E34 Movement and trajectory

Topic

Detecting and estimating motion—whether of the organism itself or of an external object—is one of the most fundamental computations performed by the visual system. The Horn-Schunck optical flow equation establishes the basic constraint underlying this detection: the premise that the intensity of a specific point in the image remains constant between two successive moments in the scene. However, this constraint is ambiguous when applied to a single edge of a specific orientation—a phenomenon known as the "aperture problem"—which makes it impossible to fully determine the motion based solely on that isolated edge.

The Lucas-Kanade estimator resolves this ambiguity by aggregating the constraint over a larger spatial window of the image and then solving for motion using a least-squares approach across that entire window. Consequently, the estimator yields robust results precisely when the image exhibits gradients in multiple directions simultaneously, such as at a corner or within a richly textured area of ​​the scene. The vertebrate visual system implements a computation very similar to the Lucas-Kanade estimator, operating across the hierarchy of visual areas established by another model in this framework. The presence of gradients in multiple directions within the spatial window results in a more robust and precise motion estimate according to the Lucas-Kanade method. "Time-to-impact," on the other hand, is a particularly useful optical invariant for tasks such as controlling a landing, avoiding an obstacle, or capturing prey; this invariant does not require separate knowledge of the distance to the observed object—a parameter established by another model in this set—or the speed of approach to that object, but rather relies solely on the object's angular expansion, which is directly observable on the organism's retina. Consequently, a bee uses this time-to-impact to regulate its landing speed, whereas a hawk or a locust uses it to capture moving prey with precision. As the object's angular expansion on the retina accelerates, the time-to-impact—according to this optical invariant—becomes progressively shorter.

Meanwhile, the Reichardt model serves as the motion detector implemented within an insect's visual system via specialized neurons; it compares the signal from a retinal receptor with a time-delayed signal from a neighboring receptor, such that the product of the two signals detects a spatiotemporal correlation characteristic of motion in a specific direction. Thus, the Reichardt model is directionally selective by design and is free from the contrast ambiguities that might otherwise affect the Lucas-Kanade estimator mentioned earlier. When the neighboring retinal receptor receives the delayed signal at the precise time interval corresponding to a specific direction of motion, the product of the two signals—according to the Reichardt model—reaches its maximum value for that specific direction. Consequently, this model—characterized by the specific motion detection and time-to-impact parameters involved—establishes proportional navigation as the guidance law that maximizes the probability of intercepting a moving target; this law adjusts the pursuer's lateral acceleration in direct proportion to the rotation rate of the line of sight connecting the two, meaning that if the line of sight is still rotating, the moving target is not yet on a path to successful interception, and this lateral correction is therefore necessary. Dragonflies and hawks employ this same proportional navigation, utilizing an internal gain very close to the theoretical optimum for this guidance law. Furthermore, if the line of sight between the pursuer and the moving target rotates more rapidly, the necessary corrective lateral acceleration—dictated by this proportional navigation—must also increase.

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