Hyperspectral Unmixing based on Constrained Nonnegative Matrix Factorization via Approximate L0

Abstract

Hyperspectral unmixing is estimating the endmembers and corresponding abundance fractions in a mixed pixel. In the past decade, NMF have been intensively studied to hyperspectral unmixing. As an important constraint for NMF, sparsity could be modeled making use of the L0 regularize. Unfortunately, the L0 regularize is an N-P hard. In this paper, we uses a novel approximate L0 sparsity constraint (which we name AL0-NMF), we propose a project gradient algorithm for AL0 -NMF. The experimental based on synthetic and real data demonstrate the effectiveness of the propose method. Introduction Hyperspectral data is a spectal image cube, containing hundreds of spectral bands and spatial information .Owing to the low spatial resolution of the sensor, there are many mixed pixels in the remote sensing image. Hyperspectral unmixing which decomposes mixed pixels into a collection of constituent spectra, or endmembers, and the corresponding abundance fractions is often used to preprocess hyperspectral data [1].In the past few decades, many hyperspectral unmixing algorithms have been propose under the linear mixing model(LMM) such as vertex component analysis (VCA) [2], independent component analysis (ICA) [3], etc. As a widely used method of blind source separation (BBS), nonnegative matrix factorization (NMF) [4] can adopted to solve the hyperspectral unmixing. Unfortunately, due to the objective function of NMF is nonconvexity, a lot of local minimum occur. An easy but effective solution to reduce the problem is to introduce further constraints into the NMF algorithm. Now, some researchers used sparsity constraint on basic NMF, like [5]. Regularization method are usually utilized to define the sparsity constraint on the abundance matrix of the endmember. In general, L1 norm is widely used to instead of L0 norm. But for L1 regularization, it will be always a constraint due to the abundances suffer from the sum to one constraint. The L0 regularization can yield a sparser results, while it is an NP-hard. In this paper, we introduce a smoothed function to approximate the L0 norm, which can enforce the sparsity of endmember abundances and avoid to solve the NP-hard problem. In this paper, we introduce a new L0 regularization into NMF (AL0-NMF) to enforce the sparsity of abundance matrix. We used projected gradient methods in [6] to ensure convergence. In this approach, the SNC is embedded in the parameter update process. Through synthetic and real hyperspectral data experiments results, we observe our algorithm is state of the art. The rest of this paper is organized as follows. In Section 2, brief introduces AL0-NMF model of the hyperspectral unmixing. Section 3 give the algorithm of hyperspectral unmixing Section 4 show the results of synthetic and real hyperspectral data. Sections 5 draws conclusions. AL0-NMF Unmixing Model LMM and NMF. The LMM can be written as follows: X AS E = + (1) International Conference on Information Sciences, Machinery, Materials and Energy (ICISMME 2015) © 2015. The authors Published by Atlantis Press 921 Where L N X R × ∈ denotes the hyperspectral data, L P A R × ∈ denotes the endmember signatures, P N S R × ∈ denotes the endmember abundances, P N E R × ∈ denotes the additive noise. And L, N, P denotes the number of bands, the number of pixels and the number of endmembers in remote sensing image, respectively. Since NMF can lead to the part-based linear representations of the nonnegative high-dimensional data, it has received a lot of attention. NMF aims to decompose one nonnegative matrix X into two low-rank nonnegative matrices A and S .So we can solve the NMF problem as an optimization problem by minimizing the Euclidean distance. Cosidering the Abundance Nonnegative Constraint (ANC) and sum to one, the cost function is as follows: