IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 7, NO. 3, JULY 2010 455
Mixture Analysis by Multichannel
Hopfield Neural Network
Shaohui Mei, Mingyi He, Member, IEEE, Zhiyong Wang, Member, IEEE, and Dagan Feng, Fellow, IEEE
Abstract—Due to the spatial-resolution limitation, mixed pixels
containing energy reflected from more than one type of ground
objects are widely present in remote sensing images, which often
results in inefficient quantitative analysis. To effectively decom-
pose such mixtures, a fully constrained linear unmixing algorithm
based on a multichannel Hopfield neural network (MHNN) is
proposed in this letter. The proposed MHNN algorithm is actu-
ally a Hopfield-based architecture which handles all the pixels
in an image synchronously, instead of considering a per-pixel
procedure. Due to the synchronous unmixing property of MHNN,
a noise energy percentage (NEP) stopping criterion which uti-
lizes the signal-to-noise ratio is proposed to obtain optimal re-
sults for different applications automatically. Experimental results
demonstrate that the proposed multichannel structure makes the
Hopfield-based mixture analysis feasible for real-world applica-
tions with acceptable time cost. It has also been observed that the
proposed MHNN-based mixture-analysis algorithm outperforms
the other two popular linear mixture-analysis algorithms and that
the NEP stopping criterion can approach optimal unmixing results
adaptively and accurately.
Index Terms—Hopfield neural network (HNN), linear mixture
model (LMM), mixed pixel unmixing, mixture analysis, remote
sensing.
I. INTRODUCTION
P
IXELS in a remote sensing image often contain spectral
information of more than one type of ground objects. Such
kind of pixels are known as mixed pixels or mixtures. The wide
presence of mixtures not only influences the performance of im-
age classification and target recognition but also is an obstacle
to the quantitative analysis of remote sensing images [1]. There-
fore, mixed pixel unmixing or mixture analysis is proposed to
obtain precise distribution of each ground object. Generally, i n
an image, the spectral signatures of typical constituent ground
Manuscript received December 31, 2008; revised May 3, 2009, August 2,
2009, and November 9, 2009. Date of publication March 1, 2010; date of
current version April 29, 2010. This work was supported in part by the National
Natural Science Foundation of China under Key Project 60736007 and in part
by Chinese Scholarship Council, ARC, PolyU, and NPU Grants.
S. Mei is with the Department of Electronics and Information Engineering,
Northwestern Polytechnical University, Xi’an 710129, China, and also with the
School of Information Technologies, The University of Sydney, Sydney, NSW
2006, Australia (e-mail: meishaohui@gmail.com).
M. He is with the Department of Electronics and Information Engineer-
ing, Northwestern Polytechnical University, Xi’an 710129, China (e-mail:
myhe@nwpu.edu.cn).
Z. Wang is with the School of Information Technologies, The University of
Sydney, Sydney, NSW 2006, Australia (e-mail: zhiyong@it.usyd.edu.au).
D. Feng is with the School of Information Technologies, The University of
Sydney, Sydney, NSW 2006, Australia, and also with the Department of Elec-
tronic and Information Engineering, The Hong Kong Polytechnic University,
Kowloon, Hong Kong (e-mail: feng@it.usyd.edu.au).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/LGRS.2009.2039114
objects are known as endmembers, and their corresponding
proportions in mixtures are known as abundance.Themain
concern of this letter is to estimate the fractional abundance of
pure ground objects in mixtures.
The linear mixture model (LMM) has been widely uti-
lized in mixture analysis. Generally, two crucial constraints,
namely, the sum-to-one constraint and the nonnegative con-
straint, must be imposed to ensure that the fractional abun-
dance is physically meaningful. The Fully Constrained Least
Square (FCLS) algorithm was proposed to meet these t wo
constraints simultaneously by using an iterative method to solve
the sum-to-one constrained LMM [2]. The Gradient Descent
Maximum Entropy (GDME) algorithm [3] can also meet these
two constraints simultaneously based on the classical maximum
entropy principle. Barnard and Casasent designed an optical
neural network by using the Hopfield minimization procedure
which can also fully address the mixed-pixel problem [4].
A Hopfield neural network (HNN) has also been constructed
for mixed-pixel fully constrained linear unmixing [5]. The
HNN-based algorithm improves the unmixing performance by
the nonlinear operator of neurons which confines its searching
space into a unit hypercube of the whole space. In addition,
some nonlinear processing in its searching strategy also makes
it more efficient to find the optimal solution.
However, these algorithms, which can be considered as per-
pixel unmixing algorithms, consider each pixel in an image
independently. It is extremely time consuming for some per-
pixel algorithms to be utilized in real-world applications. For
example, it usually takes several hours for a hyperspectral
image containing tens of thousands of pixels by HNN-based
mixture analysis [5]. Although the parallel implementation of
HNN on multiprocessor systems may make it feasible for real-
world applications, the hardware cost is prohibitively high since
tens of thousands of processors are required if the unmixing
procedure of each pixel is implemented on one processor.
Therefore, in this letter, by exploiting the inner connection of
HNN-based mixture analysis for different pixels in an image
[5], a novel multichannel HNN (MHNN) is constructed to
decompose all the pixels synchronously on a single processor.
Based on such an implementation, the computational time has
been greatly shortened. In order to achieve optimal solutions
for different images, a noise energy percentage (NEP) stopping
criterion is also proposed to adaptively approach the best un-
mixing results of MHNN, other than selecting the number of
iterations empirically.
II. MHNN
Let r =(r
1
,r
2
,...,r
b
)
T
be the spectral vector of a mixture,
in which b is the number of bands. Let M =(m
1
, m
2
,...,m
c
)
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