Spatially-global integration of closed contours by means of shortest-path in a log-polar representation

Transcription

1 Spatially-global integration of closed contours by means of shortest-path in a log-polar representation Terry Kwon and Kunal Agrawal, Yunfeng Li, Zygmunt Pizlo Support: National Eye Institute Purdue University 1

2 Outline Introduction Model Experiments Discussion 2

3 Introduction 3

4 Motivation Integrating fragmented, closed contours is difficult because there are many spurious contours. 4

5 Synthetic image illustrating the nature of the problem.

6 Prior models Most prior models, like Sha shua & Ullman s (1988) structural saliency, keep growing the contour based on local smoothness criteria. But these models do not guarantee that the most salient contour will be a closed curve. How to use local computations and guarantee closure? Change the problem s representation and use spatially-global mechanism. 6

7 Log-polar mapping of the retina in area V1 Tootell et al. (1982) A circle on the retina maps into an almost straight line in area V1. 7

8 Log-polar representation A circle on the retina maps into a straight line in area V1. A convex curve on the retina maps into a smooth open curve in area V1. One semi-axis on the retina has two representations in area V1. So, by solving the shortest path problem between a point and itself in the area V1, we (i.e., the visual system) are likely to detect a closed, 8 smooth and convex curve on the retina.

9 Examples of log-polar representation Three squares whose centers coincide with the center of the retina are represented by identical curves in the log-polar coordinates except for a translation along the horizontal or vertical axis. The log-polar representation is deformed by translation on the retina. But, local angles are always preserved (Conformal mapping). 9

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16 Shortest path model (Model SP) Generate a random polygon Fragment the contour Perturb the orientation of edges Add noise edges Transform back to the Cartesian coordinate Find the shortest (least-cost) path Transform into the log-polar representation 16

17 How the shortest path model works Convert contours to the log-polar representation. The origin of the polar coordinate system is the center of the image (fixation point). Existing edges have cost zero. Interpolated edges have cost equal to the Euclidean distance in the log-polar representation. It is cheaper to go through existing contours. Find the shortest (least cost) path from the starting point to itself. Transform it back to the Cartesian (retinal) coordinates. 17

18 Shortest path model (Model SP) misses concave parts of concave curves Object s contour Fragmented contour Fragmented contour with noise Reconstructed contour Shortest path in LPR Log-polar representation (LPR) 18

19 Local interpolation as the front-end helps (Model LI-SP) Fragmented contour Fragmented contour with noise Local Interpolation Reconstructed contour Shortest path in LPR Log-polar representation (LPR) 19

20 Experiment: presented at VSS last year Subjects drew the contours using a stylus. Their contours were very similar to the contours drawn by the model. 20

21 Experiment: presented at VSS this year The effect of the fixation position on detection of curves. 21

22 Effect of fixation points A closed curve is mapped to an open curve as long as the fixation point is inside. If the fixation point is outside, then the closed curve is projected to another closed curve in the log-polar representation. In this case, the shortest path in area V1 will not detect a closed curve on the retina. The model can find the shortest path only when the fixation position is inside. 22

23 Method 4 subjects (one naïve) Egg-like (oval) stimuli: x ±0.04x + y2 4 2 = 1 Fragmented and mixed with noise. 20 of jitter level of the edges in the egg: -25 to -15 or 15 to 25 Moderate level of perturbation to minimize the local interpolation based on smoothness 23

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25 Procedure We used a signal detection experiment. Each trial started with a fixation cross in the center of the screen. Right after the subject pressed a middle mouse button, the stimulus was shown for 100 msec. The subject was asked to report the direction of the egg with the mouse. A beep was sounded after an incorrect response. 25

26 Types of stimuli Question: How much difference exists between central and peripheral viewing of a closed curve? Confounding factor: Retinal eccentricity To control for this factor, we used big eggs for the central viewing and small eggs for peripheral viewing. 26

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28 Types of stimuli Another confounding factor: number of samples To control for this factor, we used sparse sampling for big eggs and dense sampling for small eggs. 28

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39 Weighted Average of Subjects' Sensitivity d' Dense Sampling Sparse Sampling 1 0 Central Viewing Peripheral 39

40 More results In my poster on Saturday. 40

41 Conclusions The shortest path model in area V1 implements 4 conventional Gestalt principles of perceptual organization: closure, proximity, good continuation and convexity. Local interpolation is the front-end This mechanism operates in the human visual system only when the fixation position is inside the closed curve (center of gravity tendency). Visual system solves a difficult problem by changing the problem s representation. This part of Gestalt s contribution was emphasized by Wertheimer and by Duncker in the context of insight problem solving.

42 Thank you.