Kinetsu Abe

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.

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.

Indefinite Khler submanifolds with positive index of relative nullity

We deal with complex submanifolds in indefinite space forms. In particular, submanifolds with large index of relative nullity are emphasized. In that context, we prove cylinder theorems in terms of indefinite metrics. We also give a systematic way of constructing a family of new complete and closed indefinite complex submanifolds in the projective setting.In the appendix, we show that the method used for complex cases can be applied to real indefinite geometry. We prove real cylinder theorems including B-scrolls in the general signature. We also show two decomposition lemmas which clarify the relationships between the Hartman-Nirenberg cylinder theorem and slanted cylinder theorems in indefinite geometry.

Complex analytic curves and maximal surfaces

Maximal immersions of a surfaceM2 inton-dimensional Lorentz space which are isometric to a fixed holomorphic mapping ofM2 into complex Lorentz space are determined. The main tool is an adaption of Calabis Rigidity Theorem. Such an adaption is necessary because of the existence of degenerate hyperplanes in complex Lorentz space.

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.

On a generalization of the hopf fibration, III (Subvarieties in theC-spaces)

Analytic subvarieties inC-spaces are discussed. First, a certain kind of closed 2-formdω is constructed. Then, the subvarieties are studied by means of this 2-form.dω may be considered as the curvature form of a connection whenC-spaces are considered as the toral bundle space over an algebraic variety. This 2-form is horizontal in its nature with respect to the bundle structure and indicates, in general, how different the bundle is from the trivial bundle. Because of this twist in the bundle space, subvarieties in theC-spaces tend to inherit the same structure. In this paper, the inherited fibration structure is studied. The most concrete results are obtained when the fiber torus has complex dimension 2.