1. From cortical anisotropy to failures of 3-D shape constancy Qasim Zaidi Elias H. Cohen State University of New York College of Optometry

2. Shape Constancy Shape is the geometrical property of an object that is invariant to location, rotation and scale. The ability to perceive the shape of a rigid object as constant across viewpoints has been considered essential to perceiving objects accurately. The visual system does not discount all perspective distortions, so the shapes of many 3-D objects change with viewpoint. Can shape constancy be expected for rotations of the image plane? North viewSouth view The Future Building, Manhattan (Griffiths & Zaidi, 2000)

3. Vertical Oblique ConvexConcave Does rotating from vertical to oblique preserve perceived depth? 3-D Shape Constancy across image rotations?

4. Vertical Oblique ConvexConcave Stimuli Perspective projection of convex and concave wedges (in circular window). Experiment 1 compared 5 vertical shapes to 5 oblique shapes in depth (concave to concave & convex to convex). Shapes Sine-wave gratings 3 spatial frequencies, 1,3,6 cpd. Oriented at 90, ± 67.5, ± 45, & ± 22.5 degrees ( wrt 3D axis). Added in randomized phases to make 10 different textures per shape. Texture

5. Exp 1 Failures of 3-D shape constancy Vertical vs. Oblique comparison task. Subjects view two shapes sequentially. Which shape is greater in depth?

6. 500 msec

12. Exp 1: Shape Comparison Results The same shape was perceived to be deeper when it was oriented vertically than when it was oriented obliquely. Oblique shapes were matched to vertical shapes of 0.77 times depth of the oblique shape (S.E. =.007).

13. 3D Shape From Texture Perception of shape from texture depends on patterns of orientation flows (Li & Zaidi, 2001; 2004) Textured shape with no orientation component orthogonal to axis of curvature.

14. Is there a corresponding OB for single 2D angles? Origins of oblique bias for 3D shape Is the 3D OB explained by an OB for 2D oriented components?

15. Exp 2 Failures of 2-D angle constancy Vertical vs. Oblique comparison task. Subjects view two shapes sequentially. Which angle is sharper?

16. 500 msec

21. Exp 2: Angle Comparison Results The same angle was perceived to be sharper when it was oriented vertically than when it was oriented obliquely. Oblique angles were matched to vertical angles 4.5 ° shallower on average.

22. Predicting the 3-D depth bias from the 2-D angle bias The average ratio of perceptually equivalent 2-D slopes = 0.862 (SE =.001) Ratio of perceptually equivalent 3-D depths = 0.771 (SE =.007) 3-D depth inconstancy can be explained by anisotropy in perception of 2-D features. irrespective of h.

23. Orientation anisotropies in cat V1 cells (Li et al 2003) Oriented energy in natural images (Hansen & Essock, 2004)

24. Stimulus orientation decoded from cortical responses The probability that an orientation-tuned cell will give a spike in response to an orientation θ is determined by its tuning curve f(θ) (Sanger, 1996): The probability of the cell giving n i spikes is given by a Poisson distribution: For independently responding neurons, the probability of n i spikes each from k cells is given by the product of the probabilities:

25. Stimulus orientation decoded from cortical responses Using Bayes formula, the optimal estimate of the stimulus is the peak of the posterior probability distribution (P(θ) = Probability of θ in natural images) : Equivalently the peak of the log of the posterior: Given d i cells tuned to each orientation θ i the equation is grouped using average responses:

26. Stimulus angle decoded from cortical responses Using orientation tuned cells in V1, plus cross-orientation inhibition, we derived a matrix valued tuning function for (V4?) cells selective for angles  composed of two lines θ p and θ q : For the prior P(  ) we made the rough approximation: Finally, stimulus angles were decoded from the population responses of orientation tuned cells using an equation similar to that for orientations:

27. ASSUMPTION: Observer perceives an angle equal to the optimally decoded angle, i.e. the peak of the posterior probability distribution Stimulus angle 140º Decoded oblique angle 142º Decoded vertical angle 138º

28. From cortical anisotropy to shape inconstancy 1.We show an oblique bias for 3-D appearance. 2.The 3-D effect can be explained by an oblique bias for 2- D angles. 3.Simulations show that the anisotropy in orientation tuning of cortical neurons plus cross-orientation inhibition explains the 2-D oblique bias. 4.Anisotropy in numbers of cells predicts the opposite bias. 5.The predictions were insensitive to the prior distribution.

29. Consequences of the oblique bias for angle perception Zucker et al Fleming et al Cohen & Singh Tse

30. Conclusions 1.If the perception of 3D shape depends on the extraction of simple image features, then bias in the appearance of the image features will lead to bias in the appearance of 3D shape. 2.Variations in properties within neural populations can have direct effects on visual percepts, and need to be included in neural decoding models. REFERENCE Cohen EH and Zaidi Q Fundamental failures of shape constancy due to cortical anisotropy. Journal of Neuroscience (Under review).