Fractal image compression and the self-affinity assumption : a stochastic signal modelling perspective

Bibliography: p. 208-225.Fractal image compression is a comparatively new technique which has gained considerable attention in the popular technical press, and more recently in the research literature. The most significant advantages claimed are high reconstruction quality at low coding rates, rapid decoding, and "resolution independence" in the sense that an encoded image may be decoded at a higher resolution than the original. While many of the claims published in the popular technical press are clearly extravagant, it appears from the rapidly growing body of published research that fractal image compression is capable of performance comparable with that of other techniques enjoying the benefit of a considerably more robust theoretical foundation. . So called because of the similarities between the form of image representation and a mechanism widely used in generating deterministic fractal images, fractal compression represents an image by the parameters of a set of affine transforms on image blocks under which the image is approximately invariant. Although the conditions imposed on these transforms may be shown to be sufficient to guarantee that an approximation of the original image can be reconstructed, there is no obvious theoretical reason to expect this to represent an efficient representation for image coding purposes. The usual analogy with vector quantisation, in which each image is considered to be represented in terms of code vectors extracted from the image itself is instructive, but transforms the fundamental problem into one of understanding why this construction results in an efficient codebook. The signal property required for such a codebook to be effective, termed "self-affinity", is poorly understood. A stochastic signal model based examination of this property is the primary contribution of this dissertation. The most significant findings (subject to some important restrictions} are that "self-affinity" is not a natural consequence of common statistical assumptions but requires particular conditions which are inadequately characterised by second order statistics, and that "natural" images are only marginally "self-affine", to the extent that fractal image compression is effective, but not more so than comparable standard vector quantisation techniques

HUMAN FACE RECOGNITION BASED ON FRACTAL IMAGE CODING

Human face recognition is an important area in the field of biometrics. It has been an active area of research for several decades, but still remains a challenging problem because of the complexity of the human face. In this thesis we describe fully automatic solutions that can locate faces and then perform identification and verification. We present a solution for face localisation using eye locations. We derive an efficient representation for the decision hyperplane of linear and nonlinear Support Vector Machines (SVMs). For this we introduce the novel concept of $\rho$ and $\eta$ prototypes. The standard formulation for the decision hyperplane is reformulated and expressed in terms of the two prototypes. Different kernels are treated separately to achieve further classification efficiency and to facilitate its adaptation to operate with the fast Fourier transform to achieve fast eye detection. Using the eye locations, we extract and normalise the face for size and in-plane rotations. Our method produces a more efficient representation of the SVM decision hyperplane than the well-known reduced set methods. As a result, our eye detection subsystem is faster and more accurate. The use of fractals and fractal image coding for object recognition has been proposed and used by others. Fractal codes have been used as features for recognition, but we need to take into account the distance between codes, and to ensure the continuity of the parameters of the code. We use a method based on fractal image coding for recognition, which we call the Fractal Neighbour Distance (FND). The FND relies on the Euclidean metric and the uniqueness of the attractor of a fractal code. An advantage of using the FND over fractal codes as features is that we do not have to worry about the uniqueness of, and distance between, codes. We only require the uniqueness of the attractor, which is already an implied property of a properly generated fractal code. Similar methods to the FND have been proposed by others, but what distinguishes our work from the rest is that we investigate the FND in greater detail and use our findings to improve the recognition rate. Our investigations reveal that the FND has some inherent invariance to translation, scale, rotation and changes to illumination. These invariances are image dependent and are affected by fractal encoding parameters. The parameters that have the greatest effect on recognition accuracy are the contrast scaling factor, luminance shift factor and the type of range block partitioning. The contrast scaling factor affect the convergence and eventual convergence rate of a fractal decoding process. We propose a novel method of controlling the convergence rate by altering the contrast scaling factor in a controlled manner, which has not been possible before. This helped us improve the recognition rate because under certain conditions better results are achievable from using a slower rate of convergence. We also investigate the effects of varying the luminance shift factor, and examine three different types of range block partitioning schemes. They are Quad-tree, HV and uniform partitioning. We performed experiments using various face datasets, and the results show that our method indeed performs better than many accepted methods such as eigenfaces. The experiments also show that the FND based classifier increases the separation between classes. The standard FND is further improved by incorporating the use of localised weights. A local search algorithm is introduced to find a best matching local feature using this locally weighted FND. The scores from a set of these locally weighted FND operations are then combined to obtain a global score, which is used as a measure of the similarity between two face images. Each local FND operation possesses the distortion invariant properties described above. Combined with the search procedure, the method has the potential to be invariant to a larger class of non-linear distortions. We also present a set of locally weighted FNDs that concentrate around the upper part of the face encompassing the eyes and nose. This design was motivated by the fact that the region around the eyes has more information for discrimination. Better performance is achieved by using different sets of weights for identification and verification. For facial verification, performance is further improved by using normalised scores and client specific thresholding. In this case, our results are competitive with current state-of-the-art methods, and in some cases outperform all those to which they were compared. For facial identification, under some conditions the weighted FND performs better than the standard FND. However, the weighted FND still has its short comings when some datasets are used, where its performance is not much better than the standard FND. To alleviate this problem we introduce a voting scheme that operates with normalised versions of the weighted FND. Although there are no improvements at lower matching ranks using this method, there are significant improvements for larger matching ranks. Our methods offer advantages over some well-accepted approaches such as eigenfaces, neural networks and those that use statistical learning theory. Some of the advantages are: new faces can be enrolled without re-training involving the whole database; faces can be removed from the database without the need for re-training; there are inherent invariances to face distortions; it is relatively simple to implement; and it is not model-based so there are no model parameters that need to be tweaked

Giving eyes to ICT!, or How does a computer recognize a cow?

Het door Schouten en andere onderzoekers op het CWI ontwikkelde systeem berust op het beschrijven van beelden met behulp van fractale meetkunde. De menselijke waarneming blijkt mede daardoor zo efficiënt omdat zij sterk werkt met gelijkenissen. Het ligt dus voor de hand het te zoeken in wiskundige methoden die dat ook doen. Schouten heeft daarom beeldcodering met behulp van 'fractals' onderzocht. Fractals zijn zelfgelijkende meetkundige figuren, opgebouwd door herhaalde transformatie (iteratie) van een eenvoudig basispatroon, dat zich daardoor op steeds kleinere schalen vertakt. Op elk niveau van detaillering lijkt een fractal op zichzelf (Droste-effect). Met fractals kan men vrij eenvoudig bedrieglijk echte natuurvoorstellingen maken. Fractale beeldcodering gaat ervan uit dat het omgekeerde ook geldt: een beeld effectief opslaan in de vorm van de basispatronen van een klein aantal fractals, samen met het voorschrift hoe het oorspronkelijke beeld daaruit te reconstrueren. Het op het CWI in samenwerking met onderzoekers uit Leuven ontwikkelde systeem is mede gebaseerd op deze methode. ISBN 906196502

Data hiding in images based on fractal modulation and diversity combining

The current work provides a new data-embedding infrastructure based on fractal modulation. The embedding problem is tackled from a communications point of view. The data to be embedded becomes the signal to be transmitted through a watermark channel. The channel could be the image itself or some manipulation of the image. The image self noise and noise due to attacks are the two sources of noise in this paradigm. At the receiver, the image self noise has to be suppressed, while noise due to the attacks may sometimes be predicted and inverted. The concepts of fractal modulation and deterministic self-similar signals are extended to 2-dimensional images. These novel techniques are used to build a deterministic bi-homogenous watermark signal that embodies the binary data to be embedded. The binary data to be embedded, is repeated and scaled with different amplitudes at each level and is used as the wavelet decomposition pyramid. The binary data is appended with special marking data, which is used during demodulation, to identify and correct unreliable or distorted blocks of wavelet coefficients. This specially constructed pyramid is inverted using the inverse discrete wavelet transform to obtain the self-similar watermark signal. In the data embedding stage, the well-established linear additive technique is used to add the watermark signal to the cover image, to generate the watermarked (stego) image. Data extraction from a potential stego image is done using diversity combining. Neither the original image nor the original binary sequence (or watermark signal) is required during the extraction. A prediction of the original image is obtained using a cross-shaped window and is used to suppress the image self noise in the potential stego image. The resulting signal is then decomposed using the discrete wavelet transform. The number of levels and the wavelet used are the same as those used in the watermark signal generation stage. A thresholding process similar to wavelet de-noising is used to identify whether a particular coefficient is reliable or not. A decision is made as to whether a block is reliable or not based on the marking data present in each block and sometimes corrections are applied to the blocks. Finally the selected blocks are combined based on the diversity combining strategy to extract the embedded binary data