hyperspectral data and achieve better classification results. Document [8] uses spectral
angle and spectral information divergence to replace Euclidean distance and combine
with geodesic distance for feature extraction. Referring to the ISOMAP algorithm, the
paper [9] proposes a maximum simplex volume decomposition algorithm based on
geodesic distance. Although the intrinsic low-dimensional structure embedded in highdimensional data can be obtained by using ISOMAP, these algorithms only consider
the spectral characteristics of hyperspectral images and ignore their spatial information.
Document [9] put forward an improved N-FINDR algorithm based on manifold
learning and spatial information, which adds spatial information to the nonlinear
dimensionality reduction data of local tangent space alignment (LTSA), but only the
spectral information of hyperspectral images is considered in the dimensionality
reduction process. In this paper, the LTSA algorithm is improved by using Euclidean
distance of hyperspectral images, which combines the spatial information of hyperspectral image. The dimension of hyperspectral image is reduced nonlinearly, and the
dimension-reduced data is extracted by referring to N-FINDR algorithm. The proposed
algorithm takes full account of the nonlinear structure of hyperspectral images and the
spatial information of images and improves the accuracy of endmember extraction.
2 Local Tangent Space Arrangement Algorithm
Local tangent space arrangement algorithm (LTSA) [10] is a classical manifold
learning algorithm proposed by Zhenyue Zhang et al. For a given data set
x 1 ; x 2 ; . . .; x N
½
, x i 2 R
m , m dimension, N points, the LTSA algorithm can obtain its
embedded low-dimensional manifold y i 2 R
d
ðd\mÞ.
The specific algorithm of LTSA is as follows:
(1) Construct neighborhoods for each point x i . In the original high-dimensional input
space, the nearest k neighbor to each point is found by Euclidean distance, so knearest neighbor of x i is set as X i ¼ x i1 ; x i2 ; Á Á Á ; x ik
f
g(including x i ).
(2) Obtain local tangent space coordinates. The local tangent space coordinates H i ¼
h
ðiÞ
1 ; . . .; h
ðiÞ
k
h
i
of x i are calculated by Formula (1).
min
X k
j¼1
x ij À x þ Q i h
ðiÞ
j
2
F
¼ min X i À x1
T
k þ QH
À
Á
2
F
ð1Þ
Among them x i ¼
1
k
P k
j¼1 x ij , Q i is the matrix composed of the left singular vectors
of the largest d singular values of the centralized matrix
X i À x i 1
T
k ¼ x i1 À x i ; . . .; x ik À x i
½
. The optimal solution of the formula is
H i ¼ Q
T
i X i À x i 1
T
k
À
Á
.
(3) Obtain global low-dimensional embedded coordinates. Let Y i be the global coordinates of the corresponding data points in X i . Firstly, the alignment technique is
616
X. Tang and S. Liu
angle and spectral information divergence to replace Euclidean distance and combine
with geodesic distance for feature extraction. Referring to the ISOMAP algorithm, the
paper [9] proposes a maximum simplex volume decomposition algorithm based on
geodesic distance. Although the intrinsic low-dimensional structure embedded in highdimensional data can be obtained by using ISOMAP, these algorithms only consider
the spectral characteristics of hyperspectral images and ignore their spatial information.
Document [9] put forward an improved N-FINDR algorithm based on manifold
learning and spatial information, which adds spatial information to the nonlinear
dimensionality reduction data of local tangent space alignment (LTSA), but only the
spectral information of hyperspectral images is considered in the dimensionality
reduction process. In this paper, the LTSA algorithm is improved by using Euclidean
distance of hyperspectral images, which combines the spatial information of hyperspectral image. The dimension of hyperspectral image is reduced nonlinearly, and the
dimension-reduced data is extracted by referring to N-FINDR algorithm. The proposed
algorithm takes full account of the nonlinear structure of hyperspectral images and the
spatial information of images and improves the accuracy of endmember extraction.
2 Local Tangent Space Arrangement Algorithm
Local tangent space arrangement algorithm (LTSA) [10] is a classical manifold
learning algorithm proposed by Zhenyue Zhang et al. For a given data set
x 1 ; x 2 ; . . .; x N
½
, x i 2 R
m , m dimension, N points, the LTSA algorithm can obtain its
embedded low-dimensional manifold y i 2 R
d
ðd\mÞ.
The specific algorithm of LTSA is as follows:
(1) Construct neighborhoods for each point x i . In the original high-dimensional input
space, the nearest k neighbor to each point is found by Euclidean distance, so knearest neighbor of x i is set as X i ¼ x i1 ; x i2 ; Á Á Á ; x ik
f
g(including x i ).
(2) Obtain local tangent space coordinates. The local tangent space coordinates H i ¼
h
ðiÞ
1 ; . . .; h
ðiÞ
k
h
i
of x i are calculated by Formula (1).
min
X k
j¼1
x ij À x þ Q i h
ðiÞ
j
2
F
¼ min X i À x1
T
k þ QH
À
Á
2
F
ð1Þ
Among them x i ¼
1
k
P k
j¼1 x ij , Q i is the matrix composed of the left singular vectors
of the largest d singular values of the centralized matrix
X i À x i 1
T
k ¼ x i1 À x i ; . . .; x ik À x i
½
. The optimal solution of the formula is
H i ¼ Q
T
i X i À x i 1
T
k
À
Á
.
(3) Obtain global low-dimensional embedded coordinates. Let Y i be the global coordinates of the corresponding data points in X i . Firstly, the alignment technique is
616
X. Tang and S. Liu