Mathematics in Independent Component Analysis

214 Chapter 15. Neurocomputing, 69:1485-1501, 2006
Noisy Image Local PCA
Local ICA Local ICA + PCA
Fig. 3. Comparison of LPCA and LICA based denoising upon an image infested with
Gaussian noise. Also note an improvement in denoising power if both are applied
consecutively (Local PCA SNR = 8.8 dB, LICA SNR = 10.6 dB, LPCA and LICA
consecutively SNR = 12.6 dB). All images where denoised using a fixed number of
clusters K = 20 and a delay radius of M = 4, which results in a 49-dimensional
feature space.
aij → ai−1,j+1, (i, j = 1, ..., n) yields the following transformed image:
⎡
⎢
P −1,1 = ⎢
⎣
an,2 . . . an,n an,1
a1,2 . . . a1,n a1,1
.
an−1,2 . . . an−1,n an−1,1
.
.
⎤
⎥
⎦
(21)
Then instead of choosing a single delay dimension, we choose a delay radius
M and use all P ν with �ν� < M as delayed versions of the original signal.
The remainder of the LICA based denoising algorithm works exactly as in
the case of a 1D time series.
In figure 3 we show that this approach using the the MDL criterion to select
the number of components compared between LPCA and LICA. In addition
we see that the algorithm also works favorable if applied multiple times.
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