data subject to ... But must ensure no case 3 problems (ID stays away from those). 1. It’s not about the lasso or how you tighten the lasso, it’s about what (plural) you tighten the lasso to. Do you think you reduced mse? No: you shouldn’t have done this. Yes: Do it again. 2. This is not “LASSO”, “LARS”, .... This is Iterated Denoising (use hard thresholding!).