J. Eur. Opt. Society-Rapid Publ. 21, 31( 2025) 5
g ¼ SSIMðz 1; XÞ ¼ 2 �
^Xz 1 þ b 1 2cov X ~; z 1 þ b2
^X 2 þ z 2
1 þ b 1 var � X~:
þ varðz 1 Þþb 2
SSIM can evaluate the brightness, contrast, and structural P differences between two images. Where X~ P ¼ 1 xi, p
^X ¼ 1 xij, x np ij, isthevalueofthejth pixel in the ith
band. b 1 and b 2 are designed to prevent instability when variables approach zero in extreme situations.
Before determining the transformation matrix A, itis necessary to obtain an estimate of a T 1 XXT a 1 for subsequent processing. a T 1 XXT a 1 is equivalent to the signal value defined in this article, so it should be as large as possible. The following equation should be solved:
max a T 1 XXT a 1
s: t: a T 1 C Xa 1 ¼ 1: ð4Þ
Using the Lagrange multiplier method, we obtain: XX T a 1 ¼ cC X a 1: ð3Þ
ð5Þ
The above equation is for solving a generalized eigenvalue problem. Solved to obtain:
a T 1 XXT a 1 ¼ c ¼ 1 p
X ci: ð6Þ
Continue to determine the transformation matrix A. The signal-to-noise ratio f of z should be small, while g should be large. Extreme situations rarely occur in hyperspectral images that cause instability of variables in equation( 3). To simplify the problem, b 1 = b 2 = 0 is set, and the problem is described as:
min 1 f � g: ð7Þ s: t: a T 1 C Xa 1 ¼ 1
Where C X 2 R pp is the covariance matrix of X T. This is still a conditional extremum problem. Let:
h ¼ 1 f � g � k � aT 1 C Xa 1 � 1 ð8Þ
The above equation can be further simplified: h ¼ n c aT 1 C 4l �
Na 1 � ðl 2 þ c � 1Þðr 2 þ 1Þ aT �
1
X a
T
1 v
� � k a T 1 C
Xa 1 � 1: ð9Þ P
Where l ¼ ^X ¼ 1 xij, r 2 ¼ var � X~ ¼ 1 sum XX T; all np np, 2
X ¼ ½ x 1; x 2;... x p Š T 2 R p, v ¼ 1 sum np XXT; 2 2 R p and
C N 2 R pp is the covariance matrix of N. By taking a partial derivative of h, weobtain:
@ h
¼ 2n @ a 1 c C Na 1
4l
� Xv T þ vX ðl 2 þ c � 1Þðr 2 þ 1Þ
T a 1 � 2kC X a 1 ¼ 0: ð10Þ
Simplify to obtain: n c C 2l
N � Xv T þ v X T a ðl 2 þ c � 1Þðr 2 1 ¼ kC X a 1: þ 1Þ ð11Þ
Solve its generalized eigenvectors to obtain the transformation matrix A and the transformed image Z = AX.
2.2 Ingredient selection strategy
The components in Z are sorted in descending order of signal-to-noise ratio and spatial structural similarity. To quickly remove low signal-to-noise ratio components and improve computational efficiency, the top 10 % of components in Z are selected as the component groups for subsequent processing. The composition groups obtained are as follows:
Z l ¼ ½ z 1; z 2;...; z l Š T 2 R ln; l ¼ ½ p = 10Š: ð12Þ
Representing the original hyperspectral image as X ¼ ½ x 1; x 2;...; x l Š T 2 R np, wherex l 2 R p represents the spectral curve where the pixel is located. The filtered component group is represented by Z l ¼ ½ z 1; z 2;...; z l Š T 2
R nl; and z i 2 R l is the curve represented by pixels. The similarity between Z k l
2 R kn and X in the feature space is obtained by examining the composition of the first k components Z k l from Zl using the following equation:
|
k ¼ h k þ D k þ M k
;
3
|
ð13Þ |
|
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X n
h k ¼ arccos x
il l jx i jjl l j � arccos zk i l k 2
i
; j jjl k i j
|
ð14Þ |
i¼1
D k ¼ z k i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X n
i¼1 ðjx i � l l j�jz k i � l k 2jÞ 2; ð15Þ
M k ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X n h i 2 ðx i � l l Þ T C 1 ðx i � l l Þ�ðz k i � l k 2Þ T C k 2ð zk i � l k 2Þ: i¼1
Where l l ¼ 1 n
P n i¼1 x i, l k 2 ¼ 1 n
P n i¼1 ð16Þ z k i, C l 2 R pp is the covariance matrix of X, C k 2 2 Rkk is the covariance matrix of
Z k l. Calculate the feature similarity D k between Z k l( k = 1,2,3,..., l) and X in sequence, and obtain the value of t when D k reaches its minimum value, that is
t ¼ argmin k: ð17Þ
At this point, the position of Z t 2 R tn in the feature space is closest to the original hyperspectral image X.