Faculty Bibliography

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.

In this paper we discuss the designs of 2-channel orthogonal near symmetric wavelets on the one hand, and the symmetric near orthogonal wavelets on the other. In both types of wavelets we use the Grobner bases design approach and present examples of both cases.