Leonid L. Kontsevich

Bayesian adaptive estimation of psychometric slope and threshold.

We introduce a new Bayesian adaptive method for acquisition of both threshold and slope of the psychometric function. The method updates posterior probabilities in the two-dimensional parameter space of psychometric functions and makes predictions based on the expected mean threshold and slope values. On each trial it sets the stimulus intensity that maximizes the expected information to be gained by completion of that trial. The method was evaluated in computer simulations and in a psychophysical experiment using the two-alternative forced-choice (2AFC) paradigm. Threshold estimation within 2 dB (23%) precision requires less than 30 trials for a typical 2AFC detection task. To get the slope estimate with the same precision takes about 300 trials.

Predominantly extra-retinotopic cortical response to pattern symmetry

Symmetry along one or more axes is a key property of objects and biological organisms. We report on a bilateral visual region of occipital cortex that responds strongly to the presence of multiple symmetries in the viewed image. The stimuli consisted of random dots organized in fourfold and onefold mirror-symmetric patterns, against random control stimuli. The contrast between symmetric and random patterns produced negligible or inconsistent activation of the primary visual projection area V1 or of other medial occipital projection areas. However, there was strong symmetry-specific activation in extra-retinotopic lateral occipital cortex. The high level of activation in this region of cortex may represent part of a general class of computations that require integration of information across a large span of the visual field.

What makes Mona Lisa smile

To study the ability of humans to read subtle changes in facial expression, we applied reverse correlation technique to reveal visual features that mediate understanding of emotion expressed by the face. Surprising findings were that (1) the noise added to a test face image had profound effect on the facial expression and (2) in almost every instance the new expression was meaningful. To quantify the effect, we asked naïve observers to rank the face of Mona Lisa superimposed with noise, based on their perception of her emotional state along the sad/happy dimension. Typically, a hundred trials (with 10 or more samples for each rank category) were sufficient to reveal areas altering the facial expression, which is about two orders of magnitude less than in the other reverse correlation studies. Moreover, the perception of smiling in the eyes was solely attributable to a configurational effect projecting from the mouth region.

Separating the effects of response nonlinearity and internal noise psychophysically.

A psychophysical method is proposed to separate the contrast dependence of internal response and its noise. The resulting contrast relationships represent a signature of the visual processing stage that limits the human observers performance. The method was applied to contrast discrimination for sustained and transient Gabor patches with a 3 cycle/deg spatial carrier. For both stimulus types the predominant noise was found to be multiplicative with a power exponent of 0.76-0.85 and the source of this noise preceded by an accelerating signal transducer with a power of 2-2.7. These exponents combine to account for the classic compressive power of about 0.4 for the signal-to-noise ratio in contrast discrimination. The estimated transducer acceleration suggests that there is a direct computation of contrast energy in the visual cortex.

The specificity of cortical region KO to depth structure

Functional MRI studies have identified a cortical region designated as KO between retinotopic areas V3A/B and motion area V5 in human cortex as particularly responsive to motion-defined or kinetic borders. To determine the response of the KO region to more general aspects of structure, we used stereoscopic depth borders and disparate planes with no borders, together with three stimulus types that evoked no depth percept: luminance borders, line contours and illusory phase borders. Responses to these stimuli in the KO region were compared with the responses in retinotopically defined areas that have been variously associated with disparity processing in neurophysiological and fMRI studies. The strongest responses in the KO region were to stimuli evoking perceived depth structure from either disparity or motion cues, but it showed negligible responses either to luminance-based contour stimuli or to edgeless disparity stimuli. We conclude that the region designated as KO is best regarded as a primary center for the generic representation of depth structure rather than any kind of contour specificity.

MECHANISMS OF STEREOSCOPIC PROCESSING: STEREOATTENTION AND SURFACE PERCEPTION IN DEPTH RECONSTRUCTION

Consideration of the range of phenomena from studies of human stereopsis suggests that a five-stage model is required to provide a complete account of the processes involved, within which any stereoattention mechanism must operate. The information from the disparity field of the optical projections to the two eyes (stage 1) goes to a set of parallel Keplerian arrays of disparity detectors, each array selective for a different spatiotemporal property of the visual images (stage 2). Global interactions produce a cyclopean depth image that is cleaned of the spurious ghost images in the Keplerian arrays (stage 3) and that may then be processed for its (hypercyclopean) form elements (stage 4). Finally, there must be a stage of integration of the stereoscopic depth cues with monocular and kinesthetic depth cues to form the overall map of perceived distance (stage 5). The fact that multiple cyclopean surfaces may be perceived as transparent implies that the stereoscopic system is not limited by a singular-surface constraint. However, it is unclear whether multiple surfaces can be seen simultaneously or whether only one surface is seen at a time by a selective-attention process, with the others perceived as a purely inchoate (qualitative) depth impression. New experiments on cueing of ambiguous stereocorrugations by singular flat planes suggest that selective stereoattention is a powerful mechanism. In fact, the results show that attention can be focused not just in horopteral planes but in a variety of depth configurations. Moreover, this attention focus may act as a tracking mechanism to allow perception of smooth cyclopean stereomotion, which has a frequency response up to ~ 5 Hz (in contrast to the ~15 Hz limit for detecting planar disparity shifts as jerky appearance and disappearance effects). Finally, the spatial limits of stereosurface reconstruction are explored with cyclopean targets to show some interesting asymmetries of the surface-wrapping process that may represent object-oriented constraints on depth reconstruction.

Extended Concepts of Occipital Retinotopy

Retinotopic mapping is a key property of organization of occipital cortex, predominantly on the medial surface but increasingly being identified in lateral and ventral regions. The retinotopic organization of early visual areas V1-3 is well established, although anatomical landmarks can help to resolve ambiguities in poorly-defined functional maps. New morphing techniques are now available to define the metric mappings quantitatively within each retinotopic area. In the dorsal occipital regions, there is fair agreement that area V3A should be split into separate V3A and V3B maps, and that beyond them lies a further area, V7. We specify the eccentricity mapping of both V3B and V7 for the first time, showing how the latter is roughly parallel to the meridional mapping and offering formal accounts of such paradoxical behavior. In ventral occipital cortex, we support the analysis of Zeki and Bartels (1) and Wade et al. (2) that V4 maps the full hemifield, and show the existence of two more areas, a ventromedial map of the lower quadrant, emphasizing the upper vertical meridian, and an adjacent area with a dominant foveal representation. In lateral cortex, the motion area defined by a motion localizer shows pronounced retinotopy, particularly in the eccentricity parameter. A dorsolateral map between the motion area and V3B, which represents the lower quadrant with an emphasis the foveal part of the lower vertical meridian, may be a counterpart to the ventromedial map.

Analysis of stereothresholds for stimuli below 2.5 c/deg

We analyze published data on disparity detection thresholds for a wide range of conditions. This type of detection changes behavior dramatically at the spatial frequency of 2.5 c/deg; above this frequency threshold remains constant while below it threshold grows at a uniform rate. Many other types of threshold, such as upper disparity limits for depth perception and threshold amplitudes for stereo and monocular motion, show similar behavior. These data lead to the postulate that there are no foveal stereo channels peaking below 2.5 c/deg, so that foveal stimuli in the whole range below 2.5 c/deg are processed by a single channel tuned to this frequency. Consequently, disparity detection thresholds at frequencies below this frequency are controlled by the single parameter of effective contrast in the 2.5 c/deg channel, whose output depends jointly on the contrast and spatial frequency of the stimuli. We develop this idea to explain the relations between spatial and contrast tuning functions for disparity thresholds. To validate our conclusions, we describe an experiment with difference-of-Gaussian stimuli over a range of interocular widths and contrast differences. For a dichoptic width ratio of 2:1, the dichoptic contrast ratio required to minimize disparity detection thresholds was 1:4, just as predicted by the model.

Distraction of attention and the slope of the psychometric function

The influential uncertainty model [J. Opt. Soc. Am. A 2, 1508 (1985)] attributes nonlinear contrast sensitivity near threshold to the inability of the observer to discriminate between the signal from stimulated locations and the noise from nonstimulated locations. We introduce an alternative interpretation, the distraction model, to describe the behavior of an observer who knows exactly which location was stimulated but may miss the test stimulus because attention has been distracted by irrelevant (noise) signals. For any stimulus sample, the observer is assumed to be certain of whether this sample is relevant or irrelevant to the stimulus. The non-linear effects predicted by the distraction model without uncertainty are similar to those predicted by the uncertainty model.

Reconstruction of shape from shading in color images

We propose a method for shape reconstruction from color shades produced by multiple chromatic light sources. The linear relation between surface-normal vectors and three-dimensional response vectors for a uniformly colored and illuminated region of a surface can be reconstructed in two steps. In the first step a quadratic form of metric in response space induced from a natural metric in normal space is reconstructed. At this stage proper image segmentation can be obtained. In the second step an exact mapping from response space into the space of surface normals is reconstructed. The matrix for this mapping is one of the square roots of the quadratic-form matrix that satisfies the integrability constraint. The method is in all respects much simpler than existing methods for solving the depth-from-shading task for monochromatic images.