6. Layers Image Compression and Reconstruction by Fuzzy Transforms

Recently we proved that fuzzy transforms (\(F\)-transforms) are useful in coding/decoding images, showing that the resulting peak-signal-to-noise-ratio (PSNR) is better than the one obtained using fuzzy relation equations and comparable with that obtained using the JPEG method. Recently some authors have explored a new image compression/reconstruction technique: the range interval [0,1] is partitioned in a finite number of subintervals of equal width in such a way that each subinterval corresponds to a image-layer of pixels. Each image-layer is coded using the direct \(F\)-transform, and afterwards all the inverse \(F\)-transforms are put together to reconstruct the whole initial image. We modify slightly this process: the pixels of the original image are normalized [15] with respect to the length of the gray scale, and thus are seen as a fuzzy matrix \(R\), which we divide into (possibly square) submatrices \(R_{B}\), called blocks. Hence we divide [0,1] into subintervals by adopting the quantile method, so that each subinterval contains the same number of normalized pixels of every block \(R_{B}\), then we apply the \(F\)-transforms to each block-layer. In terms of quality of the reconstructed image, our method is better than that one based on the standard \(F\)-transforms.