The goal of this paper is to study a mathematical framework of 2D object shape modeling and learning for middle level vision problems, such as image segmentation and perceptual organization. For this purpose, we pursue generic shape models which characterize the most common features of 2D object shapes. In this paper, shape models are learned from observed natural shapes based on a minimax entropy learning theory (Zhu and Mumford 1997, Zhu, Wu and Mumford 1997)[31, 32]. The learned shape models are Gibbs distributions dened on Markov random elds (MRFs). The neighborhood structures of these MRFs correspond to Gestalt laws {co-linearity, co-circularity, proximity, parallelism, and symmetry. Thus both contour-based and region-based features are accounted for. Stochastic Markov chain Monte Carlo (MCMC) algorithms are proposed for learning and model verication. Furthermore, this paper provides a quantitative measure for the so-called non-accidental statistics, and thus justies some empi...

An important element of learning from examples is the extraction of patterns and regularities from data. This paper investigates the structure of patterns in data defined over discrete features, i.e. features with two or more qualitatively distinct values. Any such pattern can be algebraically decomposed into a spectrum of component patterns, each of which is a simpler or more atomic ‘‘regularity.’ ’ Each component regularity involves a certain number of features, referred to as its degree. Regularities of lower degree represent simpler or more coarse patterns in the original pattern, while regularities of higher degree represent finer or more idiosyncratic patterns. The full spectral breakdown of a pattern into component regularities of minimal degree, referred to as its power series, expresses the original pattern in terms of the regular rules or patterns it obeys, amounting to a kind of ‘‘theory’ ’ of the pattern. The number of regularities at various degrees necessary to represent the pattern is tabulated in its power spectrum, which expresses how much of a pattern’s structure can be explained by regularities of various levels of complexity. A weighted mean of the pattern’s spectral power gives a useful numeric summary of its overall complexity, called its algebraic complexity. The basic theory of algebraic decomposition is extended in several ways, including algebraic accounts of the typicality of individual objects within concepts, and estimation of the power series from noisy data. Finally some relations between these algebraic quantities and empirical data are discussed.

Scenes are composed of numerous objects, textures and colors which are arranged in a variety of spatial layouts. This presents the question of how visual complexity is represented by a cognitive system. In this paper, we aim to study the representation of visual complexity for real-world scene images. Is visual complexity a perceptual property simple enough so that it can be compressed along a unique perceptual dimension? Or is visual complexity better represented by a multi-dimensional space? Thirty-four participants performed a hierarchical grouping task in which they divided scenes into successive groups of decreasing complexity, describing the criteria they used at each stage. Half of the participants were told that complexity was related to the structure of the image whereas the instructions in the other half were unspecified. Results are consistent with a multi-dimensional representation of visual complexity (quantity of objects, clutter, openness, symmetry, organization, variety of colors) with task constraints modulating the shape of the complexity space (e.g. the weight of a specific dimension).

Two basic problems in image interpretation are: a) determining which interpretations are the most plausible amoungst many possibilities; and b) controlling the search for plausible interpretations. We address these issues using a Bayesian approach, with the plausibility ordering and search pruning based on the posterior probabilities of interpretations. However, due to the need for detailed quantitative prior probabilities and the need to evaluate complex integrals over various conditional distributions, a full Bayesian approach is currently impractical except in tightly constrained domains. To circumvent these difficulties we introduce the notion of qualitative probabilistic analysis. In particular, given spatial and contrast resolution parameters, we consider only the asymptotic order of the posterior probability for any interpretation as these resolutions are made finer. We introduce this approach for a simple card-world domain, and present computational results for blocks-world ima...

This paper presents a logic-based approach to grouping and perceptual organization, called Minimal Model theory, and presents efficient methods for computing interpretations in this framework. Grouping interpretations are first defined as logical structures, built out of atomic qualitative scene descriptors ("regularities") that are derived from considerations of non-accidentalness. These interpretations can then be partially ordered by their degree of regularity or constraint (measured numerically by their logical depth). The Genericity Constraint---the principle that interpretations should minimize coincidences in the observed configuration---dictates that the preferred interpretation will be the minimum in this partial order, i.e. the interpretation with maximum depth. This maximum-depth interpretation, also called the minimal model or minimal interpretation, is in a sense the "simplest" (algebraically minimal) interpretation available of the image configuration. As a side-effect, the "most salient" or most structured part of the scene can be identified, as the maximum-depth subtree of the minimal model. An efficient (O(n )) method for computing the minimal interpretation is presented, along with examples. Computational experiments show that the algorithm performs well under a wide range of parameter settings.

Perceptual grouping is the process by which elements in the visual image are aggregated into larger and more complex structures, i.e., “objects. ” This paper reports a study of the spatial factors and time-course of the development of objects over the course of the first few hundred milliseconds of visual processing. The methodology uses the now well-established idea of an “object benefit ” for certain kinds of tasks (here, faster within-object than between-objects probe comparisons) to test what the visual system in fact treats as an object at each point during processing. The study tested line segment pairs in a wide variety of spatial configurations at a range of exposure times, in each case measuring the strength of perceptual grouping as reflected in the magnitude of the object benefit. Factors tested included nonaccidental properties such as collinearity, cotermination, and parallelism; contour relatability; Gestalt factors such as symmetry and skew symmetry, and several others, all tested at fine (25 msec) time-slices over the course of processing. The data provide detailed information about the comparative strength of these factors in inducing grouping at each point in processing. The result is a vivid picture of the chronology of object formation, as objects progressively coalesce, with fully bound visual objects completed by about 200 msec of processing. The organization of the initially inchoate visual field into coherent units or “objects, ” called perceptual grouping or binding, is of fundamental importance in visual perception, influencing the perception of lightness (Adelson, 1993; Gilchrist, 1977), motion (Shimojo, Silverman, & Nakayama, 1988), and recognition of objects (Biederman, 1987). Much has been learned about the grouping factors originally identified by the Gestaltists, such as spatial

Object-based models of visual attention purport to explain why it is easier to process information within one object or perceptual group than across two or more groups. Perceptual groups are generally defined in terms of Gestalt grouping principles. These models of attention have been used to explain the phenomenon of cognitive tunneling within Heads-Up Displays (HUDs), on the assumption that the symbology of a Heads-Up Display (HUD) in a cockpit forms a single perceptual group and the outside scene forms another. Despite extensive empirical support, object-based models have various shortcomings. In particular, the use of Gestalt grouping principles to define the notion of objects does not allow for an operational measure of what an object is to the visual system. Also, the Gestalt principles do not allow for a systematic distinction between spatial and object-based mechanisms of attention. Finally, it is generally assumed that Gestalt grouping occurs preattentively, whereas there is evidence that perceptual grouping requires attentional resources. The proposed line of research aims to develop an account of object-based attention that does not rely on these premises. Rather, it is assumed that the cost of dividing attention between

Human observers are remarkably proficient at extracting the causal structure of both natural and artifactual worlds on the basis of extremely impoverished observations, but a rigorous theory of how they achieve this is elusive. This paper investigates the formal structure of this problem, collapsing the distinction between human and automatic inference about complex systems, and considering an abstract observer in an abstract world. We introduce a structure algebra, a reduced description logic that is designed to be biased towards the recovery of causal structure in much the same way as are human observers. We illustrate how the algebra captures human intuitions about the "natural" interpretation of certain canonic types of observed property covariation. Finally we propose a formal criterion for the adequacy of a given description language to capture algebraically the "true" causal structure of a particular closed world.

Discussions of the foundations of perceptual inference have often centered on 2 governing principles, the likelihood principle and the simplicity principle. Historically, these principles have usually been seen as opposed, but contemporary statistical (e.g., Bayesian) theory tends to see them as consistent, because for a variety of reasons simpler models (i.e., those with fewer dimensions or free parameters) make better predictors than more complex ones. In perception, many interpretation spaces are naturally hierarchical, meaning that they consist of a set of mutually embedded model classes of various levels of complexity, including simpler (lower dimensional) classes that are special cases of more complex ones. This article shows how such spaces can be regarded as algebraic structures, for example, as partial orders or lattices, with interpretations ordered in terms of dimensionality. The natural inference rule in such a space is a kind of simplicity rule: Among all interpretations qualitatively consistent with the image, draw the one that is lowest in the partial order, called the maximum-depth interpretation. This interpretation also maximizes the Bayesian posterior under certain simplifying assumptions, consistent with a unification of simplicity and likelihood principles. Moreover, the algebraic approach brings out the compositional structure inherent in such spaces, showing how perceptual interpretations are composed from a lexicon of primitive perceptual descriptors.

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