Abstract-We study tight wavelet frames associated with given refinable functions which are obtained with the unitary extension principles. All possible solutions of the corresponding matrix equations are found. It is proved that the problem of the extension may be always solved with two framelets. In particular, if symbols of the refinable functions are polynomials (rational functions), then the corresponding framelets with polynomial (rational) symbols can be found.
The authors present an algorithm for very low bit-rate video coding that combines new ideas in motion estimation, wavelet filter design, and wavelet-based coding techniques. A new motion compensation technique using image warping and overlapped block motion compensation is proposed to reduce temporal redundancies in a given image sequence. This combined motion model has the advantage of representing more complex motion than simple block matching schemes. To further improve the quality of the temporal prediction, an adaptive grid with variable density according to the varying motion activity of a given scene is generated. An adaptively switched high-quality texture interpolation is employed to cope with the problem of fractional displacements in such a way that both objective and subjective reconstruction quality is improved. Spatial decorrelation of the motion compensated residual images is performed using an one-parametric family of biorthogonal infinite impulse response (IIR) wavelet filters coupled with the highly efficient pre-coding scheme of 'partitioning, aggregation and conditional coding' (PACC). Experimental results demonstrate significant improvements in objective quality of 1.0-2.3 dB PSNR in comparison to the H.263+ test model TMN10 using advanced coding options. In addition, the authors' intracoding method provides a performance gain of 0.5 dB PSNR on the average for a test suite of various still images when compared to the emerging still image coding standard JPEG-2000
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