WEIGHTED OVERCOMPLETE DENOISING Onur G. Guleryuz Epson Palo Alto Laboratory Palo Alto, CA (Please view in full screen mode to see animations)

Overview Signal in additive, i.i.d., Gaussian noise scenario. Consider standard denoising with overcomplete transforms and thresholding. Denoised estimates are suboptimally combined to form an average. Optimal combination as an adaptive linear estimation problem for each pixel. Simulation results with DCTs and wavelets. Conclusion.  Definition, assumptions, why it works…  Three solutions ( no explicit statistics required ! )  One solution is based on number of zero coefficients.  Form of equivalent adaptive linear denoising filters.

Notation : N-dimensional signal : signal corrupted with additive noise : estimate of given

Denoising with Overcomplete Transforms and Thresholding : linear transform : transform step : thresholding step : denoised estimate extensive literature : combination stepthis paper

Basic Principles for a Single Transform and Hard-Thresholding 0N-1 k Transform Domain x(n) X(k) 0N-1 k + W(k) c(k) = +w(n)y(n)= Signal Domain x(n)X(k) Main Assumption:Sparse Decomposition +T -T0N-1 k c(k) ^ Denoised

Overcomplete Transforms Denoise …

Basic Idea of this Paper - 1 Give more weight to sparse blocks (I will determine weights optimally).

Basic Idea of this Paper - 2 Suppose only DCT-DC terms remain after thresholding. Optimally determine weights everywhere.

Main Derivation Assumption: Thresholding works! Denoising removes mostly noise. Optimally determine for n=1,…,N to minimize conditional mse. Solutions will be independent of explicit statistics. No covariance/modeling assumptions, etc. Reminder box

Main Derivation with Hard-Thresholding Thresholding works! Sparse decompositions (signal only has components in significant sets.) Significant sets. Reminder box Optimization:

Main Derivation with Hard-Thresholding

Solution is only a function of and the significant sets

Solutions : only the diagonal terms of needs basis functions of the transforms and cross scalar products depending on the needs basis functions of the transforms only needs the spatial support of the basis functions of the transforms for block transforms, diagonal entries are 1/(number of nonzero coefficients in each block) full solution diagonal solution significant- only solution

Properties of Solutions fully overcomplete 8x8 DCTs (256x256) voronoi The full solution is sensitive to model failures.

standardfull diagonalsignif.-only

standard diagonal

Equivalent Adaptive Linear Filters Reminder box pixel 1 pixel 2 pixel 3 At each pixel we adaptively determine a linear filter for denoising.

Equivalent Adaptive Linear Filters pixel 1pixel 2pixel 3 full solution diagonal solution standard solution

Equivalent Adaptive Linear Filters pixel 1pixel 2pixel 3 full solution diagonal solution standard solution

Simulation Results fully overcomplete 8x8 DCTs (1280x960) teapot

standard diagonal signif.-only

Simulation Results fully overcomplete 3 level wavelets (Daubechies orthonormal D8 bank) (1280x960) teapot

Simulation Results fully overcomplete 3 level wavelets (Daubechies orthonormal D8 bank) Lena (512x512) Barbara (512x512)

Conclusion Instead of blindly combining denoised estimates, form the optimal combination.  Also true for denoising with naturally overcomplete transforms like complex wavelets. Statistically “clean” formulation, no dependence on explicit statistics. Easily generalized to other forms of thresholding (additional degree of freedom). Better, more sophisticated thresholding is expected to improve results.