Rate-Distortion Optimized Tree-Structured Compression Algorithms for Piecewise Polynomial Images

This paper presents novel coding algorithms based on tree-structured segmentation, which achieve the correct asymp- totic 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 be- havior (D(R) ~ C02^-c1 R), thus improving over classic wavelet schemes. We also prove that the computational complexity of the scheme is of 0(NlogN). 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 0(NlogN). 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

Rate-distortion optimized geometrical image processing

Since geometrical features, like edges, represent one of the most important perceptual information in an image, efficient exploitation of such geometrical information is a key ingredient of many image processing tasks, including compression, denoising and feature extraction. Therefore, the challenge for the image processing community is to design efficient geometrical schemes which can capture the intrinsic geometrical structure of natural images. This thesis focuses on developing computationally efficient tree based algorithms for attaining the optimal rate-distortion (R-D) behavior for certain simple classes of geometrical images, such as piecewise polynomial images with polynomial boundaries. A good approximation of this class allows to develop good approximation and compression schemes for images with strong geometrical features, and as experimental results show, also for real life images. We first investigate both the one dimensional (1-D) and two dimensional (2-D) piecewise polynomials signals. For the 1-D 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 and is called prune-join algorithm. This allows to achieve the correct exponentially decaying R-D behavior, D(R) ~ 2-cR, thus improving over classical wavelet schemes. We also show that the computational complexity of the scheme is of O(N logN). We then extend this scheme to the 2-D case using a quadtree, which also achieves an exponentially decaying R-D behavior, for the piecewise polynomial image model, with a low computational cost of O(N logN). Again, the key is an R-D optimized prune and join strategy. We further analyze the R-D performance of the proposed tree algorithms for piecewise smooth signals. We show that the proposed algorithms achieve the oracle like polynomially decaying asymptotic R-D behavior for both the 1-D and 2-D scenarios. Theoretical as well as numerical results show that the proposed schemes outperform wavelet based coders in the 2-D case. We then consider two interesting image processing problems, namely denoising and stereo image compression, in the framework of the tree structured segmentation. For the denoising problem, we present a tree based algorithm which performs denoising by compressing the noisy image and achieves improved visual quality by capturing geometrical features, like edges, of images more precisely compared to wavelet based schemes. We then develop a novel rate-distortion optimized disparity based coding scheme for stereo images. The main novelty of the proposed algorithm is that it performs the joint coding of disparity information and the residual image to achieve better R-D performance in comparison to standard block based stereo image coder

Graph Spectral Image Processing

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation

Tree-Structured Nonlinear Adaptive Signal Processing

In communication systems, nonlinear adaptive filtering has become increasingly popular in a variety of applications such as channel equalization, echo cancellation and speech coding. However, existing nonlinear adaptive filters such as polynomial (truncated Volterra series) filters and multilayer perceptrons suffer from a number of problems. First, although high Order polynomials can approximate complex nonlinearities, they also train very slowly. Second, there is no systematic and efficient way to select their structure. As for multilayer perceptrons, they have a very complicated structure and train extremely slowly Motivated by the success of classification and regression trees on difficult nonlinear and nonparametfic problems, we propose the idea of a tree-structured piecewise linear adaptive filter. In the proposed method each node in a tree is associated with a linear filter restricted to a polygonal domain, and this is done in such a way that each pruned subtree is associated with a piecewise linear filter. A training sequence is used to adaptively update the filter coefficients and domains at each node, and to select the best pruned subtree and the corresponding piecewise linear filter. The tree structured approach offers several advantages. First, it makes use of standard linear adaptive filtering techniques at each node to find the corresponding Conditional linear filter. Second, it allows for efficient selection of the subtree and the corresponding piecewise linear filter of appropriate complexity. Overall, the approach is computationally efficient and conceptually simple. The tree-structured piecewise linear adaptive filter bears some similarity to classification and regression trees. But it is actually quite different from a classification and regression tree. Here the terminal nodes are not just assigned a region and a class label or a regression value, but rather represent: a linear filter with restricted domain, It is also different in that classification and regression trees are determined in a batch mode offline, whereas the tree-structured adaptive filter is determined recursively in real-time. We first develop the specific structure of a tree-structured piecewise linear adaptive filter and derive a stochastic gradient-based training algorithm. We then carry out a rigorous convergence analysis of the proposed training algorithm for the tree-structured filter. Here we show the mean-square convergence of the adaptively trained tree-structured piecewise linear filter to the optimal tree-structured piecewise linear filter. Same new techniques are developed for analyzing stochastic gradient algorithms with fixed gains and (nonstandard) dependent data. Finally, numerical experiments are performed to show the computational and performance advantages of the tree-structured piecewise linear filter over linear and polynomial filters for equalization of high frequency channels with severe intersymbol interference, echo cancellation in telephone networks and predictive coding of speech signals

An Improved Image Compression Algorithm Based on Daubechies- Wavelets with Arithmetic Coding

In this paper, we present image compression techniques to utilizing the visual redundancy and investigated. To effectively define and utilize image compression context for natural image is difficult problem. Inspired by recent research in the advancements of image compression techniques, we propose Daubechies-Wavelet with arithmetic coding towards the improvement over visual quality rather than spatial wise fidelity. Image compression using Daubechies-Wavelet with arithmetic coding is quite simple and good technique of compression to produce better compression results. In this image compression technique we first apply Daubechies-Wavelet transform then 2D Walsh-Wavelet transform on each kxk where (k=2n) block of the low frequency sub band. Split all values from each transformed block kxk followed by applying arithmetic coding for image compress. Index Terms-Image Compression, Daubechies-Wavelet, Arithmetic codin