A COMPARATIVE STUDY ON THE METHODS FOR ESTIMATION OF MIXING RATIO WITHIN A PIXEL 
Yasunori Terayama, 
1 Honjo, Saga-si, 
Yoko Ueda, Kohei Arai 
(Dept. Information Science, Science and Engineering Faculty, Saga University) 
Saga 840 Japan 
and Masao Matsumoto 
(Faculty of Computer Science and Systems Engineering. Kyushu Institute of Technology) 
680-4 Kawazu, Iizuka-si, 
ABSTRACT 
Fukuoka 820 Japan 
A comparative study has been made with the following three methods for estimation of mixing ratio within a 
pixel. The first one is the well known least square method by means of Generalized Inverse Matrix. The second 
the Least SQuare method minimizing the square error of 
one is Maximum LikeliHood method and 
estimated Mixing ratio. 
last one is 
It was found that the estimation accuracy of the Maximum LikeliHood method is 
superior to the other methods in the case for the simulated data with S/N ratio of up to about 28. 
KEY WORDS: Mixed Pixel, Estimation of Mixing Ratio, 
1. INTRODUCTION 
There are many classification methods of remortly 
sensed imagery data. Many of them put label in a 
pixel basis. Even for the pixels consist of plural 
categories, ordinary classification method give one 
category to the pixels. Then, it was considered that 
information of mixing ratio within a pixel was 
taken out without abandoning that information. 
In remortly sensed image, estimation of partial 
class mixing ratio within a pixel have the 
significant role for land coverage. For example, 
cloud coverage estimation based on category 
decomposition is useful for making products of sea 
surface temperature. 
Category decomposition give us the way get that 
proportion from a mixed pixel (MIXEL). 
In this paper, we picked up the three methods for 
estimation of category proportion within a pixel 
which are proposed. Comparing these estimation 
accuracies with simulation data consists of the 
mixels added nomal distributed random number. 
2. THE METHODS OF CLASS MIXING RATIO ESTIMATION 
2.1 Least square method by means of Generalized 
Inverse Matrix 
  
Let an observed vector be I with the 
dimensionality M, mixing ratio or proportion vector 
be B with the number of classes N and the matrix 
representing the spectral response of all classes 
be A. 
I=AB (1) 
I: (1.1... In)? (2) 
A= {Mo Ao. s Ma (3) 
lA d. erm Aon | 
benedés endi (dedita | 
JA con s Ann | 
B= (3.8... Bu)? (4) 
Since I is given vector, if matrix A is assumed, 
then vector B is determined under constraint that 
minimizing norm of estimation error Ei, 
| S E;? [--nmin, E- I - A B (5) 
986 
Inverse Problem Solving, Cloud Coverage. 
Bs (A A A I (6) 
Then, under constraint that the sum of mixing 
ratio has 1.0, equation (6) is solved the following 
analyticaly. 
1 =u A 71 
BA IS: ——————— (ACA UT (7) 
ut (AA) I 
At this point, u is a vector has factors with all of 
1.05 
2.2 Maximam LikeliHood method 
  
If independent variables Xi (i=1,2,...., N) follows 
normal distribution as a function of N(ui,o ;?), 
the following equation is shown X and N. 
X Sarit noi ÉD a: 0181) (8) 
ist i=1 int 
So that when the response of spectral reflectance 
is independent of each other in the classes, the 
following equation is conducted on the observed 
vector I and N. 
M 
I5 9A Bí. NCA:// B, BÀ ZB (9) 
iz 
Aim Ave Arn, SR Ain ) (10) 
7 = dias ( 0O::°, oi^... oin? ) (11) 
The probability P; is shown equation(12) when I; is 
observed in the i band and proportion vector has B. 
Then probability P that the observed vector is I 
and the proportion vector is B is shown the next 
equation(13), 
ftn C) M be pj CI. 1615 
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Sup CO) EY dN EN ad