Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 842017, 17 pages

http://dx.doi.org/10.1155/2015/842017

## Backtracking-Based Simultaneous Orthogonal Matching Pursuit for Sparse Unmixing of Hyperspectral Data

^{1}College of Astronautics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China^{2}State Key Laboratory of Integrated Service Networks, Xidian University, Xi’an 710071, China

Received 12 November 2014; Revised 3 April 2015; Accepted 3 April 2015

Academic Editor: Kishin Sadarangani

Copyright © 2015 Fanqiang Kong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Sparse unmixing is a promising approach in a semisupervised fashion by assuming that the observed signatures of a hyperspectral image can be expressed in the form of linear combination of only a few spectral signatures (endmembers) in an available spectral library. Simultaneous orthogonal matching pursuit (SOMP) algorithm is a typical simultaneous greedy algorithm for sparse unmixing, which involves finding the optimal subset of signatures for the observed data from a spectral library. But the numbers of endmembers selected by SOMP are still larger than the actual number, and the nonexisting endmembers will have a negative effect on the estimation of the abundances corresponding to the actual endmembers. This paper presents a variant of SOMP, termed backtracking-based SOMP (BSOMP), for sparse unmixing of hyperspectral data. As an extension of SOMP, BSOMP incorporates a backtracking technique to detect the previous chosen endmembers’ reliability and then deletes the unreliable endmembers. Through this modification, BSOMP can select the true endmembers more accurately than SOMP. Experimental results on both simulated and real data demonstrate the effectiveness of the proposed algorithm.

#### 1. Introduction

With the rapid development of space technology, hyperspectral remote sensing image has gained more and more attention in many application domains, such as environmental monitoring, target detection, material identification, mineral exploration, and military surveillance. However, due to the low spatial resolution of the hyperspectral imaging sensor, each pixel in the hyperspectral image often contains a mixture of several different materials. In order to deal with the problem of spectral mixing, hyperspectral unmixing is used to decompose each pixel’s spectrum to identify and quantify the fractional abundances of the pure spectral signatures or endmembers in each mixed pixel [1, 2]. There are two basic models of hyperspectral unmixing used to analyze the mixed pixel problem: the linear mixture model and the nonlinear mixture model. The linear mixture model assumes that each mixed pixel can be expressed as a linear combination of endmembers weighted by their corresponding abundances [3]. The linear mixture model has been widely applied for spectral unmixing, due to its computational tractability and flexibility. Under the linear mixture model, the traditional linear spectral unmixing algorithms based on geometry [4–8], statistics [5, 9], and nonnegative matrix factorization [10–12] have been proposed. However, some of these methods [10–12] are unsupervised and could extract virtual endmembers with no physical meaning. In [5], the presence in the data of at least one pure pixel per endmember is assumed. If the pure pixel assumption is not fulfilled because of the inadequate spatial resolution and the microscopic mixture of distinct materials, the unmixing results will not be accurate and the unmixing process is a rather challenging task.

Sparse unmixing, as a semisupervised method, has been proposed to overcome this challenge. It assumes that the observed image can be expressed as a linear combinations of spectral signatures from a spectral library that is known in advance [3, 13]. But the number of spectral signatures in the spectral library is much larger than the number of endmembers in the hyperspectral image; the sparse unmixing model is combinatorial and difficult to find a unique, stable, and optimal solution. Fortunately, sparse linear regression techniques can be used [14, 15] to solve it.

Several sparse regression techniques, such as greedy algorithms (GAs) [16–19] and convex relaxation methods [20–23], are usually adopted to solve the sparse unmixing problem. Convex relaxation methods, such as SUnSAL [20], SUnSAL-TV [21], and CLSUnSAL [22], use the alternating direction method of multipliers to efficiently solve the norm sparse regression problem which can decompose a complex problem into several simpler ones. Convex relaxation methods can obtain the global sparse optimization solution and are more sophisticated than the Gas; however they are far more complicated than the GAs. The GAs, such as OMP [16], SP [17], and CGP [18, 19], adopt one or more potential endmembers from the spectral library in each iteration that explains the largest correlation between the current residual of one input signal and the spectral library. The GAs can get an approximate solution for the norm problem without smoothing the penalty function and have low computational complexity. However the endmembers selection criterion of GAs is not optimal in the sense of minimizing the residual of the new approximation; it means that a nonexisting endmember once selected into the supporting set will never be deleted. So the GAs tend to be trapped into the local optimum and are likely to miss some of the actual endmembers. To solve the local optimal solutions problem of GAs, several representative simultaneous greedy algorithms (SGAs) are presented, including simultaneous subspace pursuit (SSP) [24], subspace matching pursuit (SMP) [24], and simultaneous orthogonal matching pursuit (SOMP) [25]. These SGAs divide the whole hyperspectral image into several blocks and pick some potential endmembers from the spectral library in each block. Then the endmembers picked in each block are associated as the endmembers sets of the whole hyperspectral data. Finally, the abundances are estimated using the whole hyperspectral data with the obtained endmembers sets. The SGAs have the same low computational complexity as the GAs and can find the actual endmembers far more accurately than the GAs. The SGAs adopt a block-processing strategy to efficiently solve the local optimal problem, but the endmembers picked in all the blocks are not all actual endmembers and the nonexisting endmembers will affect the estimation of the abundances corresponding to the actual endmembers. To solve the drawback of the SGAs, RSFoBa [26] combinates a forward greedy step and a backward greedy step, which can select the actual endmembers more accurately than the SGAs.

Inspired by the existing SGA methods, we propose a sparse unmixing algorithm termed backtracking-based simultaneous orthogonal matching pursuit (BSOMP) in this paper. Similar to SOMP and SMP, it uses a block-processing strategy to select some potential endmembers and adds them to the estimated endmembers set, which divides the whole hyperspectral image into several blocks and picks some potential endmembers from the spectral library in each block. Furthermore, BSOMP incorporates a backtracking process to detect the previous chosen endmembers’ reliability and then deletes the unreliable endmember from the estimated endmembers set in each iteration. Through this modification, BSOMP can identify the true endmembers set more accurately than the other considered SGA methods.

The remainder of the paper is organized as follows. Section 2 introduces the simultaneous sparse unmixing model. In Section 3, we present the proposed BSOMP algorithm and give out the theoretical analysis for the algorithm. Section 4 provides a quantitative comparison between BSOMP and previous sparse unmixing algorithms, using both simulated hyperspectral and real hyperspectral data. Finally, we conclude in Section 5.

#### 2. Simultaneous Sparse Unmixing Model

The linear sparse unmixing model assumes that the observed spectrum of a mixed pixel is a linear combination of a few spectral signatures presented in a known spectral library. Let denote the measured spectrum vector of a mixed pixel with bands, , where is the number of signatures in the library ; denote a spectral library; then the linear sparse unmixing model can be expressed as follows [23]:where denotes the fractional abundance vector with regard to the library and is the noise. Considering physical constraints, abundance nonnegativity constraint (ANC) and abundance sum-to-one constraint (ASC) are imposed on the linear sparse model as follows: where is the th element of .

The sparse unmixing problem can be expressed as follows:where (called the norm) denotes the number of nonzero atoms in and is the tolerated error due to the noise and model error. It is worth mentioning that we do not explicitly add the ASC in (3), because the hyperspectral libraries generally contain only nonnegative components, and the nonnegativity of the sources automatically imposes a generalized ASC [23].

The simultaneous sparse unmixing model assumes that several input signals can be expressed in the form of different linear combinations of the same elementary signals. This means that all the pixels in the hyperspectral image are constrained to share the same subset of endmembers selected from the spectral library. Then we can use SGA methods for sparse unmixing; the sparse unmixing model in (1) becomeswhere denote the hyperspectral data matrix with bands and mixed pixels, denote a spectral library, denotes the fractional abundance matrix, each column of which corresponds with the abundance fractions of a mixed pixel, and is the noise matrix.

Under the simultaneous sparse unmixing model, the simultaneous sparse unmixing problem can be expressed as follows:where denotes the Frobenius norm of , is the number of nonzero rows in matrix [25], and is the tolerated error due to the noise and model error.

It should be noted that the model in (5) is reasonable because there should be only a few rows with nonzero entries in the abundance matrix in light of only a small number of endmembers in the hyperspectral image, compared with the dimensionality of the spectral library [22].

#### 3. Backtracking-Based SOMP

In this section, we first present our new algorithm, BSOMP, for sparse unmixing of hyperspectral data. Then, a theoretical analysis of the algorithm will be given. Then, we give a convergence theorem for the proposed algorithm.

##### 3.1. Statement of Algorithm

The whole process of using BSOMP for sparse unmixing of hyperspectral data is summarized in Algorithm 1. The algorithm includes three main parts: SOMP for endmember selection, backtracking processing, and abundance estimation. In the first part, we adopt a block-processing strategy for SOMP to efficiently select endmembers. This strategy divides the whole hyperspectral image into several blocks. Then, in each block, SOMP will pick several potential endmembers from the spectral library and add them to the estimated endmembers set. In the second part, we utilize a backtracking strategy to remove some endmembers chosen wrongly from the estimated endmembers set in previous processing and identify the true endmembers set more accurately. The backtracking processing is halted when the maximum total correlation between an endmember in the spectral library and the residual drops below threshold . Finally, the abundances are estimated using the obtained endmembers set under the constraint of nonnegativity.