Computer Graphics Group at Johns Hopkins University

	Metric-Aware Processing of Spherical Imagery M. Kazhdan and H. Hoppe SIGGRAPH Asia (2010) Abstract: Processing spherical images is challenging. Because no spherical parameterization is globally uniform, an accurate solver must account for the spatially varying metric. We present the first efficient metric-aware solver for Laplacian processing of spherical data. Our approach builds on the commonly used equirectangular parameterization, which provides differentiability, axial symmetry, and grid sampling. Crucially, axial symmetry lets us discretize the Laplacian operator just once per grid row. One difficulty is that anisotropy near the poles leads to a poorly conditioned system. Our solution is to construct an adapted hierarchy of finite elements, adjusted at the poles to maintain derivative continuity, and selectively coarsened to bound element anisotropy. The resulting elements are nested both within and across resolution levels. A streaming multigrid solver over this hierarchy achieves excellent convergence rate and scales to huge images. We demonstrate applications in reaction-diffusion texture synthesis and panorama stitching and sharpening.
	Closed-form Blending of Local Symmetries D. Ghosh, N. Amenta, and M. Kazhdan Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (July 2010) Abstract: We present a closed-form solution for the symmetrization problem, solving for the optimal deformation that reconciles a set of local bilateral symmetries. Given as input a set of point-pairs which should be symmetric, we first compute for each local neighborhood a transformation which would produce an approximate bilateral symmetry. We then solve for a single global symmetry which includes all of these local symmetries, while minimizing the deformation within each local neighborhood. Our main motivation is the symmetrization of digitized fossils, which are often deformed by a combination of compression and bending. In addition, we use the technique to symmetrize articulated models.
	A User-Assisted Approach to Visualizing Multidimensional Images J. Lawrence, S. Arietta, M. Kazhdan, D. Lepage, and C. O'Hagan IEEE Transactions on Visualizations and Computer Graphics (TVCG) (2010) Abstract: We present a new technique for fusing together an arbitrary number of aligned images into a single color or intensity image. We approach this fusion problem from the context of Multidimensional Scaling (MDS) and describe an algorithm that preserves the relative distances between pairs of pixel values in the input (vectors of measurements) as perceived differences in a color image. The two main advantages of our approach over existing techniques are that it can incorporate user constraints into the mapping process and it allows adaptively compressing or exaggerating features in the input in order to make better use of the output’s limited dynamic range. We demonstrate these benefits by showing applications in various scientific domains and comparing our algorithm to previously proposed techniques.
	A Statistical Approach for Achievable Dose Querying in IMRT Planning P. Simari, B. Wu, R. Jacques, A. King, T. McNutt, R. Taylor, and M. Kazhdan Medical Image Computing and Computer Assisted Intervention (MICCAI) (September 2010) Abstract: The task of IMRT planning, particularly in head-and-neck cancer, is a difficult one, often requiring days of work from a trained dosimetrist. One of the main challenges is the prescription of achievable target doses that will be used to optimize a treatment plan. This work explores a data-driven approach in which effort spent on past plans is used to assist in the planning of new patients. Using a database of treated patients, we identify the features of patient geometry that are correlated with received dose and use these to prescribe target dose levels for new patients. We incorporate our approach in a quality-control system, identifying patients with organs that received a dose significantly higher than the one recommended by our method. For all these patients, we have found that a replan using our predicted dose results in noticeable sparing of the organ without compromising dose to other treatment volumes.
	Organization of Data in Non-Convex Spatial Domains E. Perlman, R. Burns, M. Kazhdan, R. Murphy, W. Ball, and N. Amenta Scientific and Statistical Database Management Conference (SSDBM) (2010) Abstract: We present a technique for organizing data in spatial databases with non-convex domains based on an automatic characterization using the medial-axis transform (MAT). We define a tree based on the MAT and enumerate its branches to partition space and define a linear order on the partitions. This ordering clusters data in a manner that respects the complex shape of the domain. The ordering has the property that all data down any branch of the medial axis, regardless of the geometry of the sub-region, are contiguous on disk. Using this data organization technique, we build a system to provide efficient data discovery and analysis of the observational and model data sets of the Chesapeake Bay Environmental Observatory (CBEO). On typical CBEO workloads in which scientists query contiguous substructures of the Chesapeake Bay, we improve query processing performance by a factor of two when compared with orderings derived from space filling curves.
	Distributed Gradient-Domain Processing of Planar and Spherical Images M. Kazhdan, D. Surendran, and H. Hoppe ACM SIGGRAPH Transactions on Graphics (2010) Abstract: Gradient-domain processing is widely used to edit and combine images. In this paper we extend the framework in two directions. First, we adapt the gradient-domain approach to operate on a spherical domain, to enable operations such as seamless stitching, dynamic-range compression, and gradient-based sharpening over spherical imagery. An efficient streaming computation is obtained using a new spherical parameterization with bounded distortion and localized boundary constraints. Second, we design a distributed solver to efficiently process large planar or spherical images. The solver partitions images into bands, streams through these bands in parallel within a networked cluster, and schedules computation to hide the necessary synchronization latency. We demonstrate our contributions on several datasets including the Digitized Sky Survey, a terapixel spherical scan of the night sky.
	Patient Geometry-Driven Information Retrieval for IMRT Treatment Plan Quality Control B. Wu, F. Ricchetti, G. Sanguineti, M. Kazhdan, P. Simari, M. Chuang, R. Taylor, R. Jacques, and T. McNutt Medical Physics (December 2009, Vol. 36, No. 12), pages 5497--5505 Abstract: Intensity modulated radiation therapy IMRT treatment plan quality depends on the planner’s level of experience and the amount of time the planner invests in developing the plan. Planners often unwittingly accept plans when further sparing of the organs at risk OARs is possible. The authors propose a method of IMRT treatment plan quality control that helps planners to evaluate the doses of the OARs upon completion of a new plan.
	Multi-objective Shape Segmentation and Labeling P. Simari, D. Nowrouzezahrai, E. Kalogerakis, K. Singh Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (July 2009), pages 1415--1425 Abstract: Shape segmentations designed for different applications show significant variation in the composition of their parts. In this paper, we introduce the segmentation and labeling of shape based on the simultaneous optimization of multiple heterogenous objectives that capture application-specific segmentation criteria. We present a number of efficient objective functions that capture useful shape adjectives (compact, flat, narrow, perpendicular, etc.) Segmentation descriptions within our framework combine multiple such objective functions with optional labels to define each part. The optimization problem is simplified by proposing weighted Voronoi partitioning as a compact and continuous parametrization of spatially embedded shape segmentations. Separation of spatially close but geodesically distant parts is made possible using multi-dimensional scaling prior to Voronoi partitioning. Optimization begins with an initial segmentation found using the centroids of a k-means clustering of surface elements. This partition is automatically labeled to optimize heterogeneous part objectives and the Voronoi centers and their weights optimized using Generalized Pattern Search. We illustrate our framework using several diverse segmentation applications: consistent segmentations with semantic labels, bounding volume hierarchies for path tracing, and automatic rig and clothing transfer between animation characters.
	Estimating the Laplace-Beltrami Operator by Restricting 3D Functions (Slides, Source and Executables) M. Chuang, L. Luo, B. Brown, S. Rusinkiewicz, and M. Kazhdan Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (July 2009), pages 1475--1484 Abstract: We present a novel approach for computing and solving the Poisson equation over the surface of a mesh. As in previous approaches, we define the Laplace-Beltrami operator by considering the derivatives of functions defined on the mesh. However, in this work, we explore a choice of functions that is decoupled from the tessellation. Specifically, we use basis functions (second-order tensor-product B-splines) defined over 3D space, and then restrict them to the surface. We show that in addition to being invariant to mesh topology, this definition of the Laplace-Beltrami operator allows a natural multiresolution structure on the function space that is independent of the mesh structure, enabling the use of a simple multigrid implementation for solving the Poisson equation.
	Parallel Poisson Surface Reconstruction M. Bolitho, M. Kazhdan, R. Burns, and H. Hoppe Inernational Symposium on Visual Computing (November 2009) Abstract: In this work we describe a parallel implementation of the Poisson Surface Reconstruction algorithm based on multigrid domain decomposition. We compare implementations using different models of data-sharing between processors and show that a parallel implementation with distributed memory provides the best scalability. Using our method, we are able to parallelize the reconstruction of models from one billion data points on twelve processors across three machines, providing a ninefold speedup in running time without sacrificing reconstruction accuracy.
	A Shape Relationship Descriptor for Radiation Therapy Planning M. Kazhdan, P. Simari, T. McNutt, B. Wu, R. Jaques, M. Chuang, and R. Taylor Medical Image Computing and Computer Assisted Intervention (MICCAI, September 2009) Abstract: In this paper we address the challenge of matching patient geometry to facilitate the design of patient treatment plans in radiotherapy. To this end we propose a novel shape descriptor, the Overlap Volume Histogram, which provides a rotation and translation invariant representation of a patient’s organs at risk relative to the tumor volume. Using our descriptor, it is possible to accurately identify database patients with similar constellations of organ and tumor geometries, enabling the transfer of treatment plans between patients with similar geometries. We demonstrate the utility of our method for such tasks by outperforming state of the art shape descriptors in the retrieval of patients with similar treatment plans. We also preliminarily show its potential as a quality control tool by demonstrating how it is used to identify an organ at risk whose dose can be significantly reduced.
	Symmetry Restoration by Stretching M. Kazhdan, N. Amenta, S. Gu, D. Wiley, and B. Hamann Canadian Conference on Computational Geometry (August 2009) Abstract: We consider restoring the bilateral symmetry of an object which has been deformed by compression. This problem arises in paleontology, where symmetric bones are compressed in the process of fossilization. Our input is a user-selected set P of point-pairs on the deformed object, which are assumed to be mirror-images in some undeformed set AP, with some added noise. We carefully formulate the problem, and give a closed-form solution.
	Streaming Multigrid for Gradient-Domain Operations on Large Images (Slides, Supplemental Images, St James, Source and Executables) M. Kazhdan and H. Hoppe SIGGRAPH (August 2008) Abstract: We introduce a new tool to solve the large linear systems arising from gradient-domain image processing. Specifically, we develop a streaming multigrid solver, which needs just two sequential passes over out-of-core data. This fast solution is enabled by a combination of three techniques: (1) use of second-order finite elements (rather than traditional finite differences) to reach sufficient accuracy in a single V-cycle, (2) temporally blocked relaxation, and (3) multi-level streaming to pipeline the restriction and prolongation phases into single streaming passes. A key contribution is the extension of the B-spline finite-element method to be compatible with the forward-difference gradient representation commonly used with images. Our streaming solver is also efficient for inmemory images, due to its fast convergence and excellent cache behavior. Remarkably, it can outperform spatially adaptive solvers that exploit application-specific knowledge. We demonstrate seamless stitching and tone-mapping of gigapixel images in about an hour on a notebook PC.
	Organizing and Indexing Non-Convex Regions E. Perlman, R. Burns, and M. Kazhdan Demonstration Paper International Conference on Very Large Data Bases (VLDB) (2008), pages 1500--1503 Abstract: We demonstrate data indexing and query processing techniques that improve the efficiency of comparing, correlating, and joining data contained in non-convex regions. We use computational geometry techniques to automatically characterize the region of space from which data are drawn, partition the region based on that characterization, and create an index from the partitions. Our motivating application performs distributed data analysis queries among federated database sites that store scientific data sets from the Chesapeake Bay. Our preliminary findings indicate that these techniques often reduce the number of I/Os needed to serve a query by a factor of five -- depending on the geometry of the query region.
	Multilevel Streaming for Out-of-Core Surface Reconstruction (Slides, Source and Executables) M. Bolitho, M. Kazhdan, R. Burns, and H. Hoppe Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (July 2007), pages 69--78 Abstract: Reconstruction of surfaces from huge collections of scanned points often requires out-of-core techniques, and most such techniques involve local computations that are not resilient to data errors. We show that a Poisson-based reconstruction scheme, which considers all points in a global analysis, can be performed efficiently in limited memory using a streaming framework. Specifically, we introduce a multilevel streaming representation, which enables efficient traversal of a sparse octree by concurrently advancing through multiple streams, one per octree level. Remarkably, for our reconstruction application, a sufficiently accurate solution to the global linear system is obtained using a single iteration of cascadic multigrid, which can be evaluated within a single multi-stream pass. We demonstrate scalable performance on several large datasets.
	Unconstrained Isosurface Extraction on Arbitrary Octrees (Slides, Source and Executables) M. Kazhdan, A. Klein, K. Dalal, and H. Hoppe Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (July 2007), pages 125--133 Abstract: This paper presents a novel algorithm for generating a watertight level-set from an octree. We show that the level- set can be efficiently extracted regardless of the topology of the octree or the values assigned to the vertices. The key idea behind our approach is the definition of a set of binary edge-trees derived from the octree’s topology. We show that the edge-trees can be used define the positions of the isovalue-crossings in a consistent fashion and to resolve inconsistencies that may arise when a single edge has multiple isovalue-crossings. Using the edge-trees, we show that a provably watertight mesh can be extracted from the octree without necessitating the refinement of nodes or modification of their values.
	An Approximate and Efficient Method for Optimal Rotation Alignment of 3D Models M. Kazhdan IEEE Transactions on Pattern Analysis and Machine Intelligence (July 2007, Vol. 29, No. 7), page 1221--1229 Abstract: In many shape analysis applications, the ability to find the best rotation that aligns two models is an essential first step in the analysis process. In the past, methods for model alignment have either used normalization techniques, such as PCA alignment, or have performed an exhaustive search over the space of rotation to find the best optimal alignment. While normalization techniques have the advantage of efficiency, providing a quick method for registering two shapes, they are often imprecise and can give rise to poor alignments. Conversely, exhaustive search is guaranteed to provide the correct answer, but even using efficient signal processing techniques, this type of approach can be prohibitively slow. In this paper we present a new method for aligning two 3D shapes. We show that the method is markedly faster than existing approaches based on efficient signal processing and we provide registration results demonstrating that the alignments obtained using our method have a high degree of precision and are markedly better then those obtained using normalization.
	Poisson Surface Reconstruction (Slides, Source and Executables) M. Kazhdan, M. Bolitho, and H. Hoppe Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (June 2006), pages 61--70 Abstract: We show that surface reconstruction from oriented points can be cast as a spatial Poisson problem. This Poisson formulation considers all the points at once, without resorting to heuristic spatial partitioning or blending, and is therefore highly resilient to data noise. Unlike radial basis function schemes, our Poisson approach allows a hierarchy of locally supported basis functions, and therefore the solution reduces to a well conditioned sparse linear system. We describe a spatially adaptive multiscale algorithm whose time and space complexities are proportional to the size of the reconstructed model. Experimenting with publicly available scan data, we demonstrate reconstruction of surfaces with greater detail than previously achievable.
	Reconstruction of Solid Models from Oriented Point Sets (Slides, Source and Executables ) M. Kazhdan Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (July 2005), pages 73--82 Abstract: In this paper we present a novel approach to the surface reconstruction problem that takes as its input an oriented point set and returns a solid, water-tight model. The idea of our approach is to use Stokes' Theorem to compute the characteristic function of the solid model (the function that is equal to one inside the model and zero outside of it). Specifically, we provide an efficient method for computing the Fourier coefficients of the characteristic function using only the surface samples and normals, we compute the inverse Fourier transform to get back the characteristic function, and we use iso-surfacing techniques to extract the boundary of the solid model. The advantage of our approach is that it provides an automatic, simple, and efficient method for computing the solid model represented by a point set without requiring the establishment of adjacency relations between samples or iteratively solving large systems of linear equations. Furthermore, our approach can be directly applied to models with holes and cracks, providing a method for hole-filling and zippering of disconnected polygonal models.
	Visualization of Time-Varying Curvilinear Grids Using a 3D Warp Texture Y. Chen, J. D. Cohen, and Subodh Kumar Proceedings of Vision, Modeling, and Visualization 2005 (10th International Fall Workshop). 9 pages Abstract: We present a novel scheme to interactively visualize time-varying scalar fields defined on a curvilinear grid. We create a 3D warp texture that maps points inR3 into the grid coordinate system. At rendering time, the warping function is reconstructed at each fragment using tri-linear interpolation, and provides the 3D texture coordinates required to look up a scalar field value stored in a separate scalar texture. In essence, this approach reduces the problem of rendering a curvilinear grid to the problem of rendering a regular grid with one additional texture lookup. Because the curvilinear grid data typically lies on a regular grid in its native space, the scalar data is easily stored in a 3D texture without the need for explicit resampling. For many time-varying data sets, the warping function itself is constant over time, so only the 3D scalar texture needs to be reloaded with each time step. Thus this factorization of the problem minimizes the bandwidth requirements for time-varying playback. We demonstrate the approach on several data sets, achieving interactive performance with low approximation error.
	A Relational Debugging Engine for Graphics Pipeline N. Duca, K. Niski, J. Bilodeau, M. Bolitho, Y. Chen, and J. D. Cohen ACM Transactions on Graphics (Proceedings of SIGGRAPH 2005). 24 (3). 11 pages Abstract: We present a new, unified approach to debugging graphics software. We propose a representation of all graphics state over the course of program execution as a relational database, and produce a query-based framework for extracting, manipulating, and visualizing data from all stages of the graphics pipeline. Using an SQLbased query language, the programmer can establish functional relationships among all the data, linking OpenGL state to primitives to vertices to fragments to pixels. Based on the Chromium library, our approach requires no modification to or recompilation of the program to be debugged, and forms a superset of many existing techniques for debugging graphics software. [ video ]
	Hardware-Compatible Vertex Compression Using Quantization and Simplification B. Purnomo, J. Bilodeau, J. D. Cohen, and S. Kumar Proceedings of ACM SIGGRAPH/Eurographics Symposium on Graphics Hardware 2005. 10 pages Abstract: We present a vertex compression technique suitable for efficient decompression on graphics hardware. Given a user-specified number of bits per vertex, we automatically allocate bits to vertex attributes for quantization to maximize quality, guided by an image-space error metric. This allocation accounts for the constraints of graphics hardware by packing the quantized attributes into bins associated with the hardware's vectorized vertex data elements. We show that this general approach is also applicable if the user specifies a total desired model size. We present an algorithm that integrally combines vertex decimation and attribute quantization to produce the best quality model for a user-specified data size. Such models have an appropriate balance between the number of vertices and the number of bits per vertex. Vertex data is transmitted to and optionally stored in video memory in the compressed form. The vertices are decompressed on-the-fly using a vertex program at rendering time. Our algorithms not only work well within the constraints of current graphics hardware but also generalize to a setting where these constraints are relaxed. They apply to models with a wide variety of vertex attributes, providing new tools for optimizing space and bandwidthconstraints of interactive graphics applications.
	vLOD: High-Fidelity Walkthroughs of Large Virtual Environments J. Chhugani, B. Purnomo, S. Krishnan, J. D. Cohen, S. Venkatasubramanian, D. Johnson, and S. Kumar IEEE Transactions on Visualization and Computer Graphics, VOL. 11, NO. 1, 2005 Abstract: We present visibility computation and data organization algorithms that enable high-fidelity walkthroughs of large 3D geometric data sets. A novel feature of our walkthrough system is that it performs work proportional only to the required detail in visible geometry at the rendering time. To accomplish this, we use a precomputation phase that efficiently generates per cell vLOD: the geometry visible from a view-region at the right level of detail. We encode changes between neighboring cells’ vLODs, which are not required to be memory resident. At the rendering time, we incrementally construct the vLOD for the current view-cell and render it. We have a small CPU and memory requirement for rendering and are able to display models with tens of millions of polygons at interactive frame rates with less than one pixel screen-space deviation and accurate visibility.
	Seamless Texture Atlases B. Purnomo, J. D. Cohen, and S. Kumar Eurographics/ACM SIGGRAPH Symposium on Geometry Processing 2004 Abstract: Texture atlas parameterization provides an effective way to map a variety of color and data attributes from 2D texture domains onto polygonal surface meshes. However, the individual charts of such atlases are typically plagued by noticeable seams. We describe a new type of atlas which is seamless by construction. Our seamless atlas comprises all quadrilateral charts, and permits seamless texturing, as well as per-fragment down-sampling on rendering hardware and polygon simplification. We demonstrate the use of this atlas for capturing appearance attributes and producing seamless renderings.
	iClay: Digitizing Cuneiform J. D. Cohen, D. Snyder, D. Duncan, J. Cooper, S. Kumar, D. Hahn, Y. Chen, B. Purnomo, and J. Graettinger Virtual Reality, Archaeology and Cultural Heritage (VAST 2004) Abstract: Advances in digital technology for the graphic and textual representation of manuscripts have not, until recently, been applied to the worldʼs oldest manuscripts, cuneiform tablets. This is due in large part both to the three-dimensional nature of cuneiform tablets and to the complexity of the cuneiform script system. The Digital Hammurabi Project and the Initiative for Cuneiform Encoding announce success in encoding Sumero-Akkadian cuneiform in Unicode while also demonstrating advances in 3D scanning and visualization of cuneiform tablets, showcased by iClay, a cross-platform, Internet-deployable, Java applet that allows for the viewing and manipulation of 2D+ images of cuneiform tablets.
	On the Visualization of Time-Varying Structured Grids Using a 3D Warp Texture Y. Chen, J. D. Cohen, and S. Kumar Poster and Interactive Demos at IEEE Visualization 2004, Best Poster Award Abstract: We present a novel scheme to interactively visualize time-varying scalar fields defined on a structured grid. The underlying approach is to maximize the use of current graphics hardware by using 3D texture mapping. This approach commonly suffers from an expensive voxelization of each time-step as well as from large size of the voxel array approximating each step. Hence, in our scheme, instead of explicitly voxelizing each scalar field, we directly store each time-step as a three dimensional texture in its native form. We create the function that warps a voxel grid into the given structured grid. At rendering time, we reconstruct the function at each pixel using hardware-based trilinear interpolation. The resulting coordinates allow us to compute the scalar value at this pixel using a second texture lookup. For fixed grids, the function remains constant across time-steps and only the scalar field table needs to be re-loaded as a texture. Our new approach achieves excellent performance with relatively low texture memory requirements and low approximation error.
	Digital Preservation of Ancient Cuneiform Tablets Using 3D Scanning S. Kumar, D. Snyder, D. Dunca, J. D. Cohen, and J. Cooper Proceedings of IEEE Fourth International Conference on 3-D Digital Imaging and Modeling, 2003
	Budget Based Sampling of Parametric Surface Patches J. Chhugani, and S. Kumar ACM 3D Interactive Graphics 2003
	Successive Mapping: An Approach to Polygonal Mesh Simplification with Guaranteed Error Bounds J. D. Cohen, D. Manocha, and M. Olano International Journal of Computational Geometry and Applications. 13 (1). February 2003. pp. 61-94.
	Perceptually Guided Simplification of Lit, Textured Meshes N. Williams, D. Luebke, J. D. Cohen, M. Kelly, and B. Schubert Proceedings of 2003 ACM Symposium on Interactive 3D Graphics. pp. 113-121. (selected for cover image)
	GLOD: A Geometric Level of Detail System at the OpenGL API Level J. D. Cohen, N. Duca, D. Luebke, and B. Schubert IEEE Visualization 2003, Seattle, WA (2003), Best poster award. [Project Page]
	ISOSLIDER: A System for Interactive Exploration of IsoSurfaces J. Chhugani, S. Vishwanath, J. D. Cohen, and S. Kumar VisSym 2003. pp. 259--266.
	Level of Detail for 3D Graphics D. Luebke, M. Reddy, J. D. Cohen, A. Varshney, B. Watson, R. Huebner Morgan Kaufmann Publishing 2002. ISBN 1-55860-838-9. 432 pages [ Amazon ] [ Barnes & Noble ].
	Interactive Visualization of Unstructured Grids Using Hierarchical 3D Textures J. Leven, J. Corso, J. D. Cohen, and S. Kumar Proceedings of IEEE/SIGGRAPH Symposium on Volume Visualization and Graphics 2002. pp. 37-44 [ Video ]
	Hybrid Simplification: Combining Multi-resolution Polygon and Point Rendering J. D. Cohen, D. G. Aliaga, and W. Zhang Proceedings of IEEE Visualiztion 2001. pp. 37-44 and 539 [ Video ]
	View-Dependent Adaptive Tessellation of Spline Surfaces\ J. Chhugani, and S. Kumar ACM Symposium on 3D Interactive Graphics, 2001. pp 59-62
	Efficient Perspective-Accurate Silhouette Computation M. Pop, W. Huang, G. Barequet, C. A. Duncan, M. T. Goodrich, and S. Kumar ACM Computational Geometry, 2001. pp 60-68