Justin Bisceglio

Isotopic approximations and interval solids

Given a nonsingular compact two-manifold F without boundary, we present methods for establishing a family of surfaces which can approximate F so that each approximant is ambient isotopic to F: The methods presented here offer broad theoretical guidance for a rich class of ambient isotopic approximations, for applications in graphics, animation and surface reconstruction. They are also used to establish sufficient conditions for an interval solid to be ambient isotopic to the solid it is approximating. Furthermore, the normals of the approximant are compared to the normals of the original surface, as these approximating normals play prominent roles in many graphics algorithms. The methods are based on global theoretical considerations and are compared to existing local methods. Practical implications of these methods are also presented. For the global case, a differential surface analysis is performed to find a positive number r so that the offsets Foð^rÞ of F at distances ^r are nonsingular. In doing so, a normal tubular neighborhood, FðrÞ; of F is constructed. Then, each approximant of F lies inside FðrÞ: Comparisons between these global and local constraints are given. q 2004 Elsevier Ltd. All rights reserved.

Computational topology for reconstruction of surfaces with boundary: integrating experiments and theory

We report new techniques and theory in computational topology for reconstructing surfaces with boundary. This complements and extends known techniques for surfaces without boundary. Our approach is motivated by differential geometry and differential topology. We have also conducted significant experimental work to test our resultant implementations. We discuss some problematic issues that can arise regarding the roles of the medial axis and sampling density. The crucial topics for C2 manifolds are (1) important defining properties of C2 manifolds with boundary, (2) creation of auxiliary surfaces, with emphasis near the boundary,(3) sampling density, and (4) successful practical algorithms and examples.

Unknots with highly knotted control polygons

An example is presented of a cubic Bezier curve that is the unknot (a knot with no crossings), but whose control polygon is knotted. It is also shown that there is no upper bound on the number of crossings in the control polygon for an unknotted Bezier curve. These examples complement known upper bounds on the number of subdivisions sufficient for a control polygon to be ambient isotopic to its Bezier curve.

Computational topology for isotopic surface reconstruction

New computational topology techniques are presented for surface reconstruction of 2-manifolds with boundary, while rigorous proofs have previously been limited to surfaces without boundary. This is done by an intermediate construction of the envelope (as defined herein) of the original surface. For any compact C2-manifold M embedded in R3, it is shown that its envelope is C1,1. Then it is shown that there exists a piecewise linear (PL) subset of the reconstruction of the envelope that is ambient isotopic to M, whenever M is orientable. The emphasis of this paper is upon the formal mathematical proofs needed for these extensions. (Practical application examples have already been published in a companion paper.) Possible extensions to non-orientable manifolds are also discussed. The mathematical exposition relies heavily on known techniques from differential geometry and topology, but the specific new proofs are intended to be sufficiently specialized to prompt further algorithmic discoveries.

Up close with simulated crowds

We discuss advancing the fine detail of deforming hero-quality simulated crowd agents in animated feature film production. To support character animation that is suitably framed arbitrarily close to camera, our approach uses a novel deformation system that combines simulation and hero-quality custom animation. Level-of-detail optimizations are handled at render time, and artists are only tasked with the design of a single high-quality resolution for each character asset. Key optimizations in our rig structures are outlined as they are fundamental to scalability, permitting our crowds to look good while numbering in the millions.

Populating the crowds in Ferdinand

With the help of new tools, we streamlined our review and render processes for crowds to triple our shot count on our latest show, Ferdinand. At the same time, we integrated some novel approaches to complex deformation features for cloth and facial animation, which elevated the quality of our crowd animations.

Bats, birds, and boggans: the simulated armies of epic

In Epic (2013), crowds are integral to the narrative and form a character as a whole. This required a new type of crowd at Blue Sky Studios, one that permits dynamic interaction between crowd characters and the environments around them in addition to supporting the high-resolution geometry with fur, deformation rigs, and material complexities needed for shots where the crowd is close to camera. Our crowd framework centers around the choice to separate the simulation process from the technique used to render the crowd. This meant we could use different simulators for different shots. At times, the crowd exceeded 100,000 characters, far more than in any of our previous films. To manage all this data we store only per character joint animation instead of deformed geometry. This compact format allows us to both display art direct-able representations of the crowd in real-time and to defer evaluation of the expensive parts of the rig until render time. To render the crowd with our in-house ray-tracing renderer, CGIStudio#8482;, we build a custom, optimized deformation system that supports rendering of both deformed geometry and deformed voxels.

Reconstructing surfaces using envelopes: bridging the gap between theory and practice

Reconstruction of surfaces with boundary remains a challenge. Recent theoretical advances [Abe et al. 2006] define envelopes as surfaces without boundary to approximate those with boundary. Figure 1 shows a Möbius strip on the left, denoted as M. An illustration of its envelope appears next, intuitively understood as attaching a small ball to each point of M to create a 3-manifold. The envelope of M is then the bounding surface of this 3-manifold. The images represent approximations of this process. The new theory shows that envelopes can guarantee topological preservation during reconstruction, but practical implementations are still emerging. This theory is advantageous in that it offers a terse measure of equivalence known as an ambient isotopy and expands the class of surfaces which may be reconstructed. We present progress toward development of an algorithm that robustly implements this theorem.Constructing envelope enclosures around point set data, in the absence of all other geometric information, means approximating normals and identifying boundaries. The most current published approach [Ohtake et al. 2005], uses error minimization to create an adaptive spherical covering. However, the authors state that this approach is not supported by any mathematical results and is only appropriate for certain types of input data.The approximation quality of a reconstructed surface depends on how well the estimated normals approximate the true normals of the sampled surface. Dey [Dey et al. 2005] presents a detailed understanding of this concept with a survey of techniques for estimating normals and the circumstances under which they are appropriate. A sufficient sampling density is a prerequisite for approximating normals. We adapt a relation for bounding the sampling criteria [Amenta et al. 2003] to be suitable for the construction of envelopes. The sampling criteria of a surface is gauged according to the distance to its medial axis. With envelopes, the distance to the medial axis is the radius of the envelope. Thus, not only must a sample density be carefully chosen but so must the radius of the envelope. Our investigation begins by examining criteria for selecting these variables.An experimental study in 2005 gave an understanding of the importance of sufficient sampling and proper envelope construction. Pursuant to these findings, a detailed test on the effects of varying both the envelope radius and the sample density was performed yielding new insight regarding sufficient sampling when generating envelopes. We conclude with progress towards improving a realistic trial presented in the literature. Spline surfaces, which allow envelope construction through an exact evaluation, are used in most of our experiments. This is only helpful for testing. A publication [Abe et al. 2005] demonstrates the practical difficulties of using the envelope technique on point set data including an example with the Stanford bunny. A revised version of this bunny test is shown here.