We suggest a set of complex differential operators that can be used to produce and filter dense orientation (tensor) fields for feature extraction, matching, and pattern recognition. We present results on the invariance properties of these operators, that we call symmetry derivatives. These show that, in contrast to ordinary derivatives, all orders of symmetry derivatives of Gaussians yield a remarkable invariance: They are obtained by replacing the original differential polynomial with the same polynomial, but using ordinary coordinates x and y corresponding to partial derivatives. Moreover, the symmetry derivatives of Gaussians are closed under the convolution operator and they are invariant to the Fourier transform. The equivalent of the structure tensor, representing and extracting orientations of curve patterns, had previously been shown to hold in harmonic coordinates in a nearly identical manner. As a result, positions, orientations, and certainties of intricate patterns, e.g., spirals, crosses, parabolic shapes, can be modeled by use of symmetry derivatives of Gaussians with greater analytical precision as well as computational efficiency. Since Gaussians and their derivatives are utilized extensively in image processing, the revealed properties have practical consequences for local orientation based feature extraction. The usefulness of these results is demonstrated by two applications: 1) tracking cross markers in long image sequences from vehicle crash tests and 2) alignment of noisy fingerprints.

This paper is concerned with establishing broadly-based system-theoretic foundations and practical techniques for the problem of system identification that are rigorous, intuitively clear and conceptually powerful. A general formulation is first given in which two order relations are postulated on a class of models: a constant one of complexity; and a variable one of approximation induced by an observed behaviour. An admissible model is such that any less complex model is a worse approximation. The general problem of identification is that of finding the admissible subspace of models induced by a given behaviour. It is proved under very general assumptions that, if deterministic models are required then nearly all behaviours require models of nearly maximum complexity. A general theory of approximation between models and behaviour is then developed based on subjective probability concepts and semantic information theory The role of structural constraints such as causality, locality, finite memory, etc., are then discussed as rules of the game. These concepts and results are applied to the specific problem or stochastic automaton, or grammar, inference. Computational results are given to demonstrate that the theory is complete and fully operational. Finally the formulation of identification proposed in this paper is analysed in terms of Klir’s epistemological hierarchy and both are discussed in terms of the rich philosophical literature on the acquisition of knowledge. 1

The Circle Hough Transform (CHT) has become a common method for circle detection in numerous image processing applications. Various modifications to the basic CHT operation have been suggested which include: the inclusion of edge orientation, simultaneous consideration of a range of circle radii, use of a complex accumulator array with the phase proportional to the log of radius, and the implementation of the CHT as filter operations. However, there has also been much work recently on the definition and use of invariance filters for object detection including circles. The contribution of the work presented here is to show that a specific combination of modifications to the CHT is formally equivalent to applying a scale invariant kernel operator. This work brings together these two themes in image processing which have herewith been quite separate. Performance results for applying various forms of CHT filters incorporating some or all of the available modifications, along with results from the invariance kernel, are included. These are in terms of an analysis of the peak width in the output detection array (with and without the presence of noise), and also an analysis of the peak position in terms of increasing noise levels. The results support the equivalence between the specific form of the CHT developed in this work and the invariance kernel. # 1999 Elsevier Science B.V. All rights reserved.

This thesis presents a signal representation in terms of operators. The signal is assumed to be an element of a vector space and subject to transformations of operators. The operators form continuous groups, so-called Lie groups. The representation can be used for signals in general, in particular if spatial relations are undefined, and it does not require a basis of the signal space to be useful. Special attention is given to orthogonal operator groups which are generated by antiHermitian operators by means of the exponential mapping. It is shown that the eigensystem of the group generator is strongly related to properties of the corresponding operator group. For one-parameter orthogonal operator groups, a phase concept is introduced. This phase can for instance be used to distinguish between spatially even and odd signals and, therefore, corresponds to the usual phase for multi-dimensional signals. Given one operator group that represents the variation of the signal and one operator ...

We are surrounded by surfaces that we perceive by visual means. Understanding the basic principles behind this perceptual process is a central theme in visual psychology, psychophysics and computational vision. In many of the computational models employed in the past, it has been assumed that a metric representation of physical space can be derived by visual means. Psychophysical experiments, as well as computational considerations, can convince us that the perception of space and shape has a much more complicated nature, and that only a distorted version of actual, physical space can be computed. This paper develops a computational geometric model that explains why such distortion might take place. The basic idea is that, both in stereo and motion, we perceive the world from multiple views. Given the rigid transformation between the views and the properties of the image correspondence, the depth of the scene can be obtained. Even a slight error in the rigid transformation parameters c...

this paper, a new invariant feature of two-dimensional contours is reported:

There is evidence that over a series of cortical processing stages, the visual system of primates produces a representation of objects which shows invariance with respect to, for example, translation, size, and view, as shown by recordings from single neurons in the temporal lobe (Rolls, 1992; Desimone, 1991; Tanaka et al., 1991). To clarify how such a system might learn to recognise `naturally' transformed objects, I investigate a model of cortical visual processing which incorporates a number of features of the primate visual system. The model consists of a series of layers with convergence from a limited region of the preceding layer, and mutual inhibition over a short range within a layer. The feed-forward connections provide the inputs to competitive networks, each utilising a modified Hebb-like learning rule which incorporates a temporal trace of the preceding neuronal activity. The modified Hebb-rule, called simply the trace learning rule, is aimed at enabling neurons to learn t...

v Declaration .................................... ix Acknowledgements................................ xi 1Prelude 1 TheVeryIdeaofRepresentation......................... 2 TypesofSimilarity ................................ 8 IsSimilarityIndeterminate? ........................... 11 TheRoleofSimilarityinCognition....................... 11 Summary&GeneralDiscussion......................... 14 2 Theories of Similarity 17 SimilarityDataSets................................ 17 SpatialRepresentation .............................. 21 FeaturalRepresentation.............................. 31 TreeRepresentation................................ 40 NetworkRepresentation ............................. 47 Alignment-BasedSimilarityModels....................... 48 TransformationalSimilarityModels ....................... 50 Summary&GeneralDiscussion......................... 54 i 3 On Representational Complexity 55 ApproachestoModelSelection ......................... 57 ChoosinganAdditiveClusteringRepresentation ................ 67 ChoosinganAdditiveTreeRepresentation ................... 82 ChoosingaSpatialRepresentation........................ 94 Summary&GeneralDiscussion......................... 95 4 Featural Representation 97 AMenagerieofFeaturalModels......................... 98 ClusteringModels.................................104 GeometricComplexityCriteria..........................106 AlgorithmsforFittingFeaturalModels .....................107 MonteCarloStudyI:DotheAlgorithmsWork? ................109 RepresentationsofKinshipTerms ........................117 MonteCarloStudyII:Complexity........................122 ExperimentI:Faces................................125 ExperimentII:Countries .............................1...

Humanistic computing is proposed as a new signal processing framework in which the processing apparatus is inextricably intertwined with the natural capabilities of our human body and mind. Rather than trying to emulate human intelligence, humanistic computing recognizes that the human brain is perhaps the best neural network of its kind, and that there are many new signal processing applications (within the domain of personal technologies) that can make use of this excellent but often overlooked processor. The emphasis of this paper is on personal imaging applications of humanistic computing, to take a first step toward an intelligent wearable camera system that can allow us to effortlessly capture our day-to-day experiences, help us remember and see better, provide us with personal safety through crime reduction, and facilitate new forms of communication through collective connected humanistic computing. The author’s wearable signal processing hardware, which began as a cumbersome backpackbased photographic apparatus of the 1970’s and evolved into a clothing-based apparatus in the early 1980’s, currently provides the computational power of a UNIX workstation concealed within ordinary-looking eyeglasses and clothing. Thus it may be worn continuously during all facets of ordinary day-to-day living, so that, through long-term adaptation, it begins to function as a true extension of the mind and body.

Most people would agree that the shape of an object is one of its most perceptually important attributes, and some researchers have argued that it is the primary attribute by which observers are able to recognize objects �e.g., Biederman, 1987). Given the ubiquity of this common intuition, it is somewhat puzzling to note that the concept of ``shape' ' has no formal mathematical definition that can adequately characterize its intended meaning when used colloquially. For example, almost everyone would concur that a big sphere and a small sphere both have the same shape, yet by most of the standard measures used in geometry they are quite different. The abstract nature of the concept of shape is perhaps best revealed by the perceptual classification of biological forms �e.g., see Thompson, 1942). Consider, for example, the ability of normal individuals to identify their friends and loved ones under a variety of different viewing conditions. We are able to identify people from different vantage points, and with different facial expressions, hairstyles, make-up, or clothing accessories, such as hats or jewellery. We