Theoretical predictions of the perceived motion-direction of same-spatial-frequency plaids

Abstract

When two sinewave gratings of possibly different contrasts moving at possibly different velocities in different directions are superimposed (added point by point), the resultant looks like and is called a plaid. Whereas sinewave gratings viewed in circular apertures always appear to move perpendicularly to their stripes, their true motion is ambiguous because the motion component parallel to the stripes is invisible. However, every plaid has a unique representation as a moving rigid object, it can be equivalently produced by the translation of just a single frame. Here, we restrict the analysis of plaids to a narrow range of spatial frequency channels by considering only plaids composed of equal-spatial-frequency components, viewed foveally. The two plaid sinewave components vary independently in contrast (c₁,_c2), speed (temporal frequency, f₁,f₂), and angle (θ₁,θ₂), in an enormous 6-dimensional space. The theory assumptions are: (1) The first-order system processes plaid components of different angles independently and represents each component’s direction as a 2-dimensional vector of length c_j²/mj and direction θ_j ; m_j is the component’s modulation threshold; j=1,2. (2) The third-order system’s components are: a rigid-direction plaid-component with amplitude 2c_k², c_k=minimum(c₁,c₂), plus a sinewave component with amplitude |c₁–c₂|² and direction θ_~k, c_~k=maximum(c₁,c₂). (3) For the first- and third-order motion systems, the output motion-direction vectors Θ₁,Θ₃ are the vector sums of each system’s two component-direction vectors. Data: When both f₁,f₂ ≥10Hz (≥15Hz, one observer), only the first-order system is active. When f₁=1Hz, f₂ =2Hz, the third-order system dominates. (4) The net perceived direction Θ depends on the frequency-dependent relative strength α of the first- and third-order system outputs Θ1, Θ3: Θ= Θ1+αΘ3. This one-parameter-alpha-theory captures the essence of the data in full ranges of relative contrasts and of relative angles of the components, and a wide range of temporal frequencies. Precise data fits require more parameters.