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Research Papers

# Intersection-Free and Topologically Faithful Slicing of Implicit Solid

[+] Author and Article Information
Pu Huang

Department of Mechanical and Automation Engineering,
The Chinese University of Hong Kong,
Hong Kong, PRC

Charlie C. L. Wang

Department of Mechanical and Automation Engineering,
The Chinese University of Hong Kong,
Hong Kong, PRC;
Epstein Department of Industrial and Systems Engineering,
University of Southern California,
Los Angeles, CA 90089
e-mail: cwang@mae.cuhk.edu.hk

Yong Chen

Epstein Department of Industrial and Systems Engineering,
University of Southern California,
Los Angeles, CA 90089

1Corresponding author.

Contributed by the Computers and Information Division of ASME for publication in the Journal of Computing and Information Science in Engineering. Manuscript received July 5, 2011; final manuscript received March 6, 2013; published online April 26, 2013. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 13(2), 021009 (Apr 26, 2013) (13 pages) Paper No: JCISE-11-1375; doi: 10.1115/1.4024067 History: Received July 05, 2011; Revised March 06, 2013

## Abstract

We present a robust and efficient approach to directly slicing implicit solids. Different from prior slicing techniques that reconstruct contours on the slicing plane by tracing the topology of intersected line segments, which is actually not robust, we generate contours by a topology guaranteed contour extraction on binary images sampled from given solids and a subsequent contour simplification algorithm which has the topology preserved and the geometric error controlled. The resultant contours are free of self-intersection, topologically faithful to the given $r$-regular solids and with shape error bounded. Therefore, correct objects can be fabricated from them by rapid prototyping. Moreover, since we do not need to generate the tessellated B-rep of given solids, the memory cost our approach is low—only the binary image and the finest contours on one particular slicing plane need to be stored in-core. Our method is general and can be applied to any implicit representations of solids.

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## Figures

Fig. 1

Incorrect contours generated by slicing a given model will produce a model with unwanted gaps (and/or membranes) in rapid prototyping: (a) the given Buddha model in implicit representation (actually Layered Depth-Normal Images (LDNI) [3]), (b) the correct model fabricated from contours generated by our approach, (c) the problematic object fabricated by slicing the locally self-intersected polygonal model extracted from (a) using a variation [4] of dual-contouring [5], and (d) a zoom-view of the incorrect layer. The models are fabricated by FDM.

Fig. 2

Tool path of five consecutive layers generated in InsightTM version 7.0 by slicing a polygonal model extracted from the implicit Buddha solid given in Fig. 1(a) by [4]. Each layer is in thickness of 0.01 in., and the regions of the Buddha model and the supporting structures are displayed in green and cyan respectively. Pay attention that an incorrect in/out membership classification is given on the layer with height = 1.79 in., which is caused by a local self-intersection on the polygonal model. As a result, the inside of the Buddha model is filled with supporting structure by mistake.

Fig. 3

A example of tubes in biomedical engineering: (left) the expected tubes configuration, (right) the manufactured model with incorrect topology merging two tubes that should be separated—this is very dangerous for medical treatments

Fig. 4

A flow chart of our direct slicing approach

Fig. 5

A binary image sampled from a Dragon solid: (a) the solid H is intersected by a slicing plane P, (b) the region of H¯=H∩P on the binary image, and (c) a zoom-view of the binary image, where the nodes inside H¯ are displayed in solid dots while outside nodes are shown in hollow

Fig. 6

Binary image generation of slicing in different orientations: (left) with slicing direction (0,1,0) and (right) slicing along (1,1,1). The bottom row shows the binary images generated on example slicing planes.

Fig. 7

Lookup table for the marching square method with topology preserved. Sticks are in yellow. Sampling nodes inside the solid H are shown in black while the outside nodes are displayed in white. The contour edges linking sticks are labeled as E.

Fig. 8

An example model fabricated from the contours generated by our method with r'-3: (a) the given Donna model in the LDNI representation, (b) the sliced contours of respective layers at 2.30, 3.00, and 3.66 in. heights, and (c) the resultant model fabricated by FDM

Fig. 9

The sweeping envelopes for contour edges in grids with different configurations. The shadow regions represent the sweeping envelops. Note that some vertices and edges are excluded from the sweeping envelops. These are p1, p3, p4, edge(p3,p4) for configuration 1; p1, p2, p3, p4, edge(p1,p2), edge(p3,p4) for configuration 2; p1, p3, p4, edge(p1, p3) for configuration 3; and p1, p2, p3, p4, edge(p2, p4) for configuration.

Fig. 10

A comparison among different smoothing strategies on the contour generated for the binary image region shown in Fig. 5(c): (a) the contour reconstructed by the topology preserving marching square method, (b) the shrinking contour after ordinary Laplacian smoothing, (c) the resultant contour after projection-based constrained smoothing, (d) the resultant contour after sliding-based constrained smoothing, and (e) the zoom-in view of contour vertices stuck in sub-optimal shape

Fig. 11

All the five configurations for the position of vi-1 and vi+1 on different sticks with considering rotational symmetry

Fig. 12

An illustration of contour simplification for the smoothed contour shown in Fig. 10(d): (a) the variational clustering result on the contour with different line type representing different regions, and (b) the final simplified contour after topology and distortion verification

Fig. 13

A contour that is originally intersection-free could become intersected or degenerate by replacing the curved region with line segments

Fig. 14

Sliding the six vertices on their respective sticks in order to form a single edge connecting the starting and ending vertices of this region

Fig. 15

An example of slicing the Filigree model: (a) a mesh tessellated from implicit Filigree model, (b) the contours and their corresponding tool path generated by InsightTM version 7.0 on the layer with 0.77 in. height, (c) the contours generated by our approach for the same layer and their corresponding tool path, (d) the rendered slicing contours generated by our approach

Fig. 16

An example of slicing the Truss model: (a) a mesh tessellated from implicit Truss model, (b) the contours and their corresponding tool path generated by InsightTM version 7.0 on the layer with 2.44 in. height, (c) the contours generated by our approach for the same layer and their corresponding tool path, (d) the rendered slicing contours generated by our approach

Fig. 17

A comparison between resultant contours with and without topology verification: (a) the given Buddha model in the LDNI representation, (b) a binary image sampled from the layer with 2.90 in. height, (c) the resultant contour without topology verification, (d) the resultant contour with topology verification

Fig. 18

A comparison between resultant contours with and without topology verification: (a) the given Truss model in the LDNI representation, (b) a binary image sampled from the layer with 0.30 in. height, (c) the resultant contour without topology verification, and (d) the resultant contour with topology verification

Fig. 19

An example of slicing the Hand-complex model: (a) the original Hand-complex model, (b) the rendered slicing contours, (c) the chart of maximum regional distortion error versus the clustering ratio α, (d) the chart of simplified contour edge number versus α, and (e) the chart of time consumption versus α

Fig. 20

An example of slicing the Spine model: (a) the original Spine model, (b) the rendered slicing contours, (c) the chart of maximum regional distortion error versus the clustering ratio α, (d) the chart of simplified contour edge number versus α, and (e) the chart of time consumption versus α

Fig. 21

A demonstration of our approach on BSP solid: (a) the given Rocker-arm model in BSP representation and the resultant rendered contours, (b) the binary image and corresponding contours for the layer with 1.66 in. height, and (c) the binary image and corresponding contours for the layer with 0.85 in. height

Fig. 22

A demonstration of our approach on RBF solid: (a) the given Armadillo model in adaptively supported RBF representation and the resultant rendered contours, (b) the binary image and corresponding contours for the layer with 2.90 in. height, and (c) the binary image and corresponding contours for the layer with 1.30 in. height

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