This paper introduces an anisotropic decomposition structure of a
recently introduced 3-D dual-tree discrete wavelet transform (DDWT),
and explores the applications for video denoising and coding. The
3-D DDWT is an attractive video representation because it isolates
motion along different directions in separate subbands, and thus
leads to sparse video decompositions. Our previous investigation
shows that the 3-D DDWT, compared to the standard discrete wavelet
transform (DWT), complies better with the statistical models based
on sparse presumptions, and gives better visual and numerical
results when used for statistical denoising algorithms. Our research
on video compression also shows that even with 4:1 redundancy, the
3-D DDWT needs fewer coefficients to achieve the same coding quality
(in PSNR) by applying the iterative projection-based noise shaping
scheme proposed by Kingsbury.
The proposed anisotropic DDWT extends the superiority of isotropic
DDWT with more directional subbands without adding to the
redundancy. Unlike the original 3-D DDWT which applies dyadic
decomposition along all three directions and produces isotropic
frequency spacing, it has a non-uniform tiling of the frequency
space. By applying this structure, we can improve the denoising
results, and the number of significant coefficients can be reduced
further, which is beneficial for video coding.
It is well documented that the statistical distribution of wavelet coefficients for natural images is non-Gaussian and that neighboring coefficients are highly dependent. In this paper, we propose a new multivariate non-Gaussian probability model to capture the dependencies among neighboring wavelet coefficients in the same scale. The model can be expressed as K exp(-||w||) where w is a N-element vector of wavelet coefficients and ||w|| is a convex combination of l2 norms over subspaces of RN. This model includes the commonly used independent Laplacian model as a special case but it has many more degrees of freedom. Based on this model, the corresponding non-linear threshold (shrinkage) function for denoising is derived using Bayesian estimation theory. Although this function does not have a closed-form solution, a successive substitution method can be used to numerically compute it.
The problem of image denoising has received more attention than the problem of image sharpening. In the paper, we propose that wavelet-based algorithms for image denoising can be used to perform image sharpening. Consequently, a variety of new image sharpening techniques becomes available. We examine the sharpening of natural images using an algorithm for image denoising with oriented complex 2D wavelets.