The Representation of Image Texture

This thesis explores how to represent image texture in order to obtain information about the geometry and structure of surfaces, with particular emphasis on locating surface discontinuities. Theoretical and psychophysical results lead to the following conclusions for the representation of image texture: (1) A texture edge primitive is needed to identify texture change contours, which are formed by an abrupt change in the 2-D organization of similar items in an image. The texture edge can be used for locating discontinuities in surface structure and surface geometry and for establishing motion correspondence. (2) Abrupt changes in attributes that vary with changing surface geometry ??ientation, density, length, and width ??ould be used to identify discontinuities in surface geometry and surface structure. (3) Texture tokens are needed to separate the effects of different physical processes operating on a surface. They represent the local structure of the image texture. Their spatial variation can be used in the detection of texture discontinuities and texture gradients, and their temporal variation may be used for establishing motion correspondence. What precisely constitutes the texture tokens is unknown; it appears, however, that the intensity changes alone will not suffice, but local groupings of them may. (4) The above primitives need to be assigned rapidly over a large range in an image.

ONTIC: A Knowledge Representation System for Mathematics

Ontic is an interactive system for developing and verifying mathematics. Ontic's verification mechanism is capable of automatically finding and applying information from a library containing hundreds of mathematical facts. Starting with only the axioms of Zermelo-Fraenkel set theory, the Ontic system has been used to build a data base of definitions and lemmas leading to a proof of the Stone representation theorem for Boolean lattices. The Ontic system has been used to explore issues in knowledge representation, automated deduction, and the automatic use of large data bases.

Procedures as a Representation for Data in a Computer Program for Understanding Natural Language

This paper describes a system for the computer understanding of English. The system answers questions, executes commands, and accepts information in normal English dialog. It uses semantic information and context to understand discourse and to disambiguate sentences. It combines a complete syntactic analysis of each sentence with a "heuristic understander" which uses different kinds of information about a sentence, other parts of the discourse, and general information about the world in deciding what the sentence means. It is based on the belief that a computer cannot deal reasonably with language unless it can "understand" the subject it is discussing. The program is given a detailed model of the knowledge needed by a simple robot having only a hand and an eye. We can give it instructions to manipulate toy objects, interrogate it about the scene, and give it information it will use in deduction. In addition to knowing the properties of toy objects, the program has a simple model of its own mentality. It can remember and discuss its plans and actions as well as carry them out. It enters into a dialog with a person, responding to English sentences with actions and English replies, and asking for clarification when its heuristic programs cannot understand a sentence through use of context and physical knowledge.

Representation and Detection of Shapes in Images

We present a set of techniques that can be used to represent and detect shapes in images. Our methods revolve around a particular shape representation based on the description of objects using triangulated polygons. This representation is similar to the medial axis transform and has important properties from a computational perspective. The first problem we consider is the detection of non-rigid objects in images using deformable models. We present an efficient algorithm to solve this problem in a wide range of situations, and show examples in both natural and medical images. We also consider the problem of learning an accurate non-rigid shape model for a class of objects from examples. We show how to learn good models while constraining them to the form required by the detection algorithm. Finally, we consider the problem of low-level image segmentation and grouping. We describe a stochastic grammar that generates arbitrary triangulated polygons while capturing Gestalt principles of shape regularity. This grammar is used as a prior model over random shapes in a low level algorithm that detects objects in images.

A Note on Object Class Representation and Categorical Perception

We present a novel scheme ("Categorical Basis Functions", CBF) for object class representation in the brain and contrast it to the "Chorus of Prototypes" scheme recently proposed by Edelman. The power and flexibility of CBF is demonstrated in two examples. CBF is then applied to investigate the phenomenon of Categorical Perception, in particular the finding by Bulthoff et al. (1998) of categorization of faces by gender without corresponding Categorical Perception. Here, CBF makes predictions that can be tested in a psychophysical experiment. Finally, experiments are suggested to further test CBF.

Learning-Based Approach to Real Time Tracking and Analysis of Faces

This paper describes a trainable system capable of tracking faces and facialsfeatures like eyes and nostrils and estimating basic mouth features such as sdegrees of openness and smile in real time. In developing this system, we have addressed the twin issues of image representation and algorithms for learning. We have used the invariance properties of image representations based on Haar wavelets to robustly capture various facial features. Similarly, unlike previous approaches this system is entirely trained using examples and does not rely on a priori (hand-crafted) models of facial features based on optical flow or facial musculature. The system works in several stages that begin with face detection, followed by localization of facial features and estimation of mouth parameters. Each of these stages is formulated as a problem in supervised learning from examples. We apply the new and robust technique of support vector machines (SVM) for classification in the stage of skin segmentation, face detection and eye detection. Estimation of mouth parameters is modeled as a regression from a sparse subset of coefficients (basis functions) of an overcomplete dictionary of Haar wavelets.

Learning Linear, Sparse, Factorial Codes

In previous work (Olshausen & Field 1996), an algorithm was described for learning linear sparse codes which, when trained on natural images, produces a set of basis functions that are spatially localized, oriented, and bandpass (i.e., wavelet-like). This note shows how the algorithm may be interpreted within a maximum-likelihood framework. Several useful insights emerge from this connection: it makes explicit the relation to statistical independence (i.e., factorial coding), it shows a formal relationship to the algorithm of Bell and Sejnowski (1995), and it suggests how to adapt parameters that were previously fixed.

The Individual is Nothing, the Class Everything: Psychophysics and Modeling of Recognition in Obect Classes

Most psychophysical studies of object recognition have focussed on the recognition and representation of individual objects subjects had previously explicitely been trained on. Correspondingly, modeling studies have often employed a 'grandmother'-type representation where the objects to be recognized were represented by individual units. However, objects in the natural world are commonly members of a class containing a number of visually similar objects, such as faces, for which physiology studies have provided support for a representation based on a sparse population code, which permits generalization from the learned exemplars to novel objects of that class. In this paper, we present results from psychophysical and modeling studies intended to investigate object recognition in natural ('continuous') object classes. In two experiments, subjects were trained to perform subordinate level discrimination in a continuous object class - images of computer-rendered cars - created using a 3D morphing system. By comparing the recognition performance of trained and untrained subjects we could estimate the effects of viewpoint-specific training and infer properties of the object class-specific representation learned as a result of training. We then compared the experimental findings to simulations, building on our recently presented HMAX model of object recognition in cortex, to investigate the computational properties of a population-based object class representation as outlined above. We find experimental evidence, supported by modeling results, that training builds a viewpoint- and class-specific representation that supplements a pre-existing repre-sentation with lower shape discriminability but possibly greater viewpoint invariance.

Sparse Representations of Multiple Signals

We discuss the problem of finding sparse representations of a class of signals. We formalize the problem and prove it is NP-complete both in the case of a single signal and that of multiple ones. Next we develop a simple approximation method to the problem and we show experimental results using artificially generated signals. Furthermore,we use our approximation method to find sparse representations of classes of real signals, specifically of images of pedestrians. We discuss the relation between our formulation of the sparsity problem and the problem of finding representations of objects that are compact and appropriate for detection and classification.

Notes on PCA, Regularization, Sparsity and Support Vector Machines

We derive a new representation for a function as a linear combination of local correlation kernels at optimal sparse locations and discuss its relation to PCA, regularization, sparsity principles and Support Vector Machines. We first review previous results for the approximation of a function from discrete data (Girosi, 1998) in the context of Vapnik"s feature space and dual representation (Vapnik, 1995). We apply them to show 1) that a standard regularization functional with a stabilizer defined in terms of the correlation function induces a regression function in the span of the feature space of classical Principal Components and 2) that there exist a dual representations of the regression function in terms of a regularization network with a kernel equal to a generalized correlation function. We then describe the main observation of the paper: the dual representation in terms of the correlation function can be sparsified using the Support Vector Machines (Vapnik, 1982) technique and this operation is equivalent to sparsify a large dictionary of basis functions adapted to the task, using a variation of Basis Pursuit De-Noising (Chen, Donoho and Saunders, 1995; see also related work by Donahue and Geiger, 1994; Olshausen and Field, 1995; Lewicki and Sejnowski, 1998). In addition to extending the close relations between regularization, Support Vector Machines and sparsity, our work also illuminates and formalizes the LFA concept of Penev and Atick (1996). We discuss the relation between our results, which are about regression, and the different problem of pattern classification.

Stimulus Simplification and Object Representation: A Modeling Study

Tsunoda et al. (2001) recently studied the nature of object representation in monkey inferotemporal cortex using a combination of optical imaging and extracellular recordings. In particular, they examined IT neuron responses to complex natural objects and "simplified" versions thereof. In that study, in 42% of the cases, optical imaging revealed a decrease in the number of activation patches in IT as stimuli were "simplified". However, in 58% of the cases, "simplification" of the stimuli actually led to the appearance of additional activation patches in IT. Based on these results, the authors propose a scheme in which an object is represented by combinations of active and inactive columns coding for individual features. We examine the patterns of activation caused by the same stimuli as used by Tsunoda et al. in our model of object recognition in cortex (Riesenhuber 99). We find that object-tuned units can show a pattern of appearance and disappearance of features identical to the experiment. Thus, the data of Tsunoda et al. appear to be in quantitative agreement with a simple object-based representation in which an object's identity is coded by its similarities to reference objects. Moreover, the agreement of simulations and experiment suggests that the simplification procedure used by Tsunoda (2001) is not necessarily an accurate method to determine neuronal tuning.

Dissociated Dipoles: Image representation via non-local comparisons

A fundamental question in visual neuroscience is how to represent image structure. The most common representational schemes rely on differential operators that compare adjacent image regions. While well-suited to encoding local relationships, such operators have significant drawbacks. Specifically, each filter's span is confounded with the size of its sub-fields, making it difficult to compare small regions across large distances. We find that such long-distance comparisons are more tolerant to common image transformations than purely local ones, suggesting they may provide a useful vocabulary for image encoding. . We introduce the "Dissociated Dipole," or "Sticks" operator, for encoding non-local image relationships. This operator de-couples filter span from sub-field size, enabling parametric movement between edge and region-based representation modes. We report on the perceptual plausibility of the operator, and the computational advantages of non-local encoding. Our results suggest that non-local encoding may be an effective scheme for representing image structure.

Investigating shape representation in area V4 with HMAX: Orientation and Grating selectivities

The question of how shape is represented is of central interest to understanding visual processing in cortex. While tuning properties of the cells in early part of the ventral visual stream, thought to be responsible for object recognition in the primate, are comparatively well understood, several different theories have been proposed regarding tuning in higher visual areas, such as V4. We used the model of object recognition in cortex presented by Riesenhuber and Poggio (1999), where more complex shape tuning in higher layers is the result of combining afferent inputs tuned to simpler features, and compared the tuning properties of model units in intermediate layers to those of V4 neurons from the literature. In particular, we investigated the issue of shape representation in visual area V1 and V4 using oriented bars and various types of gratings (polar, hyperbolic, and Cartesian), as used in several physiology experiments. Our computational model was able to reproduce several physiological findings, such as the broadening distribution of the orientation bandwidths and the emergence of a bias toward non-Cartesian stimuli. Interestingly, the simulation results suggest that some V4 neurons receive input from afferents with spatially separated receptive fields, leading to experimentally testable predictions. However, the simulations also show that the stimulus set of Cartesian and non-Cartesian gratings is not sufficiently complex to probe shape tuning in higher areas, necessitating the use of more complex stimulus sets.

An Equivalence Between Sparse Approximation and Support Vector Machines

In the first part of this paper we show a similarity between the principle of Structural Risk Minimization Principle (SRM) (Vapnik, 1982) and the idea of Sparse Approximation, as defined in (Chen, Donoho and Saunders, 1995) and Olshausen and Field (1996). Then we focus on two specific (approximate) implementations of SRM and Sparse Approximation, which have been used to solve the problem of function approximation. For SRM we consider the Support Vector Machine technique proposed by V. Vapnik and his team at AT&T Bell Labs, and for Sparse Approximation we consider a modification of the Basis Pursuit De-Noising algorithm proposed by Chen, Donoho and Saunders (1995). We show that, under certain conditions, these two techniques are equivalent: they give the same solution and they require the solution of the same quadratic programming problem.