Image Denoising by Data Adaptive and Non-Data Adaptive Transform Domain Denoising Method Using EEG Signal - Proceedings of All India Seminar on Biomedical Engineering 2012 (AISOBE-2012) - page 12

In addition to the basic assumption of statistically independence, by imposing
the following fundamental restrictions, the noise free ICA model can be defined if:

1. All the independent components S i , with the possible exception of one com-
ponent, must be non-Gaussian
2. The number of observed linear mixtures m must be at least as large as the
number of independent components n, i.e., m [ p
3. The matrix A must be of full column rank.

We can invert the mixing matrix as in Eq. ( 3 ):

S ¼ A 1 X ð 3 Þ

Thus, to estimate one of the independent components, we can consider a linear
combination of X i Let us denote this by Eq. ( 4 ):

Y ¼ b T X ¼ b T AS ð 4 Þ
Hence, if b were one of the rows of A -1 , this linear combination b T X would
actually equal one of the independent components. But in practice we cannot
determine such 'b' exactly because we have no knowledge of matrix A, but we can
find an estimator that gives a good approximation. In practice there are two
different measures of non-Gaussianity.

Kurtosis

The classical measure of non-Gaussianity is kurtosis or the fourth order cumulant.
It is defined by Eq. ( 5 )

ðÞ¼ Ey 4 3E f y 2 g

2

Kurt ð 5 Þ

As the variable y is assumed to be standardized we can say in Eq. ( 6 ) as:

Kurt ðÞ¼ E fg 3 ð 6 Þ

Hence the kurtosis is simply a normalized version of the fourth moment E{y4}.
For the Gaussian case the fourth moment is equal to 3 and hence kurt (y) = 0.
Thus, for Gaussian variable kurtosis is zero but for non-Gaussian random variable
it is nonzero [ 4 , 5 ].