Self-Organizing Maps and Their Applications in Image Processing, Information Organization, and Retrieval - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 405

0.5

f1(x)
f2(x)
T_opt
T_mid

0.4

0.3

0.2

0.1

0

4

-2 0 2 6

FIGURE 11.4
One-dimensional Gaussian mixture model.

the sample set distribution has the form

K

f ( x ) =

1 i f i (

x ). (11.48)

i =

Nearest mean reclassification algorithms, such as the K -means, may have a
serious shortcoming, particularly when a mixture distribution of the form of
Equation 11.48 consists of several overlapping distributions. 40 In the following
we confine ourselves to a 1D Gaussian mixture to maintain simplicity, i.e.,

f ( x ) = N ( x ; m 1 , σ) + ( 1 − ) N ( x ; m 2 , σ) , (11.49)

where N ( x ; m, σ) denotes a 1D Gaussian pdf having mean m and standard

deviation . The pdf of such a Gaussian mixture is plotted in Figure 11.4.

An important goal is to decompose a mixture into several Gaussian-like dis-
tributions. However, the clustering procedures decompose the mixture by
using a properly defined threshold. For example, the nearest mean reclassi-
fication algorithm with piecewise quadratic boundary applied to Equation
11.49 employs a threshold T opt defined by 40

σ

N ( T opt ; m 1 , σ) = ( 1 − ) N ( T opt ; m 2 , σ). (11.50)

As a result, the distribution of class 1 includes the tail of the distribution
N ( σ) N ( σ)

. Ac-
cordingly, the estimated mean values from the “truncated” distributions could

x ; m 2 , and does not include the tail of the distribution x ; m 1 ,