Cellular Automata for Efficient Image and Video Compression - Cellular Automata in Image Processing and Geometry - page 13

Fig. 1.9 Various sliding bases and their associated radial basis functions

In the above, 1 is a square matrix with all unity elements, both Y and B i , j are
matrices with the same size as the original image X and

ⓦ denotes element wise

matrix multiplication (i.e. if C a ij b ij ). Consequently the recovery

process is an iterative process given by recursive equation (1.3), with a total of at
most 2 N multiplications (using a pice-wise linear basis function, the number of
multiplications can be further reduced). The sliding basis B i , j is computed using a
radial basis kernel inspired from [20]. The detailed formulae and an example for 3
types of basis functions are given in Figure 1.9.
The sliding basis implements the equivalent of a fuzzy membership function with
a maximal value 1 on the position

= A ⓦ B then c i , j =

. Neigh-
boring pixels at distance d will correspond to the decaying amplitude of the kernel
function according to a radius parameter r to be determined. Such a radial basis
function is necessary to reconstruct the neighboring pixels that were not sampled
during the measurement process and it is expected to be in relationship with the
correlation model of the signal X .
In terms of recovery performance we estimate the PSNR (measured in dB) of
the reconstructed image Y (in this case the noise is the difference between original
and the reconstructed image) as a function of K . Such a curve may be also consid-
ered as a rate-distortion curve since the compression rate is proportional to K for a
given N .
In the following, the “Lena” image with N

( i , j ) of the current measurement ( d = 0 )

256 samples (pixels) is con-
sidered. It was found that the choice of the basis function (among the 3 types men-
tioned in Fig. 1.9) has little influence on the PSNR value. Consequently, in order to

= 256 ×