Abstract-This paper presents novel coding algorithms based on tree-structured segmentation, which achieve the correct asymptotic rate-distortion (R-D) behavior for a simple class of signals, known as piecewise polynomials, by using an R-D based prune and join scheme. For the one-dimensional case, our scheme is based on binary-tree segmentation of the signal. This scheme approximates the signal segments using polynomial models and utilizes an R-D optimal bit allocation strategy among the different signal segments. The scheme further encodes similar neighbors jointly to achieve the correct exponentially decaying R-D behavior ( ( ) 0 2 ), thus improving over classic wavelet schemes. We also prove that the computational complexity of the scheme is of ( log ). We then show the extension of this scheme to the two-dimensional case using a quadtree. This quadtree-coding scheme also achieves an exponentially decaying R-D behavior, for the polygonal image model composed of a white polygon-shaped object against a uniform black background, with low computational cost of ( log ). Again, the key is an R-D optimized prune and join strategy. Finally, we conclude with numerical results, which show that the proposed quadtree-coding scheme outperforms JPEG2000 by about 1 dB for real images, like cameraman, at low rates of around 0.15 bpp.
It is well known that wavelets provide good non-linear approximation of one-dimensional (1-D) piecewise smooth functions. However, it has been shown that the use of a basis with good approximation properties does not necessarily lead to a good compression algorithm. The situation in 2-D is much more complicated since wavelets are not good for modeling piecewise smooth signals (where discontinuities are along smooth curves). The purpose of this work is to analyze the performance of compression algorithms for 2-D piecewise smooth functions directly in a rate distortion context. We consider some simple image models and compute rate distortion bounds achievable using oracle based methods. We then present a practical compression algorithm based on optimal quadtree decomposition that, in some cases, achieve the oracle performance.
We consider a new type of monotone nonexpansive mappings in an ordered Banach space X with partial order . This new class of nonlinear mappings properly contains nonexpansive, firmly-nonexpansive and Suzuki-type generalized nonexpansive mappings and partially extends α-nonexpansive mappings. We obtain some existence theorems and weak and strong convergence theorems for the Mann iteration. Some useful examples are presented to illustrate the facts.
MSC: Primary 47H10; 54H25
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