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Sparse Matrix Factorizations for Hyperspectral Unmixing

Sparse Matrix Factorizations for Hyperspectral Unmixing. John Wright Visual Computing Group Microsoft Research Asia. Sept . 30 , 2010. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A A A A. Problem setting.

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Sparse Matrix Factorizations for Hyperspectral Unmixing

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  1. Sparse Matrix Factorizations for HyperspectralUnmixing John Wright Visual Computing Group Microsoft Research Asia Sept. 30, 2010 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAA

  2. Problem setting Goal: Recover the hyperspectral image as accurately and efficiently as possible. Important practical subproblem: given observations estimate . High spectral res, low spatial res High sptial res, RGB .

  3. Assumptions A1: Scene simplicity. There are a limited number of materials (and hence a limited number of distinct reflectances) in the scene. We represent the reflectances of these materials as the columns of an (unknown) matrix So, for each location (i,j), Ideally, the vector is 1-sparse (only one material present). A2: Sampling rate. The materials change slowly enough in space that only a few distinct materials are present in each ``pixel’’ of . Hence, with sparse. .

  4. Computationally tractable approach First estimate the basis , then use it to find the coefficients Find (A,X) such that X is sparse and For each location (i,j) in the high-resolution image, solve a small sparse coding problem Reconstruct the high-resolution hyperspectral image via: .

  5. Sparse matrix factorizations Problem: Given an observation that is a product of an unknown (possible overcomplete) basis and a set of unknown sparse coefficients , recover the pair . Harder than sparse coding against a known basis[Donoho+Elad ‘01, Candes+Tao ‘05]. Progress recently [Geng et. al. ‘10]: Appears to be exactly solvable via local minimization, provided the solution is sparse and we have seen enough measurements. Geng, Wang, Wright, On the correctness of dictionary learning algorithms, in preparation. .

  6. Sparse matrix factorizations Problem: Given , with unknown (possible overcomplete) basis , unknown sparse coefficients , recover the pair . Domain of optimization Usenonsmooth Gauss-Newton to solve Surprisingly, strong sense in which this “works”: Exact recovery Problem size Geng, Wang, Wright, On the correctness of dictionary learning algorithms, in preparation. .

  7. Numerical Results Uniform parameter settings across all images … better results are possible with adaptive choice of thresholds (e.g., RMSE 4.6 for balloons). Sorry, no pictures in these slides… will send around in a day or two. .

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