In a variety of visual domains a distinction has been drawn between "first-order" stimulus which can be measured by the standard Fourier energy model and "second-order" stimuli which can not. A first-order motion stimuli is defined by spatio-temporal variations in luminance giving rise to a Fourier spectrum oriented through the frequency space origin. Second-order stimuli, such as the motion of a contrast modulation over a texture, have the property of being characterised in the spatio-temporal Fourier domain by power spectra which are displaced away from the frequency space origin. Observers can readily perceive second-order motions and it has been postulated that there exists separate channels specifically to process second order signals. These second-order detecting mechanisms rely on a pre-processing step which carries out an explicit rectification of the stimulus to recover a luminance signal that can then be prosessed with standard Fourier energy methods.
Our approach it to try and develop a single motion mechanism which can see both first and second-order motion signals. As can be seen in the figure below our motion model is capable of extracting a coherent motion signal from a number of commonly used second-order motion sequences. It should be noted that this is exactly the same model as was applied to other luminance based (first-order) motions, for example the traffic sequence . We may therefore conclude that a separate second-order channel is not required to account for human visual performance.
Johnston A, Benton C, & McOwan P W, Induced motion at texture-defined motion boundaries Proc. Royal Soc. Lond. B Nov 1999
The above figure shows the following. For each type of second-order stimulus (a-d) the left most panel shows a single, characteristic frame from the motion sequence. The next panel shows a space-time plot of the stimulus, space horizontally, and time increasing vertically downwards. The third panel shows the temporal derivative of the space time plot. The fourth panel shows the log of speed measures obtained by applying the motion model, the recovered speed values are scaled to the full image brightness. Finally the panel on the right shows the direction of motion, which should be read with reference to the surrounding colour wheel. Each of the motion sequences is moving rightwards (in the green direction)
The second-order motion sequences shown are as follows