Secondorder Motion

So far, we have discussed motion for gray-scale backgrounds and objects, where the moving object is a region of luminance that changes its position over time. The moving object can also be defined as a region of color that changes positions over time. Luminance and color are regarded as first-order properties of the image. Motion that is produced directly as changes in position over time of these first-order image properties is also called Fourier motion. Our discussion of motion has so far assumed that the brightness and texture of the moving object's retinal image are constant over time. Humans accurately and reliably detect the motion of objects despite dramatic changes of luminance (motion through shadows in a forest) and rapid changes in the surroundings (animals running through waving grass). For conditions like theses, any luminance-based procedure, such as computer algorithms that detect motion by finding regions with matching luminance (or even contrast) over time, would be severely degraded. In these and other cases, there are other invariant properties that define the objects and their motion. Stimuli that have motion that is defined by higher order parameters such as changes in contrast are called second-order motion stimuli (also called non-Fourier motion). We define a second-order motion stimulus to be a stimulus that on average has equal first-order motion energy in opposite directions for all spatial and temporal frequencies. Equivalently, for second-order motion stimuli, the responses of motion energy units tuned to opposite directions are on average equal. Thus, purely first-order mechanisms would be unable to detect second-order motion. The ability of humans to perceive motion in second-order stimuli can provide important evidence about the type of motion mechanisms the brain uses.

Charles Chubb and George Sperling showed that humans perceive motion for many types of second-order motion stimuli. A single frame of a typical second-order motion stimulus is shown in Fig. 7A, and an x-t plot of its motion is shown in Fig. 7B. Humans reliably perceive this second-order motion stimulus to be moving. Chubb and Sperling showed that if the luminance values of certain second-order motion stimuli were subjected to a nonlinear transformation (such as rectification, taking the absolute value, or simple squaring), then the resulting stimulus could be detected by a standard first-order mechanism. The effect of applying a nonlinear transformation (absolute value) to the stimulus is shown in Fig. 7C. Some researchers have suggested that there are two separate motion pathways: one that processes first-order motion using motion energy detectors and a second that consists of a nonlinear stage followed by motion energy detectors. Others have proposed that a single pathway containing an early nonlinear-ity is sufficient to explain human perception of second-order motion. Although some experiments have found differences in the perception of first- and second-order motion stimuli, evidence is accumulating that motion perception of most second-order stimuli can be explained by a single pathway with an early nonlinearity. Many experiments have shown that there are similarities between the perception of first- and second-order motion. For instance, if a first- or second-order motion stimulus has a contrast high enough to be detected, its direction of motion can be discriminated. First- and second-order motion stimuli produce adaptation effects that are similar and are independent of which type of motion was used as the adapter and the test. They also produce similar evoked potentials. Recent work by Ethan Taub, Jonathan Victor, and Mary Conte showed that direction-discrimination and speed-discrimination thresholds of both first- and second-order motion stimuli were accurately predicted by a single pathway with an early nonlinearity. They showed that an early nonlinearity not only allows the second-order motion stimulus to be processed by first-order mechanisms but also necessarily produces other signals that behave like masks and can strongly influence motion perception. Although these effects depend strongly on the type of early nonlinearity, they likely explain why many experiments find differences between the perception of first- and second-order motion. The

Figure 7 Non-Fourier motion. (A) An x-y plot of one frame of a second-order motion stimulus. This stimulus was generated by first creating a checkerboard of small squares and randomly choosing each square to be bright or dark and then multiplying each square's contrast by a sinusoid. This produces regions of very low contrast where the sinusoid is near zero (the approximately uniform gray regions) and regions of very high contrast where the sinusoid is near positive or negative one (the high-contrast speckled regions). (B) An x-t plot showing the second-order stimulus moving rightward. (C) The same moving second-order motion stimulus after it has been put through a nonlinearity. In this case the nonlinearity is the absolute value of the contrast. This operation ignores the sign of contrast so that bright and dark squares contrast become equally bright. This nonlinearity produces a moving rectified sinusoid.

Figure 7 Non-Fourier motion. (A) An x-y plot of one frame of a second-order motion stimulus. This stimulus was generated by first creating a checkerboard of small squares and randomly choosing each square to be bright or dark and then multiplying each square's contrast by a sinusoid. This produces regions of very low contrast where the sinusoid is near zero (the approximately uniform gray regions) and regions of very high contrast where the sinusoid is near positive or negative one (the high-contrast speckled regions). (B) An x-t plot showing the second-order stimulus moving rightward. (C) The same moving second-order motion stimulus after it has been put through a nonlinearity. In this case the nonlinearity is the absolute value of the contrast. This operation ignores the sign of contrast so that bright and dark squares contrast become equally bright. This nonlinearity produces a moving rectified sinusoid.

investigation of second-order motion stimuli has thus revealed the importance of nonlinear stages in early visual processing.

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