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. 2026 Jun 24;26(6):7. doi: 10.1167/jov.26.6.7

Visual information for the identification of nonrigid motion

James T Todd 1,1
PMCID: PMC13313045  PMID: 42339982

Abstract

van Zuijlen et al. (2025) have recently argued that the perceptual distinction between rigid and non-rigid motions of a torus knot can be based on patterns of optical flow produced by systematic changes in image shading. The present commentary will present theoretical analyses and empirical demonstrations to show that the 2D vector flow field produced by smoothly shaded surfaces is relatively useless as a source of information about rigid and nonrigid motion when an object moves relative to its sources of illumination. The evidence to support this conclusion is that the nonrigid distortions can easily be identified from deforming smooth occlusion contours when all variations in shading are removed. Although a rotating shaded surface without smooth occlusions can be perceived as rigid, that perception is eliminated entirely if its flow field is presented as a pattern of moving lines without any gradients of shading. The computation of 2D image velocities is often taken for granted by researchers in visual perception, but the theoretical underpinnings of that approach are highly questionable when applied to untextured shaded surfaces that move relative to their sources of illumination.

Keywords: optical flow, shading, shape and contour

Human observers have a remarkable ability to identify different categories of motion such as bouncing, rotating, running, or melting, but little is known about the visual information on which those judgments are based, except in a few specialized contexts. One of these exceptions is the perceptual distinction between rigid and nonrigid motion. Several different computational models have been developed for distinguishing those distinct categories of motion (e.g., Hogervorst, Kappers, & Koenderink, 1996; Koenderink & van Doorn, 1991; Todd; 1982; Ullman (1977), and there have been numerous psychophysical investigations that have examined the abilities of human observers to discriminate those categories (e.g., Braunstein, Hoffman & Pollick, 1990; Choi, Feldman & Singh, 2024; Dosher, Landy, & Sperling, 1989; Jain & Zaidi, 2011; Koerfer & Lappe, 2022; Norman & Todd, 1993; Norman & Todd, 1994; Perotti, Todd & Norman, 1996; Todd, 1982; Todd, 1984; Todd, 1985; Todd & Bressan, 1990).

The most recent study to consider this issue was reported by van Zuijlen, Yang, van Assen, & Nishida (2025). Their displays depicted shaded images of a rotating torus knot, either with or without a nonrigid compression transformation of variable magnitude that was perpendicular to the direction of rotation. Similar displays are shown in Figure 1, and in Animation S1 of the supplemental materials. On each trial, observers were shown two animations of a torus knot undergoing rigid or nonrigid motion in successive intervals, and they were asked to judge which one appeared more nonrigid. They also varied the three-dimensional (3D) orientation of the axis of rotation, and the material properties of the depicted surfaces including diffuse reflections, specular reflections, and transparency. The results revealed that these manipulations had little influence on performance.

Figure 1.

Figure 1.

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Static views of a compressed and uncompressed torus knot. Animation S1 in the supplemental materials shows these objects undergoing rigid and nonrigid motion.

The ability of observers to detect nonrigid deformations under these conditions is particularly interesting because the displays violated the underlying assumptions of all the computational models cited above. The authors argue that observers’ judgments may somehow be determined by patterns of optical flow, but they provide no specific models of how that might be possible. In the present commentary I will argue that the two-dimensional (2D) vector flow field produced by smoothly shaded surfaces is relatively useless as a source of information about rigid and nonrigid motion when an object moves relative to its sources of illumination. The evidence to support this conclusion is that the nonrigid distortions can easily be identified from deforming smooth occlusion contours when all variations in shading are removed. Although a rotating shaded surface without smooth occlusions can be perceived as rigid, that perception is eliminated entirely if its optical flow is presented as a pattern of moving lines without any gradients of shading. These observations suggest that the relevant information for identifying distorting torus knots includes the optical deformations of smooth occlusion contours and systematic changes over time in the perception of 3D shape from shading.

Projective correspondence

Most of the early research on the perception of 3D structure and rigidity from motion was focused on identifiable points on an object (e.g., the vertices of polyhedra, or small patches of texture) whose optical projections can be tracked over time in a 2D image. A strong implicit assumption of that research is that there is a projective correspondence between objects and images, such that multiple views of an identifiable image feature must always correspond to the same physical point on an object's surface.

Within that context, there are a couple of different sources of information that could potentially be used to distinguish rigid and nonrigid motions in depth. One approach proposed by Ullman (1977) and Koenderink and van Doorn (1991) is to decompose the projected motion into a moving frame of reference, in which all points move along parallel trajectories. This is always possible for the projected motions of rigid rotations in 3D space with small amounts of perspective, and it is generally not possible for nonrigid deformations of an object. Another approach proposed by Todd (1982) and Hogervorst et al. (1996) highlights constraints on the global trajectories of individual points. If the motion is rigid, they must all oscillate around a common axis at the same frequency. One advantage of the first approach is that it can detect nonrigid motions from only two frames of a motion sequence, whereas the alternative models require information from a large portion of an object's trajectory. However, the trajectory-based approach can detect some non-rigid motions that would be considered as rigid by a 2-frame analysis.

The motion sequences depicted in Figure 2 and in Animation S2 of the supplemental materials are designed to exemplify this distinction. The top panel shows a sequence of still frames for a wireframe figure rotating rigidly in depth. The lower left panel shows a pattern of lines whose endpoints all oscillate around different axes. Either of the proposed models would be able to detect the nonrigid distortion in that case. The lower right panel shows a pattern of lines whose endpoints all oscillate around the same axis but at different frequencies. A trajectory-based model can detect the nonrigid distortion in that case, but a two-frame analysis cannot.

Figure 2.

Figure 2.

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Static views from three motion sequences of wireframe figures. Top row: Rigid rotation in depth. (Lower left) Nonrigid motion where the points oscillate in different directions. (Lower right) Nonrigid motion where the points oscillate in parallel directions. Videos of these events can be observed in Animation S2 of the supplemental materials.

There is some empirical evidence that human observers can exploit either of these different sources of information. Perotti et al. (1996) obtained rigidity judgments for moving patterns similar to the ones shown in Figure 2, and they also varied the number of distinct frames in the depicted motion sequences. When the displays were reduced to just two frames that alternated back and forth, the observers reliably identified the motion in the lower left panel as nonrigid, and the other motions were reliably judged as rigid. However, as the number of frames was increased to reveal a larger portion of the entire trajectory, then the motion in the lower right panel appeared more and more nonrigid.

Violations of projective correspondence

A serious problem for either of these approaches is that there are many commonly observed aspects of optical motion that do not satisfy the assumption of projective correspondence and cannot therefore be successfully analyzed using existing computational models. For example, consider an ellipsoid surface rotating rigidly in depth about its minor axis as shown in Figure 3 and Animation S3 in the supplemental materials. Note that there are three distinct features within the optical projection of this object: (1) a red stripe painted on the surface, (2) a bright white specular highlight, and (3) the smooth occlusion boundary that separates the surface from its background. As the ellipsoid rotates from left to right, the projection of the red stripe moves in one direction, but the specular highlight and the smooth occlusion contour move in the opposite direction. As shown in the animation, these three types of features all transform in a completely different manner.

Figure 3.

Figure 3.

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Static views of a rotating ellipsoid that depict three different types of optical motion from a surface reflectance contour, a smooth occlusion boundary, and a specular highlight. A video of this event can be observed in Animation S3 of the supplemental materials.

Measures of optical flow from dynamic changes in image shading were first developed in the 1980s (Horn & Schunck, 1981; Lucas & Kanade, 1981; Nagel, 1981; Nagel, 1987), and they are all based on an assumption of projective correspondence. This is often described as a brightness constancy constraint, where it is assumed that the luminance of each local surface region remains constant over time. What these analyses are designed to measure are the movements of individual pixel intensities. As long as local luminance remains constant, then the movements of pixel intensities will be a reliable measure of the projected motions of individual surface points. The problem with this approach is that the brightness constancy constraint is frequently violated in the natural environment. For the display shown in Figure 3, for example, the assumption is satisfied for the motion of the red stripe, but not for the deformations of smooth occlusion contours, diffuse shading, or specular highlights.

The existence of different types of transformations that are all superimposed in the same display raises an interesting question of how they should be perceptually represented. Should they all be mushed together into a single vector field that would be difficult to analyze by any existing model, or should they be perceptually segregated so that each type of transformation can be analyzed by a specialized processing module?

Object motion versus observer motion

Norman and Todd (1994) identified five different types of image features including surface texture, cast shadows, diffuse reflections, specular reflections, and smooth occlusion contours that produce very different patterns of optical motion. They also noted that the motions of some of these features can change dramatically depending on whether an observer is moving relative to a fixed environment or an object is moving relative to a fixed observer and pattern of illumination.

For example, the left panel of Figure 4 shows a matte surface with diffuse (Lambertian) shading, and the right panel shows the iso-luminance contours in that image that connect points with identical luminance. Patterns of iso-contours (which are also referred to as level sets) are a particularly convenient way of representing the structure of the luminance field, while stripping away the relevant information for the perception of 3D shape from shading (see Norman & Todd, 2025). They can be generated quite easily in an image editing application (e.g., Adobe Photoshop or After Effects) by posterizing an image into bands of uniform intensity and then applying an edge filter. Animation S4 of the supplemental materials shows how these patterns change over time as the camera orbits around the depicted object, or the object rotates relative to a fixed camera.

Figure 4.

Figure 4.

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A shaded image of a smoothly curved matte surface (left) and its pattern of iso-luminance contours (right). Animation S4 of the supplemental materials shows how these patterns change over time as the object rotates in depth about a vertical axis or the camera orbits around the object.

The key thing to note in these animations is that movement of the observer has no effect on diffuse shading. Thus, when the camera moves, the iso-luminance contours remain at fixed locations on the object's surface, which satisfies the assumption of projective correspondence. This is what allows shading to be baked as a texture in virtual reality displays to avoid having to perform complex lighting computations in real time. Note in the camera motion animation of the iso-luminance contours that the object appears to rotate rigidly in depth relative to the point of observation, and that provides a strong perception of 3D structure from motion.

However, when the object rotates relative to a fixed camera (and the sources of illumination) the iso-luminance contours do not remain at fixed locations on the object's surface. This creates a more complex pattern of optical motion with numerous sources and sinks that are incompatible with rigid motion in depth. The luminance flow field in that case does not provide a valid representation of how the actual object is moving relative to the observer. It is important to keep in mind that the image velocities in the shaded animation have the same overall pattern of optical flow as its corresponding contour display (although the shaded one is more densely sampled). The fact that their depicted motions are perceived quite differently provides strong evidence that the vector field of image velocities may not be a relevant source of information for the perceived rigid motion when an object moves relative to its sources of illumination.

The left panel of Figure 5 shows a metal surface with purely specular shading, and the right panel shows the iso-luminance contours in that image. Animation S5 of the supplemental materials shows how these patterns change over time as the camera orbits around the depicted object, or the object rotates relative to a fixed camera. Unlike diffuse shading, specular reflections do not remain invariant over movements of the observer. This is quite evident in the camera motion animation of the iso-luminance contours, which has a perceptually incoherent structure. For the luminance flow of specular surfaces there is no qualitative distinction between movements of an object and movements of the observer as there is for the flow of Lambertian surfaces.

Figure 5.

Figure 5.

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A shaded image of a smoothly curved metal surface (left) and its pattern of iso-luminance contours (right). Animation S5 of the supplemental materials shows how these patterns change over time as the object rotates in depth about a vertical axis or the camera orbits around the object.

Smooth occlusion contours

Another interesting aspect of optical flow that violates the assumption of projective correspondence is the deformation of an object's smooth occlusion contour, which is sometimes referred to as the rim (Koenderink, 1984). The rim is defined as the locus of points on an object whose surface normals are all perpendicular to the line of sight. The silhouette is a subset of the rim that separates an object from its background. It excludes interior points where regions of a surface are occluded by other parts of the same object. Objects that are uniformly convex have no internal occlusions so that the rim and the silhouette are the same, but that is not the case for objects that contain concave regions.

The earliest research on the perceived deformations of smooth occlusion contours was focused on objects that are uniformly convex. For example, Ernst Mach (1886/1959) observed that the silhouettes of smoothly curved objects such as eggs are perceived as a nonrigid fluid-like motion. Similar observations were reported many years later by Wallach and O'Connell (1953). However, Todd (1985) showed that the deforming silhouettes of more complex objects can produce a compelling perception of rigid rotation in depth, and subsequent studies have confirmed that deforming silhouettes provide useful information for the perception of 3D shape (e.g., Norman, Dawson & Raines, 2000; Norman et al., 2016; Norman & Raines, 2002; Norman, Todd & Orban, 2004), the axis of rotation (Pollick, Giblin, Rycroft, & Wilson, 1992), and distinguishing rigid from nonrigid motion (Norman & Todd, 1994). There are no studies of which I am aware that have examined the motions of the entire rim.

There have also been several theoretical analyses to show how it is possible to compute the 3D shape and rigidity of an object from deformations of its silhouette (Cheung, Baker & Kanade, 2005; Cipolla, 1998; Cipolla, Astrom & Giblin, 1995; Giblin, Pollick & Rycroft, 1994). Many of these analyses have restrictive conditions that are not characteristic of human perception, like requiring known observer motion relative to a fixed object. Nevertheless, they provide clear mathematical proof that deforming silhouettes provide potentially useful visual information. It is also important to note that none of these models can function effectively on a generalized flow field that combines several different types of transformations arising from the deformations of texture, diffuse shading, cast shadows, or specular highlights. They all require that the silhouette must first be isolated from those other types of optical deformations.

Let us now examine how the deformations of smooth occlusion contours may have influenced the rigidity ratings in the experiments by van Zuijlen et al. (2025). Figure 6 shows the rim of a torus knot on the left and a silhouette of the same object on the right and videos of these displays can be observed in Animation S6 of the supplemental materials. The rim animation was created using a toon renderer. The key thing to note in these animations is that the deformations of smooth occlusion contours provide sufficient information to perceptually distinguish rigid and non-rigid motion even when all variations in smooth shading have been removed. Thus, it is quite possible that the measures of luminance flow that the authors reported may have had little or no impact on the observers’ judgments.

Figure 6.

Figure 6.

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The rim of a torus knot (left) and its silhouette (right). Animation S6 of the supplemental materials shows these objects undergoing rigid and nonrigid motion.

It is interesting to note when observing Animation S6 that that there is a subtle difference between the deforming rims and silhouettes. If you stare at the silhouettes for an extended period, the direction of rotation in depth can spontaneously reverse, but that does not occur for the deforming rims. The key source of information about the direction of rotation is provided by T-junctions along the rim. That information is eliminated in the silhouettes because filling the internal regions of an object with a uniform intensity converts all visible T-junctions to V-junctions, which are much less informative.

Moving surfaces without smooth occlusions

Given that smooth occlusions provide sufficient information to determine the 3D shape and rigidity of an object, it is interesting to consider whether deformations of shading may also provide useful information. One way of examining that issue is to present the luminance flow of shaded surfaces in the absence of any visible smooth occlusion contours. That is not possible for a rotating torus knot, but it can be achieved for rotating Monge surfaces over a restricted range of orientations. The left panel of Figure 7 shows a shaded image of a slanted bumpy plane, and the right panel shows the pattern of iso-luminance contours for that image.

Figure 7.

Figure 7.

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A shaded image of a bumpy plane (left) and its pattern of iso-luminance contours (right). Animation S7 of the supplemental materials shows how these patterns change over time in several different conditions.

Animation S7 of the supplemental materials shows how these patterns change over time in three different conditions: 1) when the object rotates about a vertical axis relative to a fixed camera; 2) when the object is stationary and the camera orbits around it; and 3) when the object is undergoing a nonrigid deformation and the camera orbits around it. Note that the nonrigid motion stands out in the shaded displays because its surface features change over time, but that the stationary rigid object stands out in the contour displays because it has no sources or sinks. The similarity between the apparent motions of the two rigid shaded objects is quite remarkable because their flow fields are dramatically different. Although a close examination reveals that their shading is not identical, that has little effect on the overall percept. This suggests that the perceptual appearance of rigid rotation is based on the invariant relations among the surface features defined by shading rather than the pattern of optical flow.

The most important thing to recognize in Animation S7 is that the shaded and contoured displays have the same patterns of optical flow, yet they appear perceptually quite different when the object rotates relative to its sources of illumination. This finding suggests that there is perceptually useful information from deformations of image shading that is not adequately captured by the 2D vector field of image velocities. One possible explanation of this is that optical flow is visually represented as a higher-order tensor structure that incorporates both shading and motion, rather than a 2D vector field of image velocities. For example, qualitative changes in the apparent 3D structure of a surface might be evaluated by measuring changes in the pattern of luminance curvature over time (see Norman & Todd, 2025). Another possibility is that observers’ perceptions of rigidity may be determined by dynamic changes over time in the perception of 3D shape from shading. Neither of these approaches would require the measurement of a 2D velocity field based on the displacements of pixel intensities.

Effects of motion on the perception of material properties

Other research reported in the literature has shown that optical motion of shaded surfaces can have a large effect on the perception of 3D shape (Dövencioglu, Wijntjes, Ben-Shahar, & Doerschner, 2015; Dövencioglu, Ben-Shahar, Barla, & Doerschner, 2017) and material properties such as gloss (Doerschner, Kersten & Schrater, 2011; Doerschner et al., 2011). For example, consider the rotating metallic surface depicted in Animation S5. Note the appearance of gloss is greatly diminished when the video is paused, which suggests that motion has a significant influence on observers’ perceptions of the surface material.

Doerschner, Kersten & Schrater (2011) developed a classifier based on the histograms of image velocities that can accurately distinguish the optical flow of matte and specular objects rotating in depth, like the ones shown in Animations S4 and S5. They also presented psychophysical evidence that the performance of this classifier is highly correlated with the judgments of human observers. The difference between the optical motions of matte and specular materials can be observed by comparing the contour patterns in Animations S4 and S5. Note that the flow from the metal object is much more complex than the flow produced by the matte material, with a greater number of sources and sinks.

However, it is unlikely that the perception of gloss can be based entirely on the image velocity field. Consider the two surfaces depicted in Figure 8. The one in the left panel shows a surface with high gloss gray paint. The one in the right panel is a transformation of that image created in Adobe After Effects using the curves modifier. The linear gamma function of the original image was replaced with three cycles of a sinusoidal function. This increases the spatial frequency of the shading by a factor of three, but it preserves the mapping of pixel intensities among the different frames of an animation sequence so that the pattern of optical flow remains invariant.

Figure 8.

Figure 8.

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A shaded image of a high gloss bumpy plane (left) and a transformed version of its image intensities. Animation S8 of the supplemental materials shows how these patterns change over time as the surface rotates back and forth around a vertical axis.

Animation S8 in the supplemental materials shows the motions of both patterns of shading as the surface rotates about a vertical axis. It is quite clear in these examples that both displays have the same pattern of optical flow. However, one of them appears to depict a high gloss surface that is rotating rigidly in depth, whereas the other does not appear either rigid or glossy. This demonstration provides strong evidence that the pattern of optical flow cannot be the sole determinate for the perception of either gloss or rigidity when a smoothly shaded surface moves relative to its sources of illumination. It is likely that the visual information on which observers’ judgments are based must involve an interaction between motion and shading that cannot be adequately captured by the 2D vector flow field presented in isolation or by an incompatible combination of motion and shading.

Conclusions

Traditional models of 3D structure and rigidity from object motion have all been based on an implicit assumption that there is a projective correspondence between objects and images, such that multiple views of an identifiable image feature must always correspond to the same physical point on an object's surface. However, the luminance flow that is produced by smoothly shaded untextured surfaces does not satisfy this assumption of projective correspondence, except in the special case of matte (Lambertian) surfaces that maintain a fixed relationship with their sources of illumination (see Figure 4). There is no question that observers are able to distinguish between rigid and nonrigid motions in the natural environment, but the visual information on which those judgments are based remains to be determined. van Zuijlen et al. (2025) argue that observers’ judgments of rigidity may somehow be determined from the vector field of image velocities for shiny or translucent surfaces that move relative to their sources of illumination, but they provide no specific models of how that might be possible. The empirical evidence they present is also suspect because the 2D vector flow fields in their displays were confounded by deformations of smooth occlusion contours, and dynamic changes over time in the perception of 3D shape from shading.

In the present commentary I have described several methodological techniques that can be used to disentangle these confounding factors. One is to employ a toon renderer to present motions of an object's entire rim in the absence of any smooth shading. A second technique is to present the flow of a shaded surface as a deforming pattern of iso-luminance contours, which also eliminates the perception of 3D shape from shading. A third technique is to modify the shading of a depicted surface in a manner that alters its apparent shape and material properties without affecting its pattern of optical flow. Animations S7 and S8 of the supplemental materials provide powerful demonstrations that the 2D velocity field from a smoothly curved untextured surface provides little or no useful information for the perception of 3D shape, material properties, or the distinction between rigid and nonrigid motion when an object moves relative to its sources of illumination.

It is important to keep in mind that there are many potential techniques for representing deformations of image structure that do not require the computation of a 2D velocity field from the displacements of pixel intensities. For example, one possibility might be to measure changes in the pattern of luminance curvature over time (see Norman & Todd, 2025). It is also possible that the apparent rigidity of moving shaded surfaces is determined by dynamic changes over time in the perception of 3D shape from shading. The computation of 2D image velocities is often taken for granted by researchers in visual perception, but the theoretical underpinnings of that approach are highly questionable when applied to untextured shaded surfaces that move relative to their sources of illumination.

Supplementary Material

Supplement 1
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Supplement 2
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Supplement 3
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Supplement 4
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Supplement 5
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Supplement 6
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Supplement 8
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Acknowledgments

Supported by a grant from the National Science Foundation (BCS-2238179).

Commercial relationships: none.

Corresponding author: James T. Todd.

Email: todd.44@osu.edu.

Address: Department of Psychology, The Ohio State University, Columbus, OH 43210, USA.

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