[1305.2395] Shape Reconstruction and Recognition with Isolated Non-directional Cues
Our final thought on the problem of grouping non-directional visual cues is to further explore non-Delaunay triangulation, in particular adaptive triangulation that yields a representation comprised of visually similar triangles

Abstract: The paper investigates a hypothesis that our visual system groups visual cues
based on how they form a surface, or more specifically triangulation derived
from the visual cues. To test our hypothesis, we compare shape recognition with
three different representations of visual cues: a set of isolated dots
delineating the outline of the shape, a set of triangles obtained from Delaunay
triangulation of the set of dots, and a subset of Delaunay triangles excluding
those outside of the shape. Each participant was assigned to one particular
representation type and increased the number of dots (and consequentially
triangles) until the underlying shape could be identified. We compare the
average number of dots needed for identification among three types of
representations. Our hypothesis predicts that the results from the three
representations will be similar. However, they show statistically significant
differences. The paper also presents triangulation based algorithms for
reconstruction and recognition of a shape from a set of isolated dots.
Experiments showed that the algorithms were more effective and perceptually
agreeable than similar contour based ones. From these experiments, we conclude
that triangulation does affect our shape recognition. However, the surface
based approach presents a number of computational advantages over the contour
based one and should be studied further.

‹Figure 1: Surface model vs. contour model. The Top image is a visual stimulus shown to a subject, and the two middle pictures represent the two internal models that may be constructed internally to induce the recognition. (Introduction)Figure 5: An example shape that may be problematic for Algorithm 1. The edge pointed by the arrow, when it becomes a boundary edge after triangles below it are removed, will have the flatness of 0. Since the circumscribing circle of the associated triangle (the large one with the arrow in it) does not cross the shape, the triangle persists regardless of K. Thus, the flatness of the boundary edge remains 0, which makes removing it based on the flatness measure problematic. (Experiment 2 (Computational Study))Figure 4: An illustration of a triangulated graph and notations used. Edges in red delineate the shape. Dashed edges are removable. Red vertices are on the boundary of the current triangulated graph. Blue vertices are internal. For each boundary, there is a unique triangle associated with it. (Experiment 2 (Computational Study))Figure 8: Performance comparison between the surface based and contour based grouping algorithms. Mean values of ξ over 20 shapes are plotted at different K. The error bars are standard error of the mean. (Experiment 2)Figure 6: Four shapes (Car, Horse, Pistol, and Violin) whose results were significantly different between ΩP (Left) and ΩA (Right). (Experiment 1)Figure 7: An example of parts based shape recognition. At K = 20, parts are clearly disjoint. However, some participants were able to recognize the shape (Umbrella) correctly. (Experiment 1)Figure 9: Results of the surface based grouping algorithm at K = 50. Edges selected are shown in blue. Dashed gray edges show triangulation. The number at the top of each result is the value of ξ. (Experiment 2)Figure 10: Results of the contour based grouping algorithm at K = 50. Edges selected are shown in blue. The number at the top of each result is the value of ξ. (Experiment 2)Figure 2: Sample images of stimuli used in Experiment 1. Each row shows a particular representation, and each column shows images at different sample size (K). Row 1: ”Points”(ΩP ), Row 2: ”All Triangles”(ΩA), Row 3: ”Triangles”(ΩT ). Column 1: K = 10, Column 2: K = 20, Column 3: K = 30, Column 4: K = 100. (Methods)Figure 3: Shapes used in our experiments. (Methods)Figure 11: Application of the algorithm with a stopping condition of φ > 5. Left: Edge images obtained by Canny edge detector. Middle: Delaunay triangulation applied to the edge image. Right: Remaining triangles after boundary edges with φ > 5 are removed incrementally. (Discussion)Figure 12: Why do we see a bulge? (a) A cartoon drawing of a cat. We tend to see a bulging 3D shape in both torso and head of this drawing. (b) The 3D perception tends to persist when gaps are introduced to the boundary, (c) edge fragments are replaced with small disks, or (d) the occlusion line is removed. (Discussion)›