Certified silhouette interpolation between sampled views via aspect surfaces, and extension of silhouette-topology evolution to surfaces with flat regions

Silhouettes — the curves where sight rays graze a smooth surface — change topology as the viewpoint moves: components are born, die, split, merge and reconnect at discrete critical events. Dai, Kim, Zeng, Gu and Yau (2007) proved that on a generic smooth surface all such topological transitions of object-space silhouettes occur exactly when the viewpoint lies on the aspect surface (the union of the surface itself and the ruled surfaces swept by asymptotic directions along its parabolic curves), established global enumerative properties of silhouettes (a geodesic-curvature/cone-angle identity, a non-containment law in elliptic regions, and a four-cusp lower bound in hyperbolic regions), and built a system that visualizes silhouette evolution for meshes and splines. The authors left open the conversion of these criteria into practice: algorithms that interpolate silhouettes between sampled views with a guarantee of no unintended topological transitions — the cause of popping artifacts in non-photorealistic rendering and animation — and the extension of the theory beyond generic surfaces to surfaces with flat regions, where the classical differential-geometric analysis breaks down. The problem — certified aspect-surface-based silhouette interpolation with complexity and robustness analysis, together with the flat-region extension of the critical-event and enumerative theory — remains open.

Background

For a smooth surface $\Sigma$ in three-dimensional space and a viewpoint $v$ off the surface, the object-space silhouette is the set of surface points where the sight ray grazes the surface, $\Gamma_v=\{p\in\Sigma:\ \langle p-v,\,n(p)\rangle=0\}$, with $n(p)$ the unit surface normal. Its projected image is the apparent contour (occluding contour), the curve along which the object separates from its background in a picture. Silhouettes are a primary cue in object recognition, in shape reconstruction from multiple views (the visual hull is the maximal solid consistent with a family of silhouettes), and in non-photorealistic rendering (NPR), where line drawings of smooth objects are dominated by their silhouettes.

As the viewpoint moves, a silhouette deforms smoothly except at discrete critical events where its topology changes. The singularity-theoretic study of these transitions began with Koenderink and van Doorn's classification of the singularities of the visual mapping; catastrophe theory, as surveyed in Arnold's monograph, supplies the classical catalog of local events of projections (lip, beak-to-beak, swallowtail). The aspect-graph research program turned this into a computational object: viewpoint space is partitioned into maximal regions inside which the topological type of the image is constant, separated by visual event surfaces. Rieger extended the classification to piecewise-smooth objects and described the global geometry of view space for smooth objects, and Pae and Ponce gave algorithms (homotopy continuation, which reaches solutions of a polynomial system by deforming a known solution along a parameter path, and curve tracing) for computing structural changes of parabolic curves and aspect graphs for algebraic surfaces undergoing deformation. This line concerns mostly the projected image, and exact construction is computationally heavy.

The source paper, Dai, Kim, Zeng, Gu and Yau (2007), approached the problem intrinsically, on the surface rather than in the image. A point of a surface is elliptic or hyperbolic according to the sign of the Gaussian curvature $K$ (the product of the two principal curvatures, i.e. of the maximum and minimum normal curvatures at the point), and parabolic when $K=0$; an asymptotic direction is a tangent direction in which the normal curvature vanishes (such directions exist at parabolic and hyperbolic points only); the geodesic curvature is the curvature of a surface curve as measured within the surface; a cusp is a point where the projected contour reverses direction. Under a genericity assumption — the Gaussian curvature vanishes only along curves (the parabolic curves), not on regions, and the viewpoint avoids finitely many degenerate positions — the paper proves, among other results: (i) for a closed silhouette component $\gamma$, the integral of signed geodesic curvature along $\gamma$ equals the cone angle subtended at the viewpoint by the view cone through $\gamma$ (a Gauss–Bonnet identity on the cone); (ii) on a surface with $K>0$ everywhere, no silhouette bounding a topological disk contains another silhouette in its visible interior; (iii) on a surface with $K<0$ everywhere, a closed silhouette component that can be shrunk to a point within the surface projects with at least four cusps — a silhouette analogue of the classical four-cusp theorem for plane caustics; and (iv) the central structural result: a topological transition of the silhouettes can occur only when the viewpoint lies on the aspect surface $\Omega(\Sigma)$, defined as the union of $\Sigma$ itself and the ruled surfaces swept by the asymptotic lines through the parabolic curves. Crossing $\Sigma$ at an elliptic point births or kills a small loop; crossing at a hyperbolic point makes two arcs meet and reconnect along the asymptotic directions; the classical lip and beak-to-beak events arise at the parabolic rulings. The paper also implements a system that computes silhouettes and aspect surfaces for triangle meshes and spline surfaces and visualizes the full sequence of critical events along viewpoint trajectories.

What the paper deliberately does not deliver is the algorithmic exploitation of its own criterion. First, it gives no procedure that, given two sampled views, interpolates the silhouettes between them while provably avoiding every topological transition not certified by the aspect-surface theory — the practical question for NPR and animation, where unintended transitions appear as popping artifacts. Second, its genericity assumption excludes surfaces with flat regions, i.e. open sets on which $K\equiv 0$ (planar or developable patches — pieces such as cylinder or cone walls, which can be unrolled flat without stretching — omnipresent in CAD and real models): there the parabolic locus degenerates from curves to regions, the asymptotic-direction construction loses meaning, and, as the authors note, the classical differential-geometric machinery becomes difficult to apply.

Problem Statement

The problem, stated by the authors of the source paper as future work, consists of two linked components serving one objective: making the topological evolution of silhouettes a certifiable, computable object for the surface classes that actually occur in graphics and vision.

(1) Certified silhouette interpolation. Let $\Sigma$ be a smooth surface given in practice as a spline or triangle-mesh approximation, let $v_0,v_1$ be two sampled viewpoints, and let $\Gamma(v_0),\Gamma(v_1)$ be the corresponding object-space silhouettes. Design and analyze a practical interpolation algorithm that uses the aspect-surface characterization — where $\Omega(\Sigma)$ denotes $\Sigma$ together with the ruled surfaces swept by the asymptotic lines along the parabolic curves of $\Sigma$ — and produces a continuous one-parameter family of silhouettes connecting $\Gamma(v_0)$ to $\Gamma(v_1)$ with a topological guarantee:

• (a) whenever a viewpoint path from $v_0$ to $v_1$ avoids $\Omega(\Sigma)$, the family realizes a smooth deformation (an ambient isotopy — a continuous deformation through bijective continuous maps of the surrounding space — carrying one silhouette onto the other) with no topological transition of any silhouette component;
• (b) when $v_0$ and $v_1$ lie in different connected components of viewpoint space minus $\Omega(\Sigma)$, the interpolation passes through exactly the critical events classified by the aspect-surface theory of the source paper — birth or death of a loop when the viewpoint crosses the surface at an elliptic point, reconnection of two arcs at a hyperbolic point, and the lip and beak-to-beak events at the parabolic rulings (the individual generating lines of those ruled surfaces) — at the crossing instants and nowhere else;
• (c) for discrete inputs the algorithm must reach these decisions robustly, since $\Omega(\Sigma)$ is a measure-zero locus and meshing and floating-point error can flip classifications: the solution must include a certified or quantified-error decision procedure together with a complexity and robustness analysis.

The driving application is silhouette animation and view interpolation for non-photorealistic rendering, where unintended topological transitions of silhouettes appear as popping artifacts.

(2) Extension to nongeneric surfaces. Extend the critical-event characterization (the aspect-surface theorem) and the enumerative global properties of silhouettes — the geodesic-curvature/cone-angle identity, the non-containment law in elliptic regions, and the four-cusp lower bound in hyperbolic regions — from generic smooth surfaces to surfaces containing flat regions, i.e. open sets on which the Gaussian curvature vanishes identically, so that the parabolic locus degenerates from curves to regions and the genericity arguments fail, and, more generally, to other nongeneric features. The extension should identify the locus of critical viewpoints and the catalog of critical events for such surfaces, and should reduce to the source results wherever the genericity assumptions hold.

The source paper also leaves open, beyond the two components above, the exploration of other global properties of silhouettes.

References

1. Dai Jun-fei, Kim Junho, Zeng Hua-yi, Gu Xian-feng, Yau Shing-tung (2007). Topology evolutions of silhouettes. Journal of Zhejiang University-SCIENCE A 8(10):1671-1680. DOI 10.1631/jzus.2007.A1671. https://doi.org/10.1631/jzus.2007.A1671
2. Koenderink J.J., van Doorn A.J. (1976). The singularities of the visual mapping. Biological Cybernetics 24(1):51-59. DOI 10.1007/BF00365595. https://doi.org/10.1007/BF00365595
3. Rieger J.H. (1987). On the classification of views of piecewise smooth objects. Image and Vision Computing 5(2):91-97. DOI 10.1016/0262-8856(87)90033-3. https://doi.org/10.1016/0262-8856(87)90033-3
4. Rieger J.H. (1990). The geometry of view space of opaque objects bounded by smooth surfaces. Artificial Intelligence 44(1-2):1-40. DOI 10.1016/0004-3702(90)90097-J. https://doi.org/10.1016/0004-3702(90)90097-J
5. Pae Sung-Il, Ponce Jean (2001). On Computing Structural Changes in Evolving Surfaces and their Appearance. International Journal of Computer Vision 43(2):113-131. DOI 10.1023/A:1011170702891. https://doi.org/10.1023/A:1011170702891
6. Kalnins Robert D., Davidson Philip L., Markosian Lee, Finkelstein Adam (2003). Coherent stylized silhouettes. ACM Transactions on Graphics 22(3):856-861. DOI 10.1145/882262.882355. https://doi.org/10.1145/882262.882355
7. Isenberg Tobias, Freudenberg Bert, Halper Nick, Schlechtweg Stefan, Strothotte Thomas (2003). A developer's guide to silhouette algorithms for polygonal models. IEEE Computer Graphics and Applications 23(4):28-37. DOI 10.1109/MCG.2003.1210862. https://doi.org/10.1109/MCG.2003.1210862
8. Kohn Kathlén, Sturmfels Bernd, Trager Matthew (2018). Changing Views on Curves and Surfaces. Acta Mathematica Vietnamica 43(1):1-29. DOI 10.1007/s40306-017-0240-1 (preprint arXiv:1707.01877). https://arxiv.org/abs/1707.01877
9. Hasegawa Masaru, Kabata Yutaro, Saji Kentaro (2019). Capturing information on curves and surfaces from their projected images. arXiv:1904.08633. https://arxiv.org/abs/1904.08633 (published as: International Journal of Mathematics for Industry 12(1):2050004, 2020. DOI 10.1142/S2661335220500045. https://doi.org/10.1142/S2661335220500045)
10. Gallet Matteo, Lubbes Niels, Schicho Josef, Vršek Jan (2018). Reconstruction of surfaces with ordinary singularities from their silhouettes. arXiv:1810.05559. https://arxiv.org/abs/1810.05559 (published as: SIAM Journal on Applied Algebra and Geometry 3(3):472-506, 2019. DOI 10.1137/18M1220911. https://doi.org/10.1137/18M1220911)
11. Wang Ziyun, Chaney Kenneth, Daniilidis Kostas (2023). EvAC3D: From Event-based Apparent Contours to 3D Models via Continuous Visual Hulls. arXiv:2304.05296. https://arxiv.org/abs/2304.05296 (published as: Computer Vision – ECCV 2022, Lecture Notes in Computer Science, pp. 284-299. DOI 10.1007/978-3-031-20071-7_17. https://doi.org/10.1007/978-3-031-20071-7_17)
12. Zhuang Yueting, Chen Cheng (2007). Efficient Silhouette Extraction with Dynamic Viewpoint. Proc. IEEE 11th International Conference on Computer Vision (ICCV 2007), pp. 1-8. DOI 10.1109/ICCV.2007.4408916. https://doi.org/10.1109/ICCV.2007.4408916
13. Wu Sing-on. Smoothing the silhouettes of polyhedral meshes by boundary curve interpolation. Doctoral thesis, The University of Hong Kong. DOI 10.5353/th_b2981554. https://doi.org/10.5353/th_b2981554
14. Arnold V.I. (1986). Catastrophe Theory (2nd edition). Springer-Verlag. DOI 10.1007/978-3-642-96937-9. https://doi.org/10.1007/978-3-642-96937-9
15. Liu Chenxi, Bénard Pierre, Hertzmann Aaron, Hoshyari Shayan (2023). ConTesse: Accurate Occluding Contours for Subdivision Surfaces. ACM Transactions on Graphics 42(1):1-16. DOI 10.1145/3544778. https://doi.org/10.1145/3544778 (preprint arXiv:2111.06006. https://arxiv.org/abs/2111.06006)