## Experiment 1

### One set of regions moving

 Number of Regions BoundaryShape 2 4 8 Straight Curved-Unbiased Convex

### Both sets of regions moving

 Number of Regions BoundaryShape 2 4 8 Straight Curved-Unbiased Convex

## Experiment 2

### One set of regions moving

 Number of Regions BoundaryShape 2 4 8 Straight Curved-Unbiased Convex

### Both sets of regions moving

 Number of Regions BoundaryShape 2 4 8 Straight Curved-Unbiased Convex

## EXPERIMENT 1

(Click on any image to view the corresponding motion display.)

Convex/symmetric regions contain motionNon-convex/asymmetric regions contain motionBoth sets of regions contain motion
Convexity & Symmetry
Symmetry

## EXPERIMENT 2

(Click on any image to view the corresponding motion display.)

Coherent MotionIncoherent Motion
Convexity & Symetry
Symetry

## Information along contours

Attneave (1954) argued that information along contours is concentrated along points of locally maximal curvature. In a recent paper, we provide a formal proof of this, based on Shannon's definition of information (Feldman & Singh, Psychological Review, in press). This paper also extends Attneave's original claim by demonstrating the asymmetry in information content that arises from the sign of curvature. Whereas Attneave's analysis treated positive and negative curvatureâ€”i.e., convex and concave contour segmentsâ€”symmetrically, we show that for closed contours, regions of negative curvature carry greater information than corresponding segments of positive curvature. MATLAB code for computing and plotting information along contours (i.e., the surprisal) is included below.

The option signed = 0 treats positive and negative curvature symmetrically (consistent with Attneave), whereas the option signed = 1takes into account (for closed contours) the sign of curvature as well, yielding a different distribution of information. The two figures below demonstrate this difference; the left one is produced from the unsigned version, the right one from the signed version.

The code assumes that the contour is sampled uniformly, in counter-clockwise direction, and is defined by a pair of vectors of equal length (x and y). resolution alters the number of neighboring points the code takes into account in computing the tangent direction at any given point.

function surprisal = contourinfoplot(x, y, signed, resolution)

x = x(:); y = y(:);

if length(x) ~= length(y)
error('the x and y vectors defining the contour must have equal lengths!');
end;

N = length(x);
b = 1; % spread term for the Von Mises distribution

if (x(1) == x(N) & y(1) == y(N))
N = N-1;
open = 0;
else
open = 1;
end

normalmap = [];
surprisalmap = [];

for j = 1:N

% define the previous and next points relative to the current one
xprev = 0;
yprev = 0;
xnext = 0;
ynext = 0;

for k = 1:resolution
prev(k) = j-k;
next(k) = j+k;
if(prev(k) < 1) prev(k) = prev(k) + N; end
if(next(k) > N) next(k) = next(k) - N; end

xprev = xprev + x(prev(k));
yprev = yprev + y(prev(k));
xnext = xnext + x(next(k));
ynext = ynext + y(next(k));
end

xprev = xprev/k;
yprev = yprev/k;
xnext = xnext/k;
ynext = ynext/k;

% vector _from_ the previous point; vector _to_ the next point
vecprev = [x(j), y(j)] - [xprev, yprev];
vecnext = [xnext, ynext] - [x(j), y(j)];

% compute the magnitude of the turning angle using dot product
alpha = acos( dot(vecprev, vecnext) / (norm(vecprev)*norm(vecnext)) );

% compute the sign of turning using cross product
% (assumes a counterclockwise sampling of the contour)
cp = cross( [vecprev, 0], [vecnext,0] );
alpha = sign(cp(3))*alpha;

% compute the surprisal using -log von mises
if(signed)
surprisal = -log( exp(b*cos(alpha - (2*pi/(N/resolution))) )/( 2*pi*besseli(0,b) ) );
else
surprisal = -log( exp(b*cos(alpha) )/( 2*pi*besseli(0,b) ) );
end;

% turning angles not defined near the end points of an open curve
if(open & (j - resolution < 1 | j + resolution > N)) surprisal = 0; end

% compute the tangent and normal vectors
tangvec = vecprev + vecnext;
tangvec = tangvec/norm(tangvec);
normvec = [[0 1; -1 0]*tangvec']';

normalmap = [normalmap; normvec];
surprisalmap = [surprisalmap; surprisal];

end;

% scale the size of histogram bars
surprisalmap = (1/range(surprisalmap))*(surprisalmap(:) - min(surprisalmap));
needlesize = max(range(x), range(y))/10;

% define the histogram normal to the shape
normalmap(:,1) = surprisalmap.*normalmap(:,1);
normalmap(:,2) = surprisalmap.*normalmap(:,2);
xnormals = [x(1:N), x(1:N) + needlesize*normalmap(:,1)];
ynormals = [y(1:N), y(1:N) + needlesize*normalmap(:,2)];

% plot
figure, plot(x,y,'r'), axis equal;
hold
on, plot(xnormals', ynormals', 'k-');

## Publications

 Bayesian Hierarchical Grouping: Perceptual grouping as mixture estimationFroyen, V., Feldman, J., & Singh, M. (2015).Psychological Review, 122(4), 575-597PDF Interface Theory of PerceptionHoffman, D., Singh, M., & Prakash, C. (in press)Psychonomic Bulletin and ReviewPDF Probing the Interface Theory of Perception: Reply to CommentariesHoffman, D., Singh, M., & Prakash, C. (in press)Psychonomic Bulletin and ReviewPDF Investigating shape representation using sensitivity to part- and axis-based transformationsDenisova, K., Feldman, J., Su, X, & Singh, M. (in press)Vision Research The role of shape complexity in the detection of closed contoursWilder, J., Feldman, J., & Singh, M. (in press)Vision Research Perception of physical stability and center of mass of 3-D objectsCholewiak, S., Fleming, R. & Singh, M. (2015)Journal of Vision, 15(2):13, 1-11JOV link Contour complexity and contour detectionWilder, J., Feldman, J., & Singh, M. M. (2015)Journal of Vision, 15(6):6, 1-16.JOV link Visual representation of contour and shape Singh, M. (in press) In: J. Wagemans (Ed.), Oxford Handbook of Perceptual Organization. Oxford University Press. Preprint Perceptual grouping as Bayesian mixture estimation Feldman, J., Singh, M., & Froyen, V. (in press) In: S. Gepshtein, L. Maloney, & M. Singh (Eds.), Oxford Handbook of Computational Perceptual Organization. Oxford University Press. Preprint Transparency and Translucency Singh, M. (2014) In: K. Ikeuchi (Ed.), Computer Vision: A Reference Guide, pp. 815-819. Springer Verlag. Preprint Rotating columns: Relating structure-from-motion, accretion/deletion, and figure/ground Froyen, V., Feldman, J., & Singh, M. (2013) Journal of Vision, 13(10):6 1-12. JOV link Visual perception of the physical stability of asymmetric three-dimensional objects Cholewiak, S., Fleming, R., & Singh, M. (2013) Journal of Vision, 13(4):12, 1-13. JOV link Perceived causality can alter the perceived trajectory of apparent motion Kim, S.-H., Feldman, J., & Singh, M. (2013) Psychological Science, 24(4), 575-582. PDF Natural selection and shape perception Singh, M. & Hoffman, D. (2013) In: Shape Perception in Human and Computer Vision: An Interdisciplinary Perspective. S. Dickinson & Z. Pizlo (Eds.). Springer Verlag. Preprint Book An integrated Bayesian approach to shape representation and perceptual organization Feldman, J., Singh, M., Briscoe, E., Froyen, V., Kim, S. & Wilder, J. (2013) In: Shape Perception in Human and Computer Vision: An Interdisciplinary Perspective. S. Dickinson & Z. Pizlo (Eds.). Springer Verlag. Preprint Book A century of Gestalt psychology in visual perception: I. Perceptual grouping and figure-ground organization Wagemans, J., Elder, J., Kubovy, M., Palmer, S., Peterson, M., Singh, M., & von der Heydt, R. (2012) Psychological Bulletin, 138(6), 1172-1217. PDF Computational evolutionary perception Hoffman, D. & Singh, M. (2012) Perception, 41(9), 1073-1091. (Special Issue on the 30th anniversary of Marr's "Vision") PDF Principles of Contour Information: A response to Lim & Leek (2012) Singh, M. & Feldman, J. (2012) Psychological Review, 119(3), 678-683. PDF Curved apparent motion induced by amodal completion Kim, S.-H., Feldman, J., & Singh, M. (2012) Attention, Perception, & Psychophysics, 74(2), 350-364. PDF Perceptual models of viewpoint preference Secord, A., Lu, C., Finkelstein, A., Singh, M. & Nealen, A. (2011) ACM Transcations on Graphics, 30(5), 109:1-12. PDF Superordinate shape classificiation using natural shape statistics Wilder, J., Feldman, J., & Singh, M. (2011) Cognition, 119(3), 325-340. PDF Perceived object stability depends on multisensory estimates of gravity Barnett-Cowan, M., Fleming, R. W., Singh, M., & Buelthoff, H. H. (2011) PLoS ONE, 6(4), e19289, 1-5. PLoS ONE link Robust visual estimation as source separation Juni, M. Z., Singh, M., & Maloney, L. T. (2010) Journal of Vision, 10(14):2, 1-20. JOV link A Bayesian framework for figure-ground interpretation Froyen, V., Feldman, J., & Singh, M. (2010) Advances in Neural Information Processing Systems, 23. PDF How well do line drawings depict shape? Cole, F., Sanik, K., DeCarlo, D., Finkelstein, A., Funkhouser, T., Rusinkiewicz, S., & Singh, M. (2009) ACM Transactions on Graphics, 28(3) (proc. SIGGRAPH). PDF An experimental criterion for consistency in interpolation of partially-occluded contours Fulvio, J., Singh, M., & Maloney, L. T. (2009) Journal of Vision, 9(4):5, 1-19.JOV link Perceptual segmentation and the perceived orientation of dot clusters: The role of robust statistics Cohen, E., Singh, M., & Maloney, L. T. (2008) Journal of Vision, 8(7):6, 1-13. (Special Issue: Perceptual Organization and Neural Computation)JOV link Precision and consistency of contour interpolation Fulvio, J., Singh, M., & Maloney, L. T. (2008) Vision Research, 48, 831-849.PDF Natural decompositions of perceived transparency: Reply to Albert (2008) Anderson, B. L., Singh, M., & O'Vari, J. (2008) Psychological Review, 115, 1144-1153.PDF Geometric determinants of shape segmentation: Tests using segment identification Cohen, E. and Singh, M. (2007) Vision Research, 47, 2825-2840.Abstract | PDF The relationship between spatial pooling and attention in saccadic and perceptual tasks Cohen, E., Schnitzer, B., Gersch, T., Singh, M. and Kowler, E. (2007) Vision Research, 47, 1907-1923. Abstract | PDF Bayesian contour extrapolation: Geometric determinants of good continuation Singh, M. and Fulvio, J. (2007) Vision Research, 47, 783-798. Abstract | PDF Bayesian estimation of the shape skeleton Feldman, J. and Singh, M. (2006) Proceedings of the National Academy of Sciences, 103, 18014-18019. Abstract | PDF Perceived orientation of complex shape reflects graded part decomposition Cohen, E. H. and Singh, M. (2006) Journal of Vision, 6, 805-821. JOV link The role of part structure in the perceptual localization of a shape Denisova, K., Singh, M., & Kowler, E. (2006) Perception, 35, 1073-1087. Abstract | Perception link Contour extrapolation using probabilistic cue combination Singh, M. and Fulvio, J. M. (2006) Computer Vision and Pattern Recognition, Proceedings. Abstract | PDF | DOI link Consistency of location and gradient judgments of visually-interpolated contours Fulvio, J. M., Singh, M., & Maloney, L. T. (2006) Computer Vision and Pattern Recognition, Proceedings. Abstract | PDF | DOI link Surface geometry influences the shape of illusory contours Fulvio, J. M. and Singh, M. (2006) Acta Psychologica, 123, 20-40. (Special Issue: Michotte's heritage in perception and cognition research) Abstract | PDF Combining achromatic and chromatic cues to transparency Fulvio, J. M., Singh, M., & Maloney, L. T. (2006) Journal of Vision, 6, 760-776. JOV link Photometric determinants of perceived transparency Singh, M., and Anderson, B. L. (2006) Vision Research, 46, 879-894. Abstract | PDF The perceived transmittance of inhomogeneous surfaces and media Anderson, B. L., Singh, M., & Meng, J. (2006) Vision Research, 46, 1982-1995. Abstract |  PDF Visual extrapolation of contour geometry Singh, M., and Fulvio, J. M. (2005) Proceedings of the National Academy of Sciences, 102, 939-944. Abstract |  PDF (Supporting Information: pdf / PNAS website) Information along contours and object boundaries Feldman, J., and Singh, M. (2005) Psychological Review, 112, 243-252. Abstract | PDF What change detection tells us about the visual representation of shape Cohen, E. H., Barenholtz, E., Singh, M., & Feldman, J. (2005) Journal of Vision, 5, 313-321. JOV link Lightness constancy through transparency: Internal consistency in layered surface representations Singh, M. (2004) Vision Research, 44, 1827-1842. Abstract | PDF Modal and amodal completion generate different shapes Singh, M. (2004) Psychological Science, 15, 454-459. Abstract | PDF Computing layered surface representations: An algorithm for detecting and separating transparent overlays Singh, M., and Huang, X. (2003) Computer Vision and Pattern Recognition, Proceedings '03, Vol II, 11-18. Abstract | PDF Detection of change in shape: An advantage for concavities Barenholtz, E., Cohen, E., Feldman, J., & Singh, M. (2003) Cognition, 89, 1-9. Abstract |  PDF Vision: Form perception Hoffman, D., and Singh, M. (2002) In: Encyclopedia of Cognitive Science, Volume 4, L. Nadel (Ed.), 486-490. London: Macmillan Publishers Limited. Early computation of part structure: Evidence from visual search Xu, Y., and Singh, M. (2002) Perception and Psychophysics, 64, 1039-1054. Abstract |  PDF Toward a perceptual theory of transparency Singh, M., and Anderson, B. (2002) Psychological Review, 109, 492-519. Abstract |  PDF The interpolation of object and surface structure Anderson, B., Singh, M., & Fleming, R. (2002) Cognitive Psychology, 44, 148-190. Abstract |  PDF Perceptual assignment of opacity to translucent surfaces: The role of image blur Singh, M., and Anderson, B. (2002) Perception, 31, 531-552. Abstract | PDF Part-based representations of visual shape and implications for visual cognition Singh, M., and Hoffman, D. (2001) In: From fragments to objects: Grouping and segmentation in vision. Advances in Psychology Series,Volume 130. T. Shipley & P. Kellman (Eds.), 401-459. New York: Elsevier Science. Abstract | PDF Constructing surfaces and contours in displays of color from motion: The role of nearest neighbors Fidopiastis, C., Hoffman, D., Prophet, W., & Singh, M. (2000) Perception, 29, 567-580. Abstract | PDF Completing visual contours: The relationship between relatability and minimizing inflections Singh, M., & Hoffman, D. (1999) Perception and Psychophysics, 61, 943-951. Abstract | PDF Parsing silhouettes: The short-cut rule Singh, M., Seyranian, G., & Hoffman, D. (1999) Perception and Psychophysics, 61, 636-660. Abstract |  PDF Contour completion and relative depth: Petter's rule and support ratio Singh, M., Hoffman, D., & Albert, M. (1999) Psychological Science, 10, 423-428. Part boundaries alter the perception of transparency Singh, M., & Hoffman, D. (1998) Psychological Science, 9, 370-378.  Abstract  |  PDF Constructing and representing visual objects Singh, M., & Hoffman, D. (1997) Trends in Cognitive Sciences, 1, 98-102. Salience of visual parts Hoffman, D., & Singh, M. (1997) Cognition, 63, 29-78. Abstract | PDF