Reconstructing Surfaces Using Envelopes: Bridging the Gap between Theory and Practice

Transcription

1 Reconstructing Surfaces Using Envelopes: Bridging the Gap between Theory and Practice Justin Bisceglio Blue Sky Studios, and University of Connecuticut, Department of Computer Science T.J. Peters University of Connecuticut, Department of Mathematics, Department of Computer Science Kinetsu Abe University of Connecuticut Department of Mathematics Figure 1: The process of creating an envelope and resulting approximation. Abstract 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. This work complements and extends the previous presented theory, notably by showing examples that are beyond the typically assumed hypotheses, and presents progress toward development of an algorithm that robustly implements this theorem. 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 during the summer of 2005 contributed to 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. Introduction 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 justinb@blueskystudios.com tpeters@cse.uconn.edu abe@math.uconn.edu Figure 2: The reconstruction of a cross-shaped pipe surface.

2 Figure 3: An early experiment with a T-shaped pipe surface. The surface was modeled with three quadratic spline surfaces with coincident boundaries along the mid-section of the T-shape. Hence, the shape is easily decomposed into three components. The purpose of this experiment was to test the ability to assemble corresponding approximations that were independently reconstructed from segmented point set data using the envelope technique. Several attempts indicate that it is not a beneficial approach when naïvely performed. The top row of the image contains two perspective views of the original shape and the decomposed components with illustrative envelope points on the right. The results of a series of reconstructions are along the bottom row. From left to right, the tests were performed with arbitrarily, increasing sample densities and decreasing envelope size. While the reconstruction results improve with successively larger data sets and decreasing envelope size, the reassembled components fail to achieve an ambient isotopic relationship to the original shape. That is, the reassembled components are never water-tight along the shared boundaries of the components. Experimental Results sample rate by a factor of two was not based on any theoretical notion. The following experiments were conducted during an undergraduate summer program hosted by the Department of Mathematics at the University of Connecticut throughout the summer of This study focused on the computational topology for approximating shapes. Similar to the T-pipe experiment in Figure 3, a cross-shaped pipe was tested and demonstrates a successful water-tight reconstruction in Figure 2. In the lower portion of the image is an enlarged view of the adjoining components showing a distinct connectivity. The top of the image displays the envelope reconstruction of one of the components on the left and its associated medial axis on the right. In this experiment, the sample density did not exceed the highest sampling used in the T-pipe experiment. As the sampling rate and envelope radius were again chosen arbitrarily, the improved result lends evidence to the importance of judicious envelope construction along with sufficient surface sampling. Figure 4 details a series of reconstructions using the envelope technique on various cut cylinders. Here, we are interested in the affect the shape of a cut has along reconstructed boundary. Three different cylinders, each with an increasingly elongated boundary, are tested. The cut is created by intersecting a plane with the cylinder where the intersecting plane is rotated from the plane perpendicular to cylinder. The variation in the shape of the cut did not produce any noticeable difference with the top two cylinders. However, the lower-left cylinder shows artifacts along the boundary of the cut. Doubling the sample density improves the result in the lower-left such that the artifacts no longer exist. This is shown in the lowerright. In all tests, the envelope radius around the cylinder is constant. As with the previous experiments, the choice to increase the Figure 4: A series of reconstructions using the envelope technique on various cut cylinders. So far, all the experiments presented used heuristics in creating the envelopes with regard to sample density and envelope radius. The next experiment expands on one of the undergraduate tests in an effort to offer insight towards choosing these variables properly. Figure 5 describes several iterations of a reconstruction with varying sampling and envelope size. The upper left corner shows the test subject - a rational quadratic, trimmed, NURBS sphere. The spline surface is rendered as a shaded tessellation. As is the case with a sphere, the shape has constant curvature. The boundary regions have a slight variation in curvature. The literature identifies curvature and the medial axis as being closely related and an im-

3 Figure 5: The results of an experiment on various approximated envelopes around a punctured sphere. The highlighted medial axis approximations are displayed larger in Figure 6.

4 Figure 6: Four enlarged views from the results on the previous figure.

5 portant factor in the reconstruction process. The expectation is to observe the effects of varying envelope characteristics most notably in the areas with the highest curvature. Several different envelope enclosures were generated from the surface. The lower left quadrant lists the numeric values for these variations while the upper right quadrant displays the envelope approximations after the sample points were reconstructed using the PowerCrust software. The envelopes are arranged corresponding to their numeric values in the data table. The lower right section contains the resulting medial axis approximations to the reconstructed envelopes. Again, the ordering corresponds to their respective envelopes. The results are interesting in that the best results were not those with the highest sampling rate and smallest envelope. This seems to suggest a dependence between the two variables, and exemplifies publications which discuss the criteria for a sufficiently dense sampling in relation to the medial axis, specifically for a point x on the original surface F the distance from x to the nearest vertex on the triangulated manifold C is at most some 0.08 times the distance from x to the medial axis of F, MA(F) [Amenta et al. 2003], [Sakkalis and Peters 2003]. What remains an enigma is the cause of artifacts in some of the cases with higher sample densities. With denser sampling, the artifacts are more frequent when the envelope radius is larger and the ratio of a point s distance to its neighbors and its distance to the medial axis approaches the theoretical criteria. Although, this criteria is meant to guarantee an ambient isotopic reconstruction which does occurs in all these trials in question. Also listed in the table is the maximum distance between any two neighboring sample points. Whether these artifacts are related to a theoretical condition or circumstantial from the PowerCrust implementation is not clear and warrants further investigation. The last experiment presented revisits an earlier, practical test [Abe et al. 2005] using the Stanford bunny data which yielded unsatisfactory results. We first perform some analysis of the data set to determine a theoretically reasonable envelope model. For simplicity, we focus only on the boundary region of our sliced bunny. Creating a spline approximation of the boundary region allows the curvature to be estimated. The maximum curvature was found to be and is shown in Figure 8 indicated by the red circle. The value was then used to create a non-self-intersecting pipe surface along the spline curve [Patrikalakis and Maekawa 2002], as well as indicate an appropriate envelope radius and sampling rate. However, for this relatively simple example, the theoretical sampling rate is computationally prohibitive. Using only the original data points from the boundary and generating envelope data around the original points produced a relatively poor reconstruction. The reconstructed curve contains gaps and is extremely coarse. As the envelope data is embellished with additional envelope samples from the spline approximation the reconstruction results begins to improve. Unfortunately, augmenting the input data to achieve this modest improvement dramatically affects the computation time of the reconstructing operation. To accomplish the final reconstruction this larger data set was subdivided into many smaller sets and the reconstructions were then reassembled. An obvious issue for future research with the envelope technique is the data requirements for practical examples. Conclusions We performed several experiments and analyses to help understand the necessary criteria when using envelopes to reconstruction surfaces from point data. The experiments highlight the strengths of the envelope method as reconstruction were performed on surfaces with boundaries, segmented data sets, and non-orientable surfaces. The observations lend support to theoretical notions regarding sampling criteria and proper envelope constructions. New questions were posed concerning the presence of artifacts in some reconstructions and the need for adaptive envelope techniques or means of reducing the data requirements for computational efficiency. Figure 7: The original bunny reconstruction Acknowledgements The poster recounts some experiments and observations performed by undergraduates during the summer of 2005 with partial support from an NSF award for Research Experiences for Undergraduates. The undergraduate researches were Adam Gamzon and John Haga from the University of Connecticut and Jason Ribeiro from the University of Pennsylvania. The authors greatly value their efforts and contributions to this study. Partial funding for all authors was from NSF grants CCF , CCF and CCR Authors J. Bisceglio and T. J. Peters were also partially funded by a 2005 IBM Faculty Award. All statements in this publication are the responsibility of the authors, not of these funding sources. The authors express their appreciation for this support. References ABE, K., BISCEGLIO, J., PETERS, T. J., RUSSELL, A. C., FER- GUSON, D. R., AND SAKKALIS, T Computational topoloogy for reconstruction of surfaces with boundary: integrating experiments and theory. In IEEE International Conference on Shape Modeling and Applications, vol. 4, IEEE, AMENTA, N., PETERS, T. J., AND RUSSELL, A. C Computational topology: ambient isotopic approximation of 2- manifolds. Theoretical Computer Science 305, DEY, T. K., LI, G., AND SU, J Normal estimation for point clouds : a comparison study for a voronoi based method. In Eurographics Symposium on Point-Based Graphics, The Eurographics Association, OHTAKE, Y., BELYAEV, A., AND SEIDEL, H.-P An integrating approach to meshing scattered point data. In SPM 05: Proceedings of the 2005 ACM symposium on Solid and physical modeling, ACM Press, New York, NY, USA, ACM, PATRIKALAKIS, N. M., AND MAEKAWA, T Shape Interrogation for Computer Aided Design and Manufacturing. Springer-Verlag. SAKKALIS, T., AND PETERS, T. J Ambient isotopic approximations for surface reconstruction and interval solids. In SM 03: Proceedings of the eighth ACM symposium on Solid modeling and applications, ACM Press, New York, NY, USA, ACM,

6 Figure 8: A spline approximation of the boundary points with region of maximum curvature indicated by a red circle (top) and a the resulting envelope point cloud (bottom). The color shift in the data points indicates the ordering of the points that was defined for the spline reconstruction. The envelope was created by projecting the data points to the spline curve. Using the Frênet frame at the closest point on the curve, a set of envelope points are created around the sample point to form a circle orthogonal to the curve. The maximum curvature was used to guide both the radius and the number of samples taken from each circle. Figure 9: Several reconstructions of the boundary curve of the split bunny example. At the top is a reconstruction that was performed using the boundary envelope data from the original bunny example. The data was mirrored to create a pipe surface. The middle image was created using envelope data from the previous figure. The envelope points are centered around the original data points from the boundary of the split bunny set. The lower image is similar to the middle image except that additional envelope points have been introduced where these additional points are centered around sample points on the spline.