The Alignment of Objects With Smooth Surfaces: Error Analysis of the Curvature Method

The recognition of objects with smooth bounding surfaces from their contour images is considerably more complicated than that of objects with sharp edges, since in the former case the set of object points that generates the silhouette contours changes from one view to another. The "curvature method", developed by Basri and Ullman [1988], provides a method to approximate the appearance of such objects from different viewpoints. In this paper we analyze the curvature method. We apply the method to ellipsoidal objects and compute analytically the error obtained for different rotations of the objects. The error depends on the exact shape of the ellipsoid (namely, the relative lengths of its axes), and it increases a sthe ellipsoid becomes "deep" (elongated in the Z-direction). We show that the errors are usually small, and that, in general, a small number of models is required to predict the appearance of an ellipsoid from all possible views. Finally, we show experimentally that the curvature method applies as well to objects with hyperbolic surface patches.

Affine Matching with Bounded Sensor Error: A Study of Geometric Hashing and Alignment

Affine transformations are often used in recognition systems, to approximate the effects of perspective projection. The underlying mathematics is for exact feature data, with no positional uncertainty. In practice, heuristics are added to handle uncertainty. We provide a precise analysis of affine point matching, obtaining an expression for the range of affine-invariant values consistent with bounded uncertainty. This analysis reveals that the range of affine-invariant values depends on the actual $x$-$y$-positions of the features, i.e. with uncertainty, affine representations are not invariant with respect to the Cartesian coordinate system. We analyze the effect of this on geometric hashing and alignment recognition methods.

On the Relationship Between Generalization Error, Hypothesis Complexity, and Sample Complexity for Radial Basis Functions

In this paper, we bound the generalization error of a class of Radial Basis Function networks, for certain well defined function learning tasks, in terms of the number of parameters and number of examples. We show that the total generalization error is partly due to the insufficient representational capacity of the network (because of its finite size) and partly due to insufficient information about the target function (because of finite number of samples). We make several observations about generalization error which are valid irrespective of the approximation scheme. Our result also sheds light on ways to choose an appropriate network architecture for a particular problem.

An Algorithm for Group Formation and Maximal Independent Set in an Amorphous Computer

Amorphous computing is the study of programming ultra-scale computing environments of smart sensors and actuators cite{white-paper}. The individual elements are identical, asynchronous, randomly placed, embedded and communicate locally via wireless broadcast. Aggregating the processors into groups is a useful paradigm for programming an amorphous computer because groups can be used for specialization, increased robustness, and efficient resource allocation. This paper presents a new algorithm, called the clubs algorithm, for efficiently aggregating processors into groups in an amorphous computer, in time proportional to the local density of processors. The clubs algorithm is well-suited to the unique characteristics of an amorphous computer. In addition, the algorithm derives two properties from the physical embedding of the amorphous computer: an upper bound on the number of groups formed and a constant upper bound on the density of groups. The clubs algorithm can also be extended to find the maximal independent set (MIS) and $Delta + 1$ vertex coloring in an amorphous computer in $O(log N)$ rounds, where $N$ is the total number of elements and $Delta$ is the maximum degree.

Multiple Resolution Image Classification

Binary image classifiction is a problem that has received much attention in recent years. In this paper we evaluate a selection of popular techniques in an effort to find a feature set/ classifier combination which generalizes well to full resolution image data. We then apply that system to images at one-half through one-sixteenth resolution, and consider the corresponding error rates. In addition, we further observe generalization performance as it depends on the number of training images, and lastly, compare the system's best error rates to that of a human performing an identical classification task given teh same set of test images.

Learning World Models in Environments with Manifest Causal Structure

This thesis examines the problem of an autonomous agent learning a causal world model of its environment. Previous approaches to learning causal world models have concentrated on environments that are too "easy" (deterministic finite state machines) or too "hard" (containing much hidden state). We describe a new domain --- environments with manifest causal structure --- for learning. In such environments the agent has an abundance of perceptions of its environment. Specifically, it perceives almost all the relevant information it needs to understand the environment. Many environments of interest have manifest causal structure and we show that an agent can learn the manifest aspects of these environments quickly using straightforward learning techniques. We present a new algorithm to learn a rule-based causal world model from observations in the environment. The learning algorithm includes (1) a low level rule-learning algorithm that converges on a good set of specific rules, (2) a concept learning algorithm that learns concepts by finding completely correlated perceptions, and (3) an algorithm that learns general rules. In addition this thesis examines the problem of finding a good expert from a sequence of experts. Each expert has an "error rate"; we wish to find an expert with a low error rate. However, each expert's error rate and the distribution of error rates are unknown. A new expert-finding algorithm is presented and an upper bound on the expected error rate of the expert is derived.

Time-Frequency Representations for Speech Signals

This work addresses two related questions. The first question is what joint time-frequency energy representations are most appropriate for auditory signals, in particular, for speech signals in sonorant regions. The quadratic transforms of the signal are examined, a large class that includes, for example, the spectrograms and the Wigner distribution. Quasi-stationarity is not assumed, since this would neglect dynamic regions. A set of desired properties is proposed for the representation: (1) shift-invariance, (2) positivity, (3) superposition, (4) locality, and (5) smoothness. Several relations among these properties are proved: shift-invariance and positivity imply the transform is a superposition of spectrograms; positivity and superposition are equivalent conditions when the transform is real; positivity limits the simultaneous time and frequency resolution (locality) possible for the transform, defining an uncertainty relation for joint time-frequency energy representations; and locality and smoothness tradeoff by the 2-D generalization of the classical uncertainty relation. The transform that best meets these criteria is derived, which consists of two-dimensionally smoothed Wigner distributions with (possibly oriented) 2-D guassian kernels. These transforms are then related to time-frequency filtering, a method for estimating the time-varying 'transfer function' of the vocal tract, which is somewhat analogous to ceptstral filtering generalized to the time-varying case. Natural speech examples are provided. The second question addressed is how to obtain a rich, symbolic description of the phonetically relevant features in these time-frequency energy surfaces, the so-called schematic spectrogram. Time-frequency ridges, the 2-D analog of spectral peaks, are one feature that is proposed. If non-oriented kernels are used for the energy representation, then the ridge tops can be identified, with zero-crossings in the inner product of the gradient vector and the direction of greatest downward curvature. If oriented kernels are used, the method can be generalized to give better orientation selectivity (e.g., at intersecting ridges) at the cost of poorer time-frequency locality. Many speech examples are given showing the performance for some traditionally difficult cases: semi-vowels and glides, nasalized vowels, consonant-vowel transitions, female speech, and imperfect transmission channels.

Error Detection and Recovery for Robot Motion Planning with Uncertainty

Robots must plan and execute tasks in the presence of uncertainty. Uncertainty arises from sensing errors, control errors, and uncertainty in the geometry of the environment. The last, which is called model error, has received little previous attention. We present a framework for computing motion strategies that are guaranteed to succeed in the presence of all three kinds of uncertainty. The motion strategies comprise sensor-based gross motions, compliant motions, and simple pushing motions.

Probabilistic Solution of Inverse Problems

In this thesis we study the general problem of reconstructing a function, defined on a finite lattice from a set of incomplete, noisy and/or ambiguous observations. The goal of this work is to demonstrate the generality and practical value of a probabilistic (in particular, Bayesian) approach to this problem, particularly in the context of Computer Vision. In this approach, the prior knowledge about the solution is expressed in the form of a Gibbsian probability distribution on the space of all possible functions, so that the reconstruction task is formulated as an estimation problem. Our main contributions are the following: (1) We introduce the use of specific error criteria for the design of the optimal Bayesian estimators for several classes of problems, and propose a general (Monte Carlo) procedure for approximating them. This new approach leads to a substantial improvement over the existing schemes, both regarding the quality of the results (particularly for low signal to noise ratios) and the computational efficiency. (2) We apply the Bayesian appraoch to the solution of several problems, some of which are formulated and solved in these terms for the first time. Specifically, these applications are: teh reconstruction of piecewise constant surfaces from sparse and noisy observationsl; the reconstruction of depth from stereoscopic pairs of images and the formation of perceptual clusters. (3) For each one of these applications, we develop fast, deterministic algorithms that approximate the optimal estimators, and illustrate their performance on both synthetic and real data. (4) We propose a new method, based on the analysis of the residual process, for estimating the parameters of the probabilistic models directly from the noisy observations. This scheme leads to an algorithm, which has no free parameters, for the restoration of piecewise uniform images. (5) We analyze the implementation of the algorithms that we develop in non-conventional hardware, such as massively parallel digital machines, and analog and hybrid networks.

Finding Edges and Lines in Images

The problem of detecting intensity changes in images is canonical in vision. Edge detection operators are typically designed to optimally estimate first or second derivative over some (usually small) support. Other criteria such as output signal to noise ratio or bandwidth have also been argued for. This thesis is an attempt to formulate a set of edge detection criteria that capture as directly as possible the desirable properties of an edge operator. Variational techniques are used to find a solution over the space of all linear shift invariant operators. The first criterion is that the detector have low probability of error i.e. failing to mark edges or falsely marking non-edges. The second is that the marked points should be as close as possible to the centre of the true edge. The third criterion is that there should be low probability of more than one response to a single edge. The technique is used to find optimal operators for step edges and for extended impulse profiles (ridges or valleys in two dimensions). The extension of the one dimensional operators to two dimentions is then discussed. The result is a set of operators of varying width, length and orientation. The problem of combining these outputs into a single description is discussed, and a set of heuristics for the integration are given.