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Hyperspectral unmixing is a powerful tool for the remote sensing image mining. Nonnegative matrix factorization (NMF) has been adopted to deal with this issue, while the precision of unmixing is closely related with the local minimizers of NMF. We present two novel initialization strategies that is based on CUR decomposition, which is physically meaningful. In the experimental test, NMF with the new initialization method is used to unmix the urban scene which was captured by airborne visible/infrared imaging spectrometer (AVIRIS) in 1997, numerical results show that the initialization methods work well.

Given a non-negative matrix

Problem (1) is commonly reformulated as the following minimization problem:

Due to non-subtractive combinations of non-negative basis vector, NMF can give a simple interpretation in many areas. We are concerned with its application of analyzing data obtained using astronomical spectrometers, which provide nonnegative spectral data [

Many algorithms have been designed for solving (2). Most of these algorithms can be classified into two categories: gradient descent methods [

Since NMF is a nonconvex optimization problem, the iterative methods typically converge slowly to a local minima. It’s said that a good initialization can improve the speed and the accuracy of the solutions of many NMF algorithms as it can produce faster convergence to an improved local minima [

Originally, the factor matrices are initialized with random numbers between 0 and 1. The random initialized matrices are dense and do not provide a good estimate for U and V. In order to obtain better solutions, some researchers have tried to improve the initialization phase. Wild [

CUR decompositions approximate the data matrix M by using a small number of actual columns of M. In hyperspectral image analysis, these methods find endmembers that are already present in the data. By doing this, these methods are easily interpretable. However, the endmembers should by contained within the data. It is called the pixel purity assumption, which is a strong requisite. Applying the column selection technique of CUR in the initialization stage of NMF is called Acol. In this paper, we give two new initialization methods for NMF which is based on CUR.

Let

The spectral angle distance (SAD) describes the difference between the true endmembers and the estimated endmembers. It is defined as follows,

where

Given a random vector

Then, the entropy of w is given by

We employ the symmetrized Kullback Leibler (KL) divergence to select the columns from

where

KL divergence is a measure of information lost, when d is used to approximated w. Since the KL divergence is not symmetric and does not satisfy the triangle inequality, the following symmetrized KL divergence is considered.

Once a column from

In this section, we compare our initialization strategy with the Urban hyperspectral image, which is taken from HYper-spectral Digital Imagery Collection Experiment (HYDICE) air-borne sensors. It contains 162 clean

bands and

We will first look at the performance of four initialization techniques: random, rand columns (Acol), random selected columns from M with SAD (SADcol) and random selected columns with SKLD (SKLDcol). It is said that the active set type NMF method is a good choice to solve the spectroscopy data, so we combine the above four initialization with the active set method proposed in [

Frobenius norm of the error,

Since the random initialization has nothing to do with the measured spectra signal, the Frobenius error in the beginning is bigger than the other methods. It can be seen from

Two initialization methods for NMF are proposed in this paper. With the help of the SAD and SKLD measure, the Frobenius error is lower than the rand and Acol. Since

Method | Rand | Acol | SADcol | SKLDcol |
---|---|---|---|---|

Time | 12.5924007s | 10.3637064s | 7.5863286s | 8.1900524s |

The work was supported in part by the National Science Foundation ofChina (41271235, 10901094,11301307), The national science and technology support program(2013BAD05B06-5). The Excellent Young Scientist Foundation ofShandong Province (BS2011SF024, BS2012SF025) and Young TeacherFoundation of Shandong Agricultural University (23744).

Li Sun,Gengxin Zhao,Xinpeng Du, (2016) CUR Based Initialization Strategy for Non-Negative Matrix Factorization in Application to Hyperspectral Unmixing. Journal of Applied Mathematics and Physics,04,614-617. doi: 10.4236/jamp.2016.44068