2D Discrete Mirror Transform for Image Non-Linear Approximation

Abstract

In this paper, a new 2D transform named Discrete Mirror Transform (DMT) is presented. The DMT is computed by decomposing a signal into its even and odd parts around an optimal location in a given direction so that the signal energy is maximally split between the two components. After minimizing the information required to regenerate the original signal by removing redundant structures, the process is iterated leading the signal energy to distribute into a continuously smaller set of coefficients. The DMT can be displayed as a binary tree, where each node represents the single (even or odd) signal derived from the decomposition in the previous level. An optimized version of the DMT (ODMT) is also introduced, by exploiting the possibility to choose different directions at which performing the decomposition. Experimental simulations have been carried out in order to test the sparsity properties of the DMT and ODMT when applied on images: referring to both transforms, the results show a superior performance with respect to the popular Discrete Cosine Transform (DCT) and Discrete Wavelet Transform (DWT) in terms of non-linear approximation.

The Mirror Transform

“…Variations to this process may be considered [18], e.g., by alternating horizontal and vertical mirroring decomposition steps, or by changing arbitrarily the direction of mirroring from one step to the next. With no regard to the adopted scheme and with the right attention, the concepts can be easily extended from the 1D case, so as to preserve the constant nature of the informative support from one decomposition stage to following one.…”