Karvel K. Thornber

Segmentation of multiple salient closed contours from real images

Using a saliency measure based on the global property of contour closure, we have developed a segmentation method which identifies smooth closed contours bounding objects of unknown shape in real images. The saliency measure incorporates the Gestalt principles of proximity and good continuity that previous methods have also exploited. Unlike previous methods, we incorporate contour closure by finding the eigenvector with the largest positive real eigenvalue of a transition matrix for a Markov process where edges from the image serve as states. Element (i, j) of the transition matrix is the conditional probability that a contour which contains edge j will also contain edge i. We show how the saliency measure, defined for individual edges, can be used to derive a saliency relation, defined for pairs of edges, and further show that strongly-connected components of the graph representing the saliency relation correspond to smooth closed contours in the image. Finally, we report for the first time, results on large real images for which segmentation takes an average of about 10 seconds per object on a general-purpose workstation.

A Comparison of Measures for Detecting Natural Shapes in Cluttered Backgrounds

We propose a new measure of perceptual saliency and quantitatively compare its ability to detect natural shapes in cluttered backgrounds to five previously proposed measures. As defined in the new measure, the saliency of an edge is the fraction of closed random walks which contain that edge. The transition-probability matrix defining the random walk between edges is based on a distribution of natural shapes modeled by a stochastic motion. Each of the saliency measures in our comparison is a function of a set of affinity values assigned to pairs of edges. Although the authors of each measure define the affinity between a pair of edges somewhat differently, all incorporate the Gestalt principles of good-continuation and proximity in some form. In order to make the comparison meaningful, we use a single definition of affinity and focus instead on the performance of the different functions for combining affinity values. The primary performance criterion is accuracy. We compute false-positive rates in classifying edges as signal or noise for a large set of test figures. In almost every case, the new measure significantly outperforms previous measures.

Segmentation of salient closed contours from real images

Using a saliency measure based on the global property of contour closure, we have developed a method that reliably segments out salient contours bounding unknown objects from real edge images. The measure also incorporates the Gestalt principles of proximity and smooth continuity that previous methods have exploited. Unlike previous measures, we incorporate contour closure by finding the eigen-solution associated with a stochastic process that models the distribution of contours passing through edges in the scene. The segmentation algorithm utilizes the saliency measure to identify multiple closed contours by finding strongly-connected components on an induced graph. The determination of strongly-connected components is a direct consequence of the property of closure. We report for the first time, results on large real images for which segmentation takes an average of about 10 secs per object on a general-purpose workstation. The segmentation is made efficient for such large images by exploiting the inherent symmetry in the task.

Orientation, Scale, and Discontinuity as Emergent Properties of Illusory Contour Shape

A recent neural model of illusory contour formation is based on a distribution of natural shapes traced by particles moving with constant speed in directions given by Brownian motions. The input to that model consists of pairs of position and direction constraints, and the output consists of the distribution of contours joining all such pairs. In general, these contours will not be closed, and their distribution will not be scaleinvariant. In this article, we show how to compute a scale-invariant distribution of closed contours given position constraints alone and use this result to explain a well-known illusory contour effect.

Characterizing the distribution of completion shapes with corners using a mixture of random processes

Abstract We derive an analytic expression for the distribution of contours x(t) generated by fluctuations in x (t)=∂ x (t)/∂t due to random impulses of two limiting types. The first type are frequent but weak while the second are infrequent but strong. The result has applications in computational theories of figural completion and illusory contours because it can be used to model the prior probability distribution of short, smooth completion shapes punctuated by occasional discontinuities in orientation (i.e., corners). This work extends our previous work on characterizing the distribution of completion shapes which dealt only with the case of frequently acting weak impulses.

Computing Stochastic Completion Fields in Linear-Time Using a Resolution Pyramid

We describe a linear-time algorithm for computing the likelihood that a completion joining two contour fragments passes through any given position and orientation in the image plane. Our algorithm is a resolution pyramid based method for solving a partial differential equation characterizing a distribution of short, smooth completion shapes. The PDE consists of a set of independent advection equations in (x, y) coupled in the ϕ dimension by the diffusion equation. A previously described algorithm used a first-order, explicit finite difference scheme implemented on a rectangular grid. This algorithm required O(n3m) time for a grid of size n x n with m discrete orientations. Unfortunately, systematic error in solving the advection equations produced unwanted anisotropic smoothing in the (x, y) dimension. This resulted in visible artifacts in the completion fields. The amount of error and its dependence on B has been previously characterized. We observe that by careful addition of extra spatial smoothing, the error can be made totally isotropic. The combined effect of this error and of intrinsic smoothness due to diffusion in the ϕ dimension is that the solution becomes smoother with increasing time, i.e., the high spatial frequencies drop out. By increasing Δξ and Δt on a regular schedule, and using a second-order, implicit scheme for the diffusion term, it is possible to solve the modified PDE in O(n2m) time, i.e., time linear in the problem size. Using current hardware and for problems of typical size, this means that a solution which previously took one hour to compute can now be computed in about two minutes.

A key to fuzzy-logic inference

Abstract Classically, whether to effect inference, one uses a small set of axioms and modus ponens, or a set of rules of inference including modus ponens, one is going beyond what can be derived with the explicit operations of logic alone. Carrying this concept over to fuzzy logic we construct a fuzzy modus ponens and other rules of inference that include modus tollens and reductio ad absurdum. These in turn are based on (and greatly facilitated by) a choice for the operation of implication that preserves the (logic) symmetry implicit in its definition. Extensions including conditional quantification, cut rules (single, multiple, and implicitory), and fuzzy mathematical induction are sketched. As an example, a fuzzy-logic treatment of the Yale shooting problem is discussed. The results suggest that the implicit processes of inference, as distinct from the explicit processes of decision (control) theory and systems theory, can be effected in fuzzy logic if, as in classical logic, one ventures outside the scope of (fuzzy) logic operations.

The fidelity of fuzzy-logic inference

The concept of fidelity of inference is introduced in order to quantify the degree to which the inferred value of a consequent can be regarded as valid. This concept also enhances our understanding of other aspects of fuzzy-logic inference including transitivity, intuitive leap, rada, and coupled implicative equations. It also enables many of the results derived earlier for element-by-element implication and derived implication to be taken over to inclusion and derived inclusion. Finally, it permits this inference to be carried out in hardware using simple circuitry, and provides a natural measure of noise immunity. >

Fuzzy Knowledge and Recurrent Neural Networks: A Dynamical Systems Perspective

Hybrid neuro-fuzzy systems – the combination of artificial neural networks with fuzzy logic – are becoming increasingly popular. However, neuro-fuzzy systems need to be extended for applications which require context (e.g., speech, handwriting, control). Some of these applications can be modeled in the form of finite-state automata. This chapter presents a synthesis method for mapping fuzzy finite-state automata (FFAs) into recurrent neural networks. The synthesis method requires FFAs to undergo a transformation prior to being mapped into recurrent networks. Their neurons have a slightly enriched functionality in order to accommodate a fuzzy representation of FFA states. This allows fuzzy parameters of FFAs to be directly represented as parameters of the neural network. We present a proof the stability of fuzzy finite-state dynamics of constructed neural networks and through simulations give empirical validation of the proofs.

Characterizing the Distribution of Completion Shapes with Corners Using a Mixture of Random Processes

We derive an analytic expression for the distribution of contours x(t) generated by fluctuations in x(t) = ∂x.(t)/∂t due to stochastic impulses of two limiting types. The first type are frequent but weak while the second are infrequent but strong. The result has applications in computational theories of figurai completion and illusory contours because it can be used to model the prior probability distribution of short, smooth completion shapes punctuated by occasional discontinuities in orientation (i.e., corners). This work extends our previous work on characterizing the distribution of completion shapes which dealt only with the case of frequently acting weak impulses.