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 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 scale-invariant. 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.

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.

Neurofuzzy systems-the combination of artificial neural networks with fuzzy logic-have become useful in many application domains. However, conventional neurofuzzy models usually need enhanced representation power for applications that require context and state (e.g., speech, time series prediction, control). Some of these applications can be readily modeled as finite state automata. Previously, it was proved that deterministic finite state automata (DFA) can be synthesized by or mapped into recurrent neural networks by directly programming the DFA structure into the weights of the neural network. Based on those results, a synthesis method is proposed for mapping fuzzy finite state automata (FFA) into recurrent neural networks. Furthermore, this mapping is suitable for direct implementation in very large scale integration (VLSI), i.e., the encoding of FFA as a generalization of the encoding of DFA in VLSI systems. The synthesis method requires FFA to undergo a transformation prior to being mapped into recurrent networks. The neurons are provided with an enriched functionality in order to accommodate a fuzzy representation of FFA states. This enriched neuron functionality also permits fuzzy parameters of FFA to be directly represented as parameters of the neural network. We also prove the stability of fuzzy finite state dynamics of the constructed neural networks for finite values of network weight and, through simulations, give empirical validation of the proofs. Hence, we prove various knowledge equivalence representations between neural and fuzzy systems and models of automata.

There has been an increased interest in combining fuzzy systems with neural networks because fuzzy neural systems merge the advantages of both paradigms. On the one hand, parameters in fuzzy systems have clear physical meanings and rule-based and linguistic information can be incorporated into adaptive fuzzy systems in a systematic way. On the other hand, there exist powerful algorithms for training various neural network models. However, most of the proposed combined architectures are only able to process static input-output relationships; they are not able to process temporal input sequences of arbitrary length. Fuzzy finite-state automats (FFAs) can model dynamical processes whose current state depends on the current input and previous states. Unlike in the case of deterministic finite-state automats (DFAs), FFAs are not in one particular state, rather each state is occupied to some degree defined by a membership function. Based on previous work on encoding DFAs in discrete-time second-order recurrent neural networks, we propose an algorithm that constructs an augmented recurrent neural network that encodes a FFA and recognizes a given fuzzy regular language with arbitrary accuracy. We then empirically verify the encoding methodology by correct string recognition of randomly generated FFAs. In particular, we examine how the networks' performance varies as a function of synaptic weight strengths.

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.>