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

Shape Recognition using Partitioned Iterated Function Systems

One of approaches in pattern recognition is the use of fractal geometry. The property of the self-similarity of the fractals has been used as feature in several pattern recognition methods. In this paper we present a new fractal recognition method which we will use in recognition of 2D shapes. As fractal features we used Partitioned Iterated Function System (PIFS). From the PIFS code we extract mappings vectors and numbers of domain transformations used in fractal image compression. These vectors and numbers are later used as features in the recognition procedure using a normalized similarity measure. The effectiveness of our method is shown on two test databases. The first database was created by the author and the second one is MPEG7 CE-Shape-1PartB database

Partitioned Iterated Function Systems with Division and a Fractal Dependence Graph in Recognition of 2D Shapes

One of the approaches in pattern recognition is the use of fractal geometry. The property of self-similarity of fractals has been used as a feature in several pattern recognition methods. All fractal recognition methods use global analysis of the shape. In this paper we present some drawbacks of these methods and propose fractal local analysis using partitioned iterated function systems with division. Moreover, we introduce a new fractal recognition method based on a dependence graph obtained from the partitioned iterated function system. The proposed method uses local analysis of the shape, which improves the recognition rate. The effectiveness of our method is shown on two test databases. The first one was created by the authors and the second one is the MPEG7 CE-Shape-1 PartB database. The obtained results show that the proposed methodology has led to a significant improvement in the recognition rate

Fractals: objects to better explain our world

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2017, Núria Fagella Rabionet[en] In this paper we will look at some properties fractals and show the usefulness of one of them, the fractal dimension, for the study of natural and artificial phenomena. We will center our attention on fractals generated by Iterated Function Sets (IFS), which we will define as a system of contractive mappings on non-empty compact sets in a complete metric space. First, we will present some simple, well known, fractals, and show how to generate them with a geometrical construction. To compute these objects, we will use two distinct algorithms based on the iteration of IFS, a deterministic and a random one. We will then see an application of the fixed point theorem for IFS, named the Collage Theorem. We will show that an IFS’s attractor is unique and independent of the initial set. Moreover, we will show that both the deterministic and the random algorithms converge to the same limit: the attractor of the system. Having studied fractals generated by IFS, we will go on to look at fractal dimensions. For this purpose we will review the classic concept of dimensions, and broaden it to include non-integer dimension. We will see different types of fractal dimensions, some of which are suited to a specific type of fractals, such as the self-similarity dimension applicable to self-similar shapes, and a more general dimension, applicable to any fractal, namely, the Hausdorff-Besicovitch Dimension. We will also see the Box-counting algorithm, which approximates the Hausdorff-Besicovitch Dimension and is often used in its stead because of the complexity of calculating the dimension. We will conclude our exploration of fractal dimensions with the presentation of a small personal contribution to this area – our own version of a program which implements the box-counting algorithm on images. In the last chapter we will see some examples of the practical uses of the fractal dimension in fields of study as diverse as medicine, market research, image classification and so on. Through these examples we can appreciate the impact of fractals on our way of modeling or explaining our world

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

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

Improved Fractal Image Compression: Centered BFT with Quadtrees

Computer Scienc

Parallel implementation of fractal image compression

Thesis (M.Sc.Eng.)-University of Natal, Durban, 2000.Fractal image compression exploits the piecewise self-similarity present in real images as a form of information redundancy that can be eliminated to achieve compression. This theory based on Partitioned Iterated Function Systems is presented. As an alternative to the established JPEG, it provides a similar compression-ratio to fidelity trade-off. Fractal techniques promise faster decoding and potentially higher fidelity, but the computationally intensive compression process has prevented commercial acceptance. This thesis presents an algorithm mapping the problem onto a parallel processor architecture, with the goal of reducing the encoding time. The experimental work involved implementation of this approach on the Texas Instruments TMS320C80 parallel processor system. Results indicate that the fractal compression process is unusually well suited to parallelism with speed gains approximately linearly related to the number of processors used. Parallel processing issues such as coherency, management and interfacing are discussed. The code designed incorporates pipelining and parallelism on all conceptual and practical levels ensuring that all resources are fully utilised, achieving close to optimal efficiency. The computational intensity was reduced by several means, including conventional classification of image sub-blocks by content with comparisons across class boundaries prohibited. A faster approach adopted was to perform estimate comparisons between blocks based on pixel value variance, identifying candidates for more time-consuming, accurate RMS inter-block comparisons. These techniques, combined with the parallelism, allow compression of 512x512 pixel x 8 bit images in under 20 seconds, while maintaining a 30dB PSNR. This is up to an order of magnitude faster than reported for conventional sequential processor implementations. Fractal based compression of colour images and video sequences is also considered. The work confirms the potential of fractal compression techniques, and demonstrates that a parallel implementation is appropriate for addressing the compression time problem. The processor system used in these investigations is faster than currently available PC platforms, but the relevance lies in the anticipation that future generations of affordable processors will exceed its performance. The advantages of fractal image compression may then be accessible to the average computer user, leading to commercial acceptance

Multiscale Methods in Image Modelling and Image Processing

The field of modelling and processing of 'images' has fairly recently become important, even crucial, to areas of science, medicine, and engineering. The inevitable explosion of imaging modalities and approaches stemming from this fact has become a rich source of mathematical applications. 'Imaging' is quite broad, and suffers somewhat from this broadness. The general question of 'what is an image?' or perhaps 'what is a natural image?' turns out to be difficult to address. To make real headway one may need to strongly constrain the class of images being considered, as will be done in part of this thesis. On the other hand there are general principles that can guide research in many areas. One such principle considered is the assertion that (classes of) images have multiscale relationships, whether at a pixel level, between features, or other variants. There are both practical (in terms of computational complexity) and more philosophical reasons (mimicking the human visual system, for example) that suggest looking at such methods. Looking at scaling relationships may also have the advantage of opening a problem up to many mathematical tools. This thesis will detail two investigations into multiscale relationships, in quite different areas. One will involve Iterated Function Systems (IFS), and the other a stochastic approach to reconstruction of binary images (binary phase descriptions of porous media). The use of IFS in this context, which has often been called 'fractal image coding', has been primarily viewed as an image compression technique. We will re-visit this approach, proposing it as a more general tool. Some study of the implications of that idea will be presented, along with applications inferred by the results. In the area of reconstruction of binary porous media, a novel, multiscale, hierarchical annealing approach is proposed and investigated