Faculty Bibliography

This paper describes a new set of dyadic wavelet frames with two generators. The construction is simple, yet the wavelets cover the time-frequency plane in an arrangement that provides a higher sampling in both time and frequency. Specifically, the spectrum of the first wavelet is concentrated halfway between the spectrum of the second wavelet and the spectrum of its dilated version. In addition, the second wavelet is translated by half integers rather than whole integers in the frame construction. This arrangement leads to an expansive wavelet transform that is approximately shift invariant and has intermediate scales. The wavelet frames presented in this paper are compactly supported and have vanishing moments.

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.

K-regular two-band orthogonal filterbanks have been applied to image processing. Such filters can be extended into a case of downsampling by two and more than two filters provided that they satisfy a set of conditions. Such a setup allows for more degrees of freedom but also at the cost of higher redundancy. The latter depends directly on the number of the wavelet filters involved. Tight frame filters allow the design of smooth scaling functions and wavelets with a limited number of coefficients. Moreover, such filters are nearly shift invariant, a desirable feature in many applications. In this paper, we explore a family of symmetric tight frame finite impulse response (FIR) filters characterized by the relations H-3(z) = H-0(-z) and H-2(z) = H-1(-z). They are simple to design and exhibit a degree of near orthogonality, in addition to near shift invariance. Both properties are desirable for noise removal purposes.

This paper explores the use of a recently introduced 3-D dual-tree discrete wavelet transform (DDWT) for video coding. The 3-D DDWT is an attractive video representation because it isolates motion along different directions in separate subbands. However, it is an overcomplete transform with 8:1 or 4:1 redundancy. Based on the effectiveness of the iterative projection-based noise shaping scheme proposed by Kingsbury on reducing the number of coefficients, and our prior investigation about the correlation between subbands at the same spatial/temp oral location, both in the significance map and in actual coefficient values, a new video coding scheme using 3D DDWT is proposed. The proposed video codec does not require motion compensation and provides better performance than the 3D SPIHT codec, both objectively and subjectively, despite the fact that the raw number of coefficients resulting from the 3-D DDWT is much more than that of the conventional 3-D DWT. The proposed coder allows full scalability in spatial, temporal and quality dimensions.