Is perceptual space inherently non-Euclidean?

It is often assumed that the space we perceive is Euclidean, although this idea has been challenged by many authors. Here we show that, if spatial cues are combined as described by Maximum Likelihood Estimation, Bayesian, or equivalent models, as appears to be the case, then Euclidean geometry cannot describe our perceptual experience. Rather, our perceptual spatial structure would be better described as belonging to an arbitrarily curved Riemannian space.

A neural model for the integration of stereopsis and motion parallax in structure-from-motion.

We introduce a model for the computation of structure-from-motion based on the physiology of visual cortical areas MT and MST. The model assumes that the perception of depth from motion is related to the firing of a subset of MT neurons tuned to both velocity and disparity. The model's MT neurons are connected to each other laterally to form modulatory receptive-field surrounds that are gated by feedback connections from area MST. This allows the building up of a depth map from motion in area MT, even in absence of disparity in the input. Depth maps from motion and from stereo are combined by a weighted average at a final stage. The model's predictions for the interaction between motion and stereo cues agree with previous psychophysical data, both when the cues are consistent with each other or when they are contradictory. In particular, the model shows nonlinearities as a result of early interactions between motion and stereo before their depth maps are averaged. The two cues interact in a way that represents an alternative to the "modified weak fusion" model of depth-cue combination.

A reversed structure-from-motion effect for simultaneously viewed stereo-surfaces.

A spatially flat stimulus is perceived as varying in depth if its velocity structure is consistent with that of a three-dimensional (3D) object. This is structure from motion (SFM). We asked if the converse effect also exists. A motion-from-structure effect would skew an object's perceived velocity structure to make it more consistent with the 3D structure provided by its depth cues. This proposed phenomenon should be opposite in sign from velocity constancy and could potentially interfere with it. Previous tests of velocity constancy compared stimuli presented at different times, not simultaneously. This explains why a reversal of SFM has not been previously reported, as it is expected to appear only for simultaneous presentations. We tested this prediction using random-dot stereograms to define two adjacent moving surfaces separated in stereoscopic depth. We found that subjects did not perceive velocity constancy with either simultaneous or sequential stimulus presentations. For sequential presentations, subjects matched retinal speeds, in agreement with previous work. However, for simultaneous presentations, the nearer surface was seen as moving faster when both surfaces were moving with the same retinal speed, an effect opposite in polarity from velocity constancy and a signature of the motion-from-structure phenomenon.

Motion in depth from interocular velocity differences revealed by differential motion aftereffect.

There are two possible binocular mechanisms for the detection of motion in depth. One is based on disparity changes over time and the other is based on interocular velocity differences. It has previously been shown that disparity changes over time can produce the perception of motion in depth. However, existing psychophysical and physiological data are inconclusive as to whether interocular velocity differences play a role in motion in depth perception. We studied this issue using the motion aftereffect, the illusory motion of static patterns that follows adaptation to real motion. We induced a differential motion aftereffect to the two eyes and then tested for motion in depth in a stationary random-dot pattern seen with both eyes. It has been shown previously that a differential translational motion aftereffect produces a strong perception of motion in depth. We show here that a rotational motion aftereffect inhibits this perception of motion in depth, even though a real rotation induces motion in depth. A non-horizontal translational motion aftereffect did not inhibit motion in depth. Together, our results strongly suggest that (1) pure interocular velocity differences can produce motion in depth, and (2) the illusory changes in position from the motion aftereffect are generated relatively late in the visual hierarchy, after binocular combination.

Seeing motion in depth using inter-ocular velocity differences.

An object moving in depth produces retinal images that change in position over time by different amounts in the two eyes. This allows stereoscopic perception of motion in depth to be based on either one or both of two different visual signals: inter-ocular velocity differences, and binocular disparity change over time. Disparity change over time can produce the perception of motion in depth. However, demonstrating the same for inter-ocular velocity differences has proved elusive because of the difficulty of isolating this cue from disparity change (the inverse can easily be done). No physiological data are available, and existing psychophysical data are inconclusive as to whether inter-ocular velocity differences are used in primate vision. Here, we use motion adaptation to assess the contribution of inter-ocular velocity differences to the perception of motion in depth. If inter-ocular velocity differences contribute to motion in depth, we would expect that discriminability of direction of motion in depth should be improved after adaptation to frontoparallel motion. This is because an inter-ocular velocity difference is a comparison between two monocular frontoparallel motion signals, and because frontoparallel speed discrimination improves after motion adaptation. We show that adapting to frontoparallel motion does improve both frontoparallel speed discrimination and motion-in-depth direction discrimination. No improvement would be expected if only disparity change over time contributes to motion in depth. Furthermore, we found that frontoparallel motion adaptation diminishes discrimination of both speed and direction of motion in depth in dynamic random dot stereograms, in which changing disparity is the only cue available. The results provide strong evidence that inter-ocular velocity differences contribute to the perception of motion in depth and thus that the human visual system contains mechanisms for detecting differences in velocity between the two eyes' retinal images.

Computing relief structure from motion with a distributed velocity and disparity representation.

Recent psychophysical experiments suggest that humans can recover only relief structure from motion (SFM); i.e., an object's 3D shape can only be determined up to a stretching transformation along the line of sight. Here we propose a physiologically plausible model for the computation of relief SFM, which is also applicable to the related problem of motion parallax. We assume that the perception of depth from motion is related to the firing of a subset of MT neurons tuned to both velocity and disparity. The model MT neurons are connected to each other laterally to form modulatory interactions. The overall connectivity is such that when a zero-disparity velocity pattern is fed into the system, the most responsive neurons are not those tuned to zero disparity, but instead are those having preferred disparities consistent with the relief structure of the velocity pattern. The model computes the correct relief structure under a wide range of parameters and can also reproduce the SFM illusions involving coaxial cylinders. It is consistent with the psychophysical observation that subjects with stereo impairment are also deficient in perceiving motion parallax, and with the physiological data that the responses of direction- and disparity-tuned MT cells covary with the perceived surface order of bistable SFM stimuli.