Bent beamlets - efficient tool in image coding

In recent years spectral methods, especially wavelets theory, have gained wide popularity in digital image processing because they allow for very sparse and efficient image coding. Although they are good in analyzing one dimensional signal, they cannot properly catch line discontinuities, so they are often present in two dimensional signals, that is in images. To avoid these problems the theory of geometrical wavelets has been created in recent years, which have all advantages of wavelets and moreover, allows to catch line discontinuities properly. Beamlets are those of a wide spectrum of the new theory of geometrical wavelets. They are successfully used in many areas of digital image processing, particularly in multiresolution image coding. In the paper presents the improvement of the beamlets theory, which allows to code images in a more efficient way than in the case of the classical beamlets. Also in other areas of image processing this improved theory can be successfully used. The experiments performed on a wide spectrum of test images have confirmed great usefulness of the improved - bent beamlets. In the paper the examples of isobar image coding are also presented

Time series forecasting using wavelet and support vector machine

Master'sMASTER OF ENGINEERIN

Approximation and Compression of Piecewise Smooth Images Using a Wavelet/Wedgelet Geometric Model

Conference paperInherent to photograph-like images are two types of structures: large smooth regions and geometrically smooth edge contours separating those regions. Over the past years, efficient representations and algorithms have been developed that take advantage of each of these types of structure independently: quadtree models for 2D wavelets are well-suited for uniformly smooth images (C² everywhere), while quadtree-organized <i>wedgelet</i> approximations are appropriate for purely geometrical images (containing nothing but C² contours). This paper shows how to <i>combine</i> the wavelet and wedgelet representations in order to take advantage of both types of structure simultaneously. We show that the asymptotic approximation and rate-distortion performance of a wavelet-wedgelet representation on piecewise smooth images mirrors the performance of both wavelets (for uniformly smooth images) and wedgelets (for purely geometrical images). We also discuss an efficient algorithm for fitting the wavelet-wedgelet representation to an image; the convenient quadtree structure of the combined representation enables new algorithms such as the recent WSFQ geometric image coder

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