Consider an m $\backslash$times n gray image with xpq being the intensity of pixel $($p$,$ q$),$ p $=1,...,$ m

and q $=1,...,$ n$.$ We want to segment the image into k non$-$overlapping clusters by using

the K$-$means clustering method.

We first reshape the image pixels such that the resulting matrix is an m $$ n $=$ t vector. The

total variation of the reshaped image is defined as:

t$$

i$=1$

$($xi $\backslash$mu T $)2$

where $\backslash$mu T is the average intensity of the image.

Given k a priori, the K$-$means algorithm aims to minimize the total within$-$cluster vari$-$

ation, denoted as:

K$$

j$=1$

t$$

i$=1$

wij $($xi $\backslash$mu j $)2$

where $\backslash$mu j is the mean intensity of cluster Cj $,$ and wij $=$

$\{$

$1$ xi in Cj

$0$ xi $/$ in Cj

However, according to the Huygens Theorem, minimizing the within$-$cluster variation is

equivalent to maximizing the between$-$cluster variation. Show that the total between$-$cluster

variation can be defined as: K$$

j$=1$

tj $(\backslash$mu j $\backslash$mu T $)2$

where tj is the number of elements in the cluster Cj

$1$