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Research Papers

Topologically Enhanced Slicing of MLS Surfaces

[+] Author and Article Information
Pinghai Yang, Kang Li

Department of Mechanical, Materials and Aerospace Engineering,  Illinois Institute of Technology, Chicago, Illinois 60616qian@iit.edu

Xiaoping Qian1

Department of Mechanical, Materials and Aerospace Engineering,  Illinois Institute of Technology, Chicago, Illinois 60616qian@iit.edu

1

Corresponding author.

J. Comput. Inf. Sci. Eng 11(3), 031003 (Aug 10, 2011) (9 pages) doi:10.1115/1.3615683 History: Received April 28, 2009; Revised March 30, 2011; Published August 10, 2011; Online August 10, 2011

Growing use of massive scan data in various engineering applications has necessitated research on point-set surfaces. A point-set surface is a continuous surface, defined directly with a set of discrete points. This paper presents a new approach that extends our earlier work on slicing point-set surfaces into planar contours for rapid prototyping usage. This extended approach can decompose a point-set surface into slices with guaranteed topology. Such topological guarantee stems from the use of Morse theory based topological analysis of the slicing operation. The Morse function for slicing is a height function restricted to the point-set surface, an implicitly defined moving least-squares (MLS) surface. We introduce a Lagrangian multiplier formulation for critical point identification from the restricted surface. Integral lines are constructed to form Morse-Smale complex and the enhanced Reeb graph. This graph is then used to provide seed points for forming slicing contours, with the guarantee that the sliced model has the same topology as the input point-set surface. The extension of this approach to degenerate functions on point-set surface is also discussed.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Overview of the proposed approach. (a) Point cloud. (b) Morse function on the Mls surface. (c) Critical point generation. (d) Morse-smale complex. (e) Enhanced reeb graph. (f) Sliced model. The dots represent the maximum, (top and bottom) saddle and minimum critical points, respectively. For details, please refer to the legend shown in Fig. 4.

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Figure 2

Critical point refinement. (a) Generation of critical points (dot) by refining the candidate points (cross), geometrical configuration of the double torus surface and a slicing plane passing a critical point. (b) Distribution of ϕ(x) and g(x) on this slicing plane and the refinement process from a candidate point (dot) to the critical point (star).

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Figure 3

Rules of generating an enhanced Reeb graph from the Morse-Smale complex. (a) Rule #1 for maximum and bottom saddle. (b) Rule #2 for top saddle. (c) Grouping the lower neighbor points in Rule #2.

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Figure 4

Illustration of the generation of an enhanced Reeb graph from the Morse-Smale complex with the example of double torus

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Figure 5

Marching based slicing (a) marching normal direction nP and surface normal nS. (b) Slicing at height of torus’ saddle (c) Marching normal distribution and contour merging at saddle.

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Figure 6

Enhanced Reeb graph generation for three examples: a knot surface in the first row, a mechanical part in the second row, and a real sculpture data in the third row. In each of these three rows, the first column shows the corresponding input point data, the second column shows the resulting Morse-Smale complex, and the third column shows the resulting enhanced Reeb graph.

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Figure 7

Topology enhanced slicing for three examples. (a) The knot surface. (b) The mechanical part with sharp corners. (c) The real sculpture data with noise and holes.

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Figure 8

Topology enhanced slicing for the knot surface. (a) Iso-view of the knot data with integral lines. (b) Top view of the slice at z=0.0. (c) Top view of the slice at z=1.83.

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Figure 9

Synthetic noise data sampled from the knot surface. (a) Point cloud. (b) Rendered triangular mesh model. (c) A typical circular section with a varying levels of noises (standard deviation from 0 to 0.1).

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Figure 10

Topology enhanced slicing for the noisy unevenly distributed data. (a) Top view of the slice at z=1.83. (b) Zoom-in of (a). (c) Distance between the sliced points in (b) and the nominal model.

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Figure 11

Degenerate torus and its degenerate maximum, minimum and saddle

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Figure 12

(a) Saddle node determination. (b) Saddle nodes and wedge nodes. (c) Generated complex.

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Figure 13

Point cloud, morse complex, and reeb graph and slided model of monkey saddle point-set (48.8 K points), fertility (58.2 K points), and mechanical block (62.8 K points).

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