A Posteriori Quantization of Progressive Matching Pursuit Streams

This paper proposes a rate-distortion optimal a posteriori quantization scheme for Matching Pursuit coefficients. The a posteriori quantization applies to a Matching Pursuit expansion that has been generated off-line, and cannot benefit of any feedback loop to the encoder in order to compensate for the quantization noise. The redundancy of the Matching Pursuit dictionary provides an indicator of the relative importance of coefficients and atom indices, and subsequently on the quantization error. It is used to define a universal upper-bound on the decay of the coefficients, sorted in decreasing order of magnitude. A new quantization scheme is then derived, where this bound is used as an Oracle for the design of an optimal a posteriori quantizer. The latter turns the exponentially distributed coefficient entropy-constrained quantization problem into a simple uniform quantization problem. Using simulations with random dictionaries, we show that the proposed exponentially upper-bounded quantization (EUQ) clearly outperforms classical schemes. Stepping on the ideal Oracle-based approach, a sub-optimal adaptive scheme is then designed that approximates the EUQ but still outperforms competing quantization methods in terms of rate-distortion characteristics. Finally, the proposed quantization method is studied in the context of image coding. It performs similarly to state-of-the-art coding methods (and even better at low rates), while interestingly providing a progressive stream, very easy to transcode and adapt to changing rate constraints

Image coding using redundant dictionaries

This chapter discusses the problem of coding images using very redundant libraries of waveforms, also referred to as dictionaries. We start with a discussion of the shortcomings of classical approaches based on orthonormal bases. More specifically, we show why these redundant dictionaries provide an interesting alternative for image representation. We then introduce a special dictionary of 2-D primitives called anisotropic refinement atoms that are well suited for representing edge dominated images. Using a simple greedy algorithm, we design an image coder that performs very well at low bit rate. We finally discuss its performance and particular features such as geometric adaptativity and rate scalability

An improved decoding scheme for Matching Pursuit Streams

This work presents an improved coefficient decoding method for Matching Pursuit streams. It builds on the adaptive a posteriori quantization of coefficients, and implements an interpolation scheme that enhances the inverse quantization performance at the decoder. A class of interpolation functions is introduced, that capture the behavior of coefficients after conditional scalar quantization. The accuracy of the interpolation scheme is verified experimentally, and the novel decoding algorithm is further evaluated in image coding applications. It can be seen that the proposed method improves the rate-distortion performance by up to 0.5 dB, only by changing the reconstruction strategy at the decoder

Applications of sparse approximation in communications

Sparse approximation problems abound in many scientific, mathematical, and engineering applications. These problems are defined by two competing notions: we approximate a signal vector as a linear combination of elementary atoms and we require that the approximation be both as accurate and as concise as possible. We introduce two natural and direct applications of these problems and algorithmic solutions in communications. We do so by constructing enhanced codebooks from base codebooks. We show that we can decode these enhanced codebooks in the presence of Gaussian noise. For MIMO wireless communication channels, we construct simultaneous sparse approximation problems and demonstrate that our algorithms can both decode the transmitted signals and estimate the channel parameters

Unequal Error Protection of Atomic Image Streams

This paper presents an unequal error protection scheme for atomic image bitstreams. An atomic stream is the encoded version of a digital image, which is represented as a sum of bi-dimensional functions, as typically generated by Matching Pursuit encoders. The atomic structure of the compressed image presents an enormous advantage in terms of flexibility, since any atom of the stream can receive a different treatment, like a finely adapted protection against error. We take benefit from this property to propose a joint source and channel coding algorithm, that finely adapts the channel rate to the relative importance of the bitstream components. A fast search algorithm determines the distortion-optimal rate allocation for given bit budget and channel loss parameters. We further extend the algorithm to differentiated protection of region of interests. Simulation results show that the unequal error protection is quite efficient, even in very adverse conditions, and it clearly outperforms simple FEC schemes

Investigation of Redundant Dictionaries for Distributed Source Coding

In this project, we investigate the possibility of using the Matching Pursuit algorithm to generate image representations of a pair of correlated images for distributed source coding. We propose to use constrained dictionaries by appropriately selecting neighbouring atoms to increase the correlation between parameters, and with this, enable the application of the distributed framework

On the joint source and channel coding of atomic image streams

This paper presents an error resilient coding scheme for atomic image bitstreams, as generated by Matching Pursuit encoders. A joint source and channel coding algorithm is proposed, that takes benefit of both the flexibility in the image representation, and the progressive nature of the bitstream, in order to finely adapt the channel rate to the relative importance of the bitstream components. An optimization problem is proposed, and a fast search algorithm determines the best rate allocation for given bit budget and loss process parameters. Simulation results show that the unequal error protection is quite efficient, even in very adverse conditions, and it clearly outperforms simple FEC schemes

High flexibility scalable image coding

This paper presents a new, highly flexible, scalable image coder based on a Matching Pursuit expansion. The dictionary of atoms is built by translation, rotation and anisotropic refinement of gaussian functions, in order to efficiently capture edges in natural images. In the same time, the dictionary is invariant under isotropic scaling, which interestingly leads to very simple spatial resizing operations. It is shown that the proposed scheme compares to state-of-the-art coders when the compressed image is transcoded to a lower (octave-based) spatial resolution. In contrary to common compression formats, our bit-stream can moreover easily and efficiently be decoded at any spatial resolution, even with irrational re-scaling factors. In the same time, the Matching Pursuit algorithm provides an intrinsically progressive stream. This worthy feature allows for easy rate filtering operations, where the least important atoms are simply discarded to fit restrictive bandwidth constraints. Our scheme is finally shown to favorably compare to state-of-the-art progressive coders for moderate to quite important rate reductions

Sparse image approximation with application to flexible image coding

Natural images are often modeled through piecewise-smooth regions. Region edges, which correspond to the contours of the objects, become, in this model, the main information of the signal. Contours have the property of being smooth functions along the direction of the edge, and irregularities on the perpendicular direction. Modeling edges with the minimum possible number of terms is of key importance for numerous applications, such as image coding, segmentation or denoising. Standard separable basis fail to provide sparse enough representation of contours, due to the fact that this kind of basis do not see the regularity of edges. In order to be able to detect this regularity, a new method based on (possibly redundant) sets of basis functions able to capture the geometry of images is needed. This thesis presents, in a first stage, a study about the features that basis functions should have in order to provide sparse representations of a piecewise-smooth image. This study emphasizes the need for edge-adapted basis functions, capable to accurately capture local orientation and anisotropic scaling of image structures. The need of different anisotropy degrees and orientations in the basis function set leads to the use of redundant dictionaries. However, redundant dictionaries have the inconvenience of giving no unique sparse image decompositions, and from all the possible decompositions of a signal in a redundant dictionary, just the sparsest is needed. There are several algorithms that allow to find sparse decompositions over redundant dictionaries, but most of these algorithms do not always guarantee that the optimal approximation has been recovered. To cope with this problem, a mathematical study about the properties of sparse approximations is performed. From this, a test to check whether a given sparse approximation is the sparsest is provided. The second part of this thesis presents a novel image approximation scheme, based on the use of a redundant dictionary. This scheme allows to have a good approximation of an image with a number of terms much smaller than the dimension of the signal. This novel approximation scheme is based on a dictionary formed by a combination of anisotropically refined and rotated wavelet-like mother functions and Gaussians. An efficient Full Search Matching Pursuit algorithm to perform the image decomposition in such a dictionary is designed. Finally, a geometric image coding scheme based on the image approximated over the anisotropic and rotated dictionary of basis functions is designed. The coding performances of this dictionary are studied. Coefficient quantization appears to be of crucial importance in the design of a Matching Pursuit based coding scheme. Thus, a quantization scheme for the MP coefficients has been designed, based on the theoretical energy upper bound of the MP algorithm and the empirical observations of the coefficient distribution and evolution. Thanks to this quantization, our image coder provides low to medium bit-rate image approximations, while it allows for on the fly resolution switching and several other affine image transformations to be performed directly in the transformed domain