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.

Some Aspects of Pattern Recognition by Computer

A computer may gather a lot of information from its environment in an optical or graphical manner. A scene, as seen for instance from a TV camera or a picture, can be transformed into a symbolic description of points and lines or surfaces. This thesis describes several programs, written in the language CONVERT, for the analysis of such descriptions in order to recognize, differentiate and identify desired objects or classes of objects in the scene. Examples are given in each case. Although the recognition might be in terms of projections of 2-dim and 3-dim objects, we do not deal with stereoscopic information. One of our programs (Polybrick) identifies parallelepipeds in a scene which may contain partially hidden bodies and non-parallelepipedic objects. The program TD works mainly with 2-dimensional figures, although under certain conditions successfully identifies 3-dim objects. Overlapping objects are identified when they are transparent. A third program, DT, works with 3-dim and 2-dim objects, and does not identify objects which are not completely seen. Important restrictions and suppositions are: (a) the input is assumed perfect (noiseless), and in a symbolic format; (b) no perspective deformation is considered. A portion of this thesis is devoted to the study of models (symbolic representations) of the objects we want to identify; different schemes, some of them already in use, are discussed. Focusing our attention on the more general problem of identification of general objects when they substantially overlap, we propose some schemes for their recognition, and also analyze some problems that are met.

Structure from Stereo and Motion

Stereopsis and motion parallax are two methods for recovering three dimensional shape. Theoretical analyses of each method show that neither alone can recover rigid 3D shapes correctly unless other information, such as perspective, is included. The solutions for recovering rigid structure from motion have a reflection ambiguity; the depth scale of the stereoscopic solution will not be known unless the fixation distance is specified in units of interpupil separation. (Hence the configuration will appear distorted.) However, the correct configuration and the disposition of a rigid 3D shape can be recovered if stereopsis and motion are integrated, for then a unique solution follows from a set of linear equations. The correct interpretation requires only three points and two stereo views.

An Analog VLSI Chip for Estimating the Focus of Expansion

For applications involving the control of moving vehicles, the recovery of relative motion between a camera and its environment is of high utility. This thesis describes the design and testing of a real-time analog VLSI chip which estimates the focus of expansion (FOE) from measured time-varying images. Our approach assumes a camera moving through a fixed world with translational velocity; the FOE is the projection of the translation vector onto the image plane. This location is the point towards which the camera is moving, and other points appear to be expanding outward from. By way of the camera imaging parameters, the location of the FOE gives the direction of 3-D translation. The algorithm we use for estimating the FOE minimizes the sum of squares of the differences at every pixel between the observed time variation of brightness and the predicted variation given the assumed position of the FOE. This minimization is not straightforward, because the relationship between the brightness derivatives depends on the unknown distance to the surface being imaged. However, image points where brightness is instantaneously constant play a critical role. Ideally, the FOE would be at the intersection of the tangents to the iso-brightness contours at these "stationary" points. In practice, brightness derivatives are hard to estimate accurately given that the image is quite noisy. Reliable results can nevertheless be obtained if the image contains many stationary points and the point is found that minimizes the sum of squares of the perpendicular distances from the tangents at the stationary points. The FOE chip calculates the gradient of this least-squares minimization sum, and the estimation is performed by closing a feedback loop around it. The chip has been implemented using an embedded CCD imager for image acquisition and a row-parallel processing scheme. A 64 x 64 version was fabricated in a 2um CCD/ BiCMOS process through MOSIS with a design goal of 200 mW of on-chip power, a top frame rate of 1000 frames/second, and a basic accuracy of 5%. A complete experimental system which estimates the FOE in real time using real motion and image scenes is demonstrated.

The Grasping Problem: Toward Task-Level Programming for an Articulated Hand

This report presents a system for generating a stable, feasible, and reachable grasp of a polyhedral object. A set of contact points on the object is found that can result in a stable grasp; a feasible grasp is found in which the robot contacts the object at those contact points; and a path is constructed from the initial configuration of the robot to the stable, feasible final grasp configuration. The algorithm described in the report is designed for the Salisbury hand mounted on a Puma 560 arm, but a similar approach could be used to develop grasping systems for other robots.

Reconfigurable Architectures for General-Purpose Computing

General-purpose computing devices allow us to (1) customize computation after fabrication and (2) conserve area by reusing expensive active circuitry for different functions in time. We define RP-space, a restricted domain of the general-purpose architectural space focussed on reconfigurable computing architectures. Two dominant features differentiate reconfigurable from special-purpose architectures and account for most of the area overhead associated with RP devices: (1) instructions which tell the device how to behave, and (2) flexible interconnect which supports task dependent dataflow between operations. We can characterize RP-space by the allocation and structure of these resources and compare the efficiencies of architectural points across broad application characteristics. Conventional FPGAs fall at one extreme end of this space and their efficiency ranges over two orders of magnitude across the space of application characteristics. Understanding RP-space and its consequences allows us to pick the best architecture for a task and to search for more robust design points in the space. Our DPGA, a fine- grained computing device which adds small, on-chip instruction memories to FPGAs is one such design point. For typical logic applications and finite- state machines, a DPGA can implement tasks in one-third the area of a traditional FPGA. TSFPGA, a variant of the DPGA which focuses on heavily time-switched interconnect, achieves circuit densities close to the DPGA, while reducing typical physical mapping times from hours to seconds. Rigid, fabrication-time organization of instruction resources significantly narrows the range of efficiency for conventional architectures. To avoid this performance brittleness, we developed MATRIX, the first architecture to defer the binding of instruction resources until run-time, allowing the application to organize resources according to its needs. Our focus MATRIX design point is based on an array of 8-bit ALU and register-file building blocks interconnected via a byte-wide network. With today's silicon, a single chip MATRIX array can deliver over 10 Gop/s (8-bit ops). On sample image processing tasks, we show that MATRIX yields 10-20x the computational density of conventional processors. Understanding the cost structure of RP-space helps us identify these intermediate architectural points and may provide useful insight more broadly in guiding our continual search for robust and efficient general-purpose computing structures.

A Note on Support Vector Machines Degeneracy

When training Support Vector Machines (SVMs) over non-separable data sets, one sets the threshold $b$ using any dual cost coefficient that is strictly between the bounds of $0$ and $C$. We show that there exist SVM training problems with dual optimal solutions with all coefficients at bounds, but that all such problems are degenerate in the sense that the "optimal separating hyperplane" is given by ${f w} = {f 0}$, and the resulting (degenerate) SVM will classify all future points identically (to the class that supplies more training data). We also derive necessary and sufficient conditions on the input data for this to occur. Finally, we show that an SVM training problem can always be made degenerate by the addition of a single data point belonging to a certain unboundedspolyhedron, which we characterize in terms of its extreme points and rays.

Uncertainty Propagation in Model-Based Recognition

Building robust recognition systems requires a careful understanding of the effects of error in sensed features. Error in these image features results in a region of uncertainty in the possible image location of each additional model feature. We present an accurate, analytic approximation for this uncertainty region when model poses are based on matching three image and model points, for both Gaussian and bounded error in the detection of image points, and for both scaled-orthographic and perspective projection models. This result applies to objects that are fully three- dimensional, where past results considered only two-dimensional objects. Further, we introduce a linear programming algorithm to compute the uncertainty region when poses are based on any number of initial matches. Finally, we use these results to extend, from two-dimensional to three- dimensional objects, robust implementations of alignmentt interpretation- tree search, and ransformation clustering.