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

Centralized and distributed semi-parametric compression of piecewise smooth functions

This thesis introduces novel wavelet-based semi-parametric centralized and distributed compression methods for a class of piecewise smooth functions. Our proposed compression schemes are based on a non-conventional transform coding structure with simple independent encoders and a complex joint decoder. Current centralized state-of-the-art compression schemes are based on the conventional structure where an encoder is relatively complex and nonlinear. In addition, the setting usually allows the encoder to observe the entire source. Recently, there has been an increasing need for compression schemes where the encoder is lower in complexity and, instead, the decoder has to handle more computationally intensive tasks. Furthermore, the setup may involve multiple encoders, where each one can only partially observe the source. Such scenario is often referred to as distributed source coding. In the first part, we focus on the dual situation of the centralized compression where the encoder is linear and the decoder is nonlinear. Our analysis is centered around a class of 1-D piecewise smooth functions. We show that, by incorporating parametric estimation into the decoding procedure, it is possible to achieve the same distortion- rate performance as that of a conventional wavelet-based compression scheme. We also present a new constructive approach to parametric estimation based on the sampling results of signals with finite rate of innovation. The second part of the thesis focuses on the distributed compression scenario, where each independent encoder partially observes the 1-D piecewise smooth function. We propose a new wavelet-based distributed compression scheme that uses parametric estimation to perform joint decoding. Our distortion-rate analysis shows that it is possible for the proposed scheme to achieve that same compression performance as that of a joint encoding scheme. Lastly, we apply the proposed theoretical framework in the context of distributed image and video compression. We start by considering a simplified model of the video signal and show that we can achieve distortion-rate performance close to that of a joint encoding scheme. We then present practical compression schemes for real world signals. Our simulations confirm the improvement in performance over classical schemes, both in terms of the PSNR and the visual quality

A Review of Bandlet Methods for Geometrical Image Representation

International audienceThis article reviews bandlet approaches to geometric image repre- sentations. Orthogonal bandlets using an adaptive segmentation and a local geometric flow well suited to capture the anisotropic regularity of edge struc- tures. They are constructed with a “bandletization” which is a local orthogonal transformation applied to wavelet coeffi cients. The approximation in these bandlet bases exhibits an asymptotically optimal decay for images that are regular outside a set of regular edges. These bandlets can be used to perform image compression and noise removal. More flexible orthogonal bandlets with less vanishing moments are constructed with orthogonal grouplets that group wavelet coeffi cients alon a multiscale association field. Applying a translation invariant grouplet transform over a translation invariant wavelet frame leads to state of the art results for image denoising and super-resolution

Toward sparse and geometry adapted video approximations

Video signals are sequences of natural images, where images are often modeled as piecewise-smooth signals. Hence, video can be seen as a 3D piecewise-smooth signal made of piecewise-smooth regions that move through time. Based on the piecewise-smooth model and on related theoretical work on rate-distortion performance of wavelet and oracle based coding schemes, one can better analyze the appropriate coding strategies that adaptive video codecs need to implement in order to be efficient. Efficient video representations for coding purposes require the use of adaptive signal decompositions able to capture appropriately the structure and redundancy appearing in video signals. Adaptivity needs to be such that it allows for proper modeling of signals in order to represent these with the lowest possible coding cost. Video is a very structured signal with high geometric content. This includes temporal geometry (normally represented by motion information) as well as spatial geometry. Clearly, most of past and present strategies used to represent video signals do not exploit properly its spatial geometry. Similarly to the case of images, a very interesting approach seems to be the decomposition of video using large over-complete libraries of basis functions able to represent salient geometric features of the signal. In the framework of video, these features should model 2D geometric video components as well as their temporal evolution, forming spatio-temporal 3D geometric primitives. Through this PhD dissertation, different aspects on the use of adaptivity in video representation are studied looking toward exploiting both aspects of video: its piecewise nature and the geometry. The first part of this work studies the use of localized temporal adaptivity in subband video coding. This is done considering two transformation schemes used for video coding: 3D wavelet representations and motion compensated temporal filtering. A theoretical R-D analysis as well as empirical results demonstrate how temporal adaptivity improves coding performance of moving edges in 3D transform (without motion compensation) based video coding. Adaptivity allows, at the same time, to equally exploit redundancy in non-moving video areas. The analogy between motion compensated video and 1D piecewise-smooth signals is studied as well. This motivates the introduction of local length adaptivity within frame-adaptive motion compensated lifted wavelet decompositions. This allows an optimal rate-distortion performance when video motion trajectories are shorter than the transformation "Group Of Pictures", or when efficient motion compensation can not be ensured. After studying temporal adaptivity, the second part of this thesis is dedicated to understand the fundamentals of how can temporal and spatial geometry be jointly exploited. This work builds on some previous results that considered the representation of spatial geometry in video (but not temporal, i.e, without motion). In order to obtain flexible and efficient (sparse) signal representations, using redundant dictionaries, the use of highly non-linear decomposition algorithms, like Matching Pursuit, is required. General signal representation using these techniques is still quite unexplored. For this reason, previous to the study of video representation, some aspects of non-linear decomposition algorithms and the efficient decomposition of images using Matching Pursuits and a geometric dictionary are investigated. A part of this investigation concerns the study on the influence of using a priori models within approximation non-linear algorithms. Dictionaries with a high internal coherence have some problems to obtain optimally sparse signal representations when used with Matching Pursuits. It is proved, theoretically and empirically, that inserting in this algorithm a priori models allows to improve the capacity to obtain sparse signal approximations, mainly when coherent dictionaries are used. Another point discussed in this preliminary study, on the use of Matching Pursuits, concerns the approach used in this work for the decompositions of video frames and images. The technique proposed in this thesis improves a previous work, where authors had to recur to sub-optimal Matching Pursuit strategies (using Genetic Algorithms), given the size of the functions library. In this work the use of full search strategies is made possible, at the same time that approximation efficiency is significantly improved and computational complexity is reduced. Finally, a priori based Matching Pursuit geometric decompositions are investigated for geometric video representations. Regularity constraints are taken into account to recover the temporal evolution of spatial geometric signal components. The results obtained for coding and multi-modal (audio-visual) signal analysis, clarify many unknowns and show to be promising, encouraging to prosecute research on the subject

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

Geometric distortion measurement for shape coding: a contemporary review

Geometric distortion measurement and the associated metrics involved are integral to the rate-distortion (RD) shape coding framework, with importantly the efficacy of the metrics being strongly influenced by the underlying measurement strategy. This has been the catalyst for many different techniques with this paper presenting a comprehensive review of geometric distortion measurement, the diverse metrics applied and their impact on shape coding. The respective performance of these measuring strategies is analysed from both a RD and complexity perspective, with a recent distortion measurement technique based on arc-length-parameterisation being comparatively evaluated. Some contemporary research challenges are also investigated, including schemes to effectively quantify shape deformation