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Curvature Blindness from Polarity Breaks and Orientation Channel Fragmentation in V1

Michael Menke

Abstract

We present a mathematical model of the curvature blindness illusion in which sinusoids appear as angular zigzags when drawn with alternating contrast polarity against a gray background. The model identifies two complementary mechanisms, both operating in V1. First, polarity channel separation: simple cells are selective for contrast polarity, and lateral connections link only same polarity neurons; where the line switches from darker than background to lighter than background at each peak and trough, the encoding population changes and the lateral chain is broken, segmenting the contour into half-wavelength pieces. Second, orientation channel fragmentation: at moderate contrast, the active orientation window is narrow, and within each half-wavelength segment no single orientation channel spans the full range of edge normals; the inflection point at the center of each segment anchors a locally straight percept. Together, the two mechanisms produce a zigzag: polarity breaks supply the corners, and fragmentation straightens the segments between them.

Curvature Blindness from Polarity Breaks and Orientation Channel Fragmentation in V1

Abstract

We present a mathematical model of the curvature blindness illusion in which sinusoids appear as angular zigzags when drawn with alternating contrast polarity against a gray background. The model identifies two complementary mechanisms, both operating in V1. First, polarity channel separation: simple cells are selective for contrast polarity, and lateral connections link only same polarity neurons; where the line switches from darker than background to lighter than background at each peak and trough, the encoding population changes and the lateral chain is broken, segmenting the contour into half-wavelength pieces. Second, orientation channel fragmentation: at moderate contrast, the active orientation window is narrow, and within each half-wavelength segment no single orientation channel spans the full range of edge normals; the inflection point at the center of each segment anchors a locally straight percept. Together, the two mechanisms produce a zigzag: polarity breaks supply the corners, and fragmentation straightens the segments between them.
Paper Structure (20 sections, 6 theorems, 8 equations, 2 figures)

This paper contains 20 sections, 6 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The illusion
  3. The main ideas
  4. The Model
  5. V1 orientation tuned edge detectors
  6. The linear response at an edge
  7. Divisive normalization
  8. The active orientation window
  9. Lateral connections: orientation and polarity
  10. Curvature Blindness
  11. The Takahashi stimulus
  12. Polarity channel separation at peaks and troughs
  13. Geometry of a half wavelength segment
  14. Orientation channel fragmentation within a segment
  15. The role of inflection points
  16. ...and 5 more sections

Key Result

Proposition 2.1

At an edge with normal $\theta_n$ and Michelson contrast $c$, a neuron of matching polarity at orientation $\theta$ is active if and only if $|\theta - \theta_n| \leq \alpha(c)$, where provided $\tau_{\mathrm{eff}}(c) \leq 1$. If $\tau_{\mathrm{eff}}(c) > 1$, no neuron is active and the edge is invisible. A neuron of the opposite polarity is never active at this edge.

Figures (2)

  • Figure 1: The curvature blindness illusion (from Takahashi takahashi17). Sinusoidal curves with alternating dark/light segments appear smooth against the white and black backgrounds, but appear as angular zigzags against the gray background.
  • Figure 2: Curvature blindness generalized to a non-sinusoidal curve. The waveform is $y = 4\sin(2\pi x/\lambda) + 5\sin(6\pi x/\lambda).$ The longer segments between extrema still read as curved while the shorter segments are straight. This is consistent with the model as the longer segments have lower average curvature and hence a larger visible window allowing residual curvature to be seen.

Theorems & Definitions (15)

  • Proposition 2.1: Active orientation window
  • proof
  • Proposition 3.1: Polarity segmentation
  • proof
  • Remark 3.2: Why the illusion requires a gray background
  • Proposition 3.3: Half segment geometry
  • proof
  • Proposition 3.4: Within segment fragmentation
  • proof
  • Proposition 3.5: Inflection points anchor straight line perception
  • ...and 5 more