1 Introduction

Motion transparency refers to the problem of detecting two or more local velocities in the image plane. The circumstances that lead to motion transparency are reasonably well understood eg [ 4 , 8 , 10 ]. They may occur owing to motion parallax: when observing the outside world through a dirty window pain, the motion of shadows or perhaps more interestingly at the location of occluding motion boundaries.

A number of computer-based models have addressed the detection motion transparency [ 10 , 9 , 2 ]. Uniformly, these models extend the idea that a single-valued velocity may be represented by a plane in frequency space[ 4 ] by adding further degrees of freedom so that the underlying model can detect two or more such planes. There are, however, some concerns to these ideas. For one, the models concentrate solely upon additive motion transparency. They often ignore the issue that many physical transparencies arise from multiplicative sources[ 8 , 4 , 5 ]. A second problem occurs under degeneracy. Circumstances where there is unreliable or insufficient information to uniquely constrain the models. Here, several models may explain the image data and one is faced with the problem of selection. A special case of this degeneracy occurs with the motion of two local 1-d image signals. These signals may be combined to define a unique motion vector or they may be kept separate as in a motion transparency.

These problems are highlighted by one's percept of motion to plaid patterns. A plaid is defined by the summation of two (moving) sinusoidal gratings. The images may appear either coherent or transparent. More interestingly, plaids may be seen transparent in two different ways. There is a linear transparency where the individual sinusoids are seen to move independently, or a multiplicative transparency composed of a contrast envelope (beat) and a carrier grating[ 3 , 5 ]. The perceived differences to plaid patterns are shown in figure 1. To see why there is a difference one should consider the properties of bandpass filters. As the figure shows, two sinusoids may always be arranged so that a single bandpass filter is sensitive to both. Here, a single linear filter cannot resolve the sinusoids and they interfere. The interference pattern leads to the multiplicative appearance[ 5 ]. However, when the two sinusoids differ markedly in frequency then the sinusoids can be separated by either scale or orientation and may be seen additively.

Broadly speaking, moving plaid patterns are seen to cohere or move rigidly when the combined image velocity vector is small, and when the two 1-d sinusoids are similar in contrast, temporal and spatial frequency. Plaid patterns are also coherent if the sinusoid's orientation difference is less than 90 [ 13 ]. If these conditions are not met, then they tend to be seen transparently. Our problem is to account for these data and address the question how one chooses one model, in preference to other models that each fit the image data. Moreover, given the notion that many vision problems are common to both artificial and biological systems, the ideas presented here, although motivated by empirical research address problems likely to be faced by artificial systems.

   
Figure 1: Additive/Multiplicative Plaids. (A) and (B) show two sinusoidal gratings of equal magnitude of spatial frequency but different orientation. (C) : An image produced from the multiplication of (A) and (B). Two luminance sinusoidal gratings can be seen. (D) : An image produced by the sum of (A) and (B). A vertical carrier and horizontal beat can be seen. (E) : Fourier power spectrum to (C). The ellipse shown in dotted lines represent the bandwidth of a linear filter. The filter is sensitive to just one component. (F) : Fourier power spectrum to (D). Here, the filter responds to both components.