Sparse Matrix Factorizations for Hyperspectral Unmixing

John Wright

Visual Computing Group

Microsoft Research Asia

Sept. 30, 2010

Goal: Recover the hyperspectral image

as accurately and efficiently as possible.

Important practical subproblem: given observations

estimate .

Problem setting

.

High spectral res, low spatial res

High sptial res, RGB

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,

Assumptions

.

with sparse.

1. Find (A,X) such that X is sparse and

2. For each location (i,j) in the high-resolution image, solve a small sparse coding problem

3. Reconstruct the high-resolution hyperspectral image via:

First estimate the basis , then use it to find the coefficients

Computationally tractable approach

.

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.

Sparse matrix factorizations

.Geng, Wang, Wright, On the correctness of dictionary learning algorithms, in preparation.

Problem: Given , with unknown (possible overcomplete) basis , unknown sparse coefficients , recover the pair .

Domain of optimization

Use nonsmooth Gauss-Newton to solve

Surprisingly, strong sense in which this “works”:

Sparse matrix factorizations

.Geng, Wang, Wright, On the correctness of dictionary learning algorithms, in preparation.

Exact recovery

Problem size

Numerical Results

.

Image Balloon Beads Thread Oil painting

RMSE (L1)

5.14 9.37 8.55 4.92

RMSE (best last time)

6.6 (RGB clustering)

11.3(global)

6.8(non-local means top 30)

5.9(local window)

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.