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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
  
2. LINEAR SPECTRAL MIXTURE MODEL 
The linear spectral mixture model is a mostly simple and 
widely used approach for remotely sensed imagery to estimate 
abundance fractions of the endmembers resident in the mixed 
pixels. Suppose that L is the number of spectral bands. Let Dy 
be an Lxl column pixel vector in a multispectral or 
hyperspectral image. Let M be an Lxp endmember signature 
matrix denoted by [m, m; ... m,] where m; is an Lx1 column 
vector represented by the signature of the j-th endmember and p 
is the number of endmembers in the image scene. Let & =[ & | 
Ou 1" be a px1 abundance column vector associated 
within Dy, where Q ; denotes the fraction of the j-th signature 
present in the pixel vector Dy .A linear mixture model of Dy 
makes use of a mixing equation to model the spectral signature 
of Dy as a linear combination of m, m» ... m, with appropriate 
abundance fractions specified by & | & >... Π, as follows. 
Dr MA: (1) 
Where E is noise or can be interpreted as measurement error. 
Here, Dy will be used to represent digital numbers. In general, 
two constraints must be imposed on this model to yield an 
optimal solution. These are the abundance sum-to-one 
P 
constraint (ASC), S i G t. and. the abundance 
j= : 
nonnegativity constraint(ANC), @ ;=0 for all 1<j<p. 
3. UFCLS METHOD 
3.1 FCLS Algorithm 
FCLS algorithm is introduced firstly in order to understand 
UFCLS method, because of the latter derived from the former. 
FCLS is the abbreviation of fully constrained least squares* * 
namely, the late is the algorithm about the inversion of @ in 
the equation(1) when two constraints, ASC and ANC, must be 
imposed on it at the same time. Taking care of the ASC, we 
introduce a new signature matrix, denoted by N, and a vector S, 
defined respectively by 
óM 
N° T with Z° {11,- : 25 r (2) 
I TT 
p 
Se 
er N doa; 3) 
The utilization of Ó in (2), (3) controls the impact of the ASC. 
In this paper, the value of Ö was fixed at 1.0x107. 
If the M and Dy in (1) are replaced with N, .S respectively, the 
equation can be derived as 
S«Nao*n (4) 
[t is the solution of the equation (4) that satisfies the above ASC 
condition. In the equation (4), the values of Dy are known in a 
given image, and the values of M are supposed to be known, so 
the values of V and § are known. Consequently, solving & 
becomes to solve p unknown parameters from Le 4 equations. 
We use the least squares error as the optimal least squares 
estimate of Q' , Q ,s, for equation (4) can be obtained by 
as" NU NIS (5) 
Next, we impose ANC on model (4). Under the constraint, the 
equation (4) can not be solved analytically since ANC results in 
a set of inequalities, only an optimal solution can be obtained, 
which generates the following constrained optimization 
problem as 
Minimize LSE- (N 2 - S) (N 2 -S) st. ax >0 * 6** 
Many methods have been developed to address this problem. In 
this section, we use the FCLS method, the principles of which 
have been described by Daniel et al. (2001) and Bro et al. 
(1997), the details of implementing the FCLS algorithm are 
given below. 
1. Initialization: The components of the estimate & ;s are 
decomposed into two index sets called active set and passive set. 
While the former consists of all indices corresponding to 
negative (or zero) components in the estimate  (s, the latter 
contains all indices corresponding to positive components in the 
estimate Q 1s. Set the passive set P®={1,2,...p} and active set 
RM », k=0. 
(k) 
2. Calculate & 1s using (5). Let Upcıs* A is 
; : 2 : (k 
3. At the k-th iteration, if all components in a are 
nonnegative, the algorithm is terminated. Otherwise, continue. 
4. Let k=k+1. Move all indices in P“" that correspond to 
; : k D 
negative (or zero) components of a s to R*', and the 
resulting index sets are denoted by P® and RW, respectively. 
Create a new index set I® and set 1®% R®, 
Stet (Y nly * YX ish) a LSó) *** a LS(n) * | |'O (sug. 
a sjf 2^ XX usq are all the components of & | sin R'9. 
6. Form a steering matrix ] ^ by deleting all rows and 
columns in the matrixe A Ne * that are specified by pe. 
7. Calculate p. . DE «8! @ p(k).1f all components in 
p (9 are negative, go to step 12. Otherwise, continue. 
(. 
8. Calculate D... 
k) : ; k 
arg{max f ! and move the index in R? 
k ; 
£) to P. 
that corresponds to Di. 
9. Form another new matrix yu by deleting all columns of 
- NT IN. 9 specified by P9. 
10. Recalculate Pp © TO «8! Œ p(k) according to the 
(k ; 
changed R™, then calculate & (0° 1s° e p. 
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