Parameterization of slant and slantlet/wavelet transforms with applications

Orthogonal transforms are very useful in digital signal and image processing. The slant transform is an orthogonal one having the best compaction performance among nonsinusoidal orthogonal transforms. This work introduces a class of parametric slant transforms by defining new formulas for a _N, b_N, values necessary for recursive generation of higher-order matrices of the transform. Thus, the parametric Slant-Hadamard transform can be tuned from classical Hadamard to classical slant transform and the parametric Slant-Haar transforms including the slantlet (wavelet) transform can be tuned from classical Haar or close to classical Haar to classical slant transform. Members are referred to as constant beta and multiple betas transforms. For the generalized Wiener filtering and the relative performance measure (RMP) of a transform relative to KLT it is shown that the performance of multiple betas transform with optimal selected beta parameters in the case of the first-order Markov models is close to the KLT and DCT, and in the case of generalized correlation model (GCM) its performance is better than that of the DCT and other orthogonal transforms. The parametric slant transforms are generalized to the order of arbitrary N.; An efficient fast algorithm for the construction of the slantlet transform matrices is developed and the parametric slantlet filterbank is constructed from the slantlet transform matrices instead of constructing the matrices from the slantlet filterbank. The performance of the parametric slantlet (wavelet) transforms in signal and image denoising is analyzed.; An algorithm for the direct construction of the sequential slant transforms is also introduced. Efficient algorithms for their fast implementation are presented.; A solution to the inverse KLT problem for any orthogonal transform is developed using a classical approach. Two transforms generalized Wiener filtering is introduced where one transform is used for the forward transformation and a second different transform is used for the inverse transformation. It is shown that the performance of the two transforms generalized Wiener filtering where the Haar transform is used for the forward transformation and the parametric slant transform is used for the inverse transformation is closer to KLT than the classical one transform generalized Wiener filtering.