Contour extrapolation using probabilistic cue combination

A common approach to the problem of contour interpolation is based on the calculus of variations. The optimal interpolating contour is taken to be one that minimizes a given smoothness functional. Two important such functionals are total curvature (or bending energy) and variation in curvature. We analyzed contours extrapolated by human observers given arcs of Euler spirals that disappeared behind an occluding surface. Irrespective of whether the Euler spirals had increasing or decreasing curvature as they approached the occluding edge, visually-extrapolated contours were found to be characterized by decaying curvature. This curvature decay is modeled in terms of a Bayesian interaction between probabilistically-expressed constraints to minimize curvature and minimize variation in curvature. The analysis suggests that using fixed smoothness functionals is not appropriate for modeling human vision. Rather, the relative weights assigned to different probabilistic shape constraints may vary as a function of distance from the point(s) of occlusion. Implications are discussed for computational models of shape completion.

[1]  James J. Clark,et al.  Data Fusion for Sensory Information Processing Systems , 1990 .

[2]  David J. Field,et al.  Contour integration by the human visual system: Evidence for a local “association field” , 1993, Vision Research.

[3]  Jacqueline M. Fulvio,et al.  Surface geometry influences the shape of illusory contours. , 2006, Acta psychologica.

[4]  M. Landy,et al.  Measurement and modeling of depth cue combination: in defense of weak fusion , 1995, Vision Research.

[5]  Kanti V. Mardia,et al.  Statistics of Directional Data , 1972 .

[6]  P. Kellman,et al.  A theory of visual interpolation in object perception , 1991, Cognitive Psychology.

[7]  Paul R. Schrater,et al.  Human and Ideal Observers for Detecting Image Curves , 2003, NIPS.

[8]  I. S. Sokolnikoff.  Mathematical theory of elasticity , 1946 .

[9]  D D Hoffman,et al.  Completing visual contours: The relationship between relatability and minimizing inflections , 1999, Perception & psychophysics.

[10]  Harry G. Barrow,et al.  Interpreting Line Drawings as Three-Dimensional Surfaces , 1980, Artif. Intell..

[11]  R. Fleming,et al.  The Interpolation of Object and Surface Structure , 2002, Cognitive Psychology.

[12]  J. Feldman.  Curvilinearity, covariance, and regularity in perceptual groups , 1997, Vision Research.

[13]  J. Elder,et al.  Ecological statistics of Gestalt laws for the perceptual organization of contours. , 2002, Journal of vision.

[14]  Steven W. Zucker,et al.  Radial Projection: An Efficient Update Rule for Relaxation Labeling , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[15]  Jacqueline M. Fulvio,et al.  Visual extrapolation of contour geometry. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[16]  Manish Singh.  Modal and Amodal Completion Generate Different Shapes , 2004, Psychological science.

[17]  D. Fitzpatrick,et al.  Orientation Selectivity and the Arrangement of Horizontal Connections in Tree Shrew Striate Cortex , 1997, The Journal of Neuroscience.

[18]  Michael S. Landy,et al.  Interpolating sampled contours in 3-D: analyses of variability and bias , 2002, Vision Research.

[19]  Lance R. Williams,et al.  Stochastic Completion Fields: A Neural Model of Illusory Contour Shape and Salience , 1995, Neural Computation.

[20]  Steven W. Zucker,et al.  A Markov Process Using Curvature for Filtering Curve Images , 2001, EMMCVPR.

[21]  Michael S. Landy,et al.  Influence function for visual interpolation , 1997, Electronic Imaging.

[22]  D. Mumford.  Elastica and Computer Vision , 1994 .

[23]  Berthold K. P. Horn.  The Curve of Least Energy , 1983, TOMS.

[24]  Steven W. Zucker,et al.  Trace Inference, Curvature Consistency, and Curve Detection , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[25]  J. Feldman,et al.  Information along contours and object boundaries. , 2005, Psychological review.

[26]  W. Gerbino,et al.  Contour curvature polarity and surface interpolation , 2005, Vision Research.

[27]  S. Grossberg,et al.  Neural dynamics of form perception: boundary completion, illusory figures, and neon color spreading. , 1985, Psychological review.

[28]  W. Gerbino,et al.  Contour interpolation by vector-field combination. , 2003, Journal of vision.

[29]  G. C. Tiao,et al.  Bayesian inference in statistical analysis , 1973 .

[30]  S. Ullman,et al.  Filling-in the gaps: The shape of subjective contours and a model for their generation , 1976, Biological Cybernetics.

[31]  Demetri Terzopoulos,et al.  Snakes: Active contour models , 2004, International Journal of Computer Vision.

[32]  George E. P. Box,et al.  Bayesian Inference in Statistical Analysis: Box/Bayesian , 1992 .

[33]  Ohad Ben-Shahar,et al.  Geometrical Computations Explain Projection Patterns of Long-Range Horizontal Connections in Visual Cortex , 2004, Neural Computation.

[34]  Jeffrey S. Perry,et al.  Edge co-occurrence in natural images predicts contour grouping performance , 2001, Vision Research.

[35]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[36]  C. Gilbert,et al.  On a common circle: natural scenes and Gestalt rules. , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[37]  A. Rosenfeld,et al.  Curve Detection in a Noisy Image , 1997, Vision Research.

[38]  W. G. Bickley.  Mathematical Theory Of Elasticity , 1946, Nature.

[39]  Jacob feldman,et al.  Bayesian contour integration , 2001, Perception & psychophysics.

[40]  Benjamin B. Kimia,et al.  Euler Spiral for Shape Completion , 2003, International Journal of Computer Vision.