Lind et al. (Journal of Experimental Psychology: Human Perception and Performance, 40 (1), 83, 2014) proposed a bootstrap process that used right angles on 3D relief structure, viewed over sufficiently large continuous perspective change, to recover the scaling factor for metric shape. Wang, Lind, and Bingham (Journal of Experimental Psychology: Human Perception and Performance, 44(10), 1508-1522, 2018) replicated these results in the case of 3D slant perception. However, subsequent work by the same authors (Wang et al., 2019) suggested that the original solution could be ineffective for 3D slant and presented an alternative that used two equidistant points (a portion of the original right angle). We now describe a three-step stratified process to recover 3D slant using this new solution. Starting with 2D inputs, we (1) used an existing structure-from-motion (SFM) algorithm to derive the object’s 3D relief structure and (2) applied the bootstrap process to it to recover the unknown scaling factor, which (3) was then used to produce a slant estimate. We presented simulations of results from four previous experiments (Wang et al., 2018, 2019) to compare model and human performance. We showed that the stratified process has great predictive power, reproducing a surprising number of phenomena found in human experiments. The modeling results also confirmed arguments made in Wang et al. (2019) that an axis of mirror symmetry in an object allows observers to use the recovered scaling factor to produce an accurate slant estimate. Thus, poor estimates in the context of a lack of symmetry do not mean that the scaling factor has not been recovered, but merely that the direction of slant was ambiguous.
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The perceptual phenomenon is that distance is scaled by an unknown scaling factor along the depth dimension while being metrically specified in the projection plane. The conventional term used to describe this has been “affine” but more operations are allowed by the perceptual phenomenon than those defined by an affine mapping only. Wang, Lind, and Bingham (2018) adopted the word relief to better describe the phenomenon. See Appendix A in Wang et al. (2018) for an extended discussion of this issue.
The current implementation does not take into consideration motion in depth, i.e. is presumed. When there is motion-in-depth, the length variables can simply be scaled by that recovered from the SFM algorithm accordingly.
Perceptually, evaluating whether there has been a sufficiently large amount of perspective change relies on the change in length of the line formed between two equidistant points. Computationally, evaluating whether sufficiently large rotation has occurred relies on the value of θ. We formulated and used this particular measure of quality simply because it increases monotonically rather than oscillating around 1 as does, for instance, cos θ. Other measures of quality that are based on θ would be equally suitable.
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Statement of Significance
Empirically, the bootstrap process has been demonstrated to be an extremely replicable phenomenon through which human observers are able to perceive veridical 3D shape and slant. The current study offers a computational basis for this process, combined with simulation results from previous human experiments. Results from this study show how observers should behave when there is a need to perceive the veridical Euclidean structure of the surrounding environment.
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Wang, X.M., Lind, M. & Bingham, G.P. Bootstrapping a better slant: A stratified process for recovering 3D metric slant. Atten Percept Psychophys 82, 1504–1519 (2020). https://doi.org/10.3758/s13414-019-01860-y
- Bootstrap process
- Geographical slant perception
- Affine geometry