# Rephael Wenger: Publications

 Isosurfaces: Geometry, Topology, and Algorithms R. Wenger.  A.K. Peters/CRC Press, 2013. "Isosurfaces: Geometry, Topology, and Algorithms" is the book I wrote on isosurfaces.  Topics include the Marching Cubes algorithm and variants, dual contouring algorithms, multilinear interpolation, multiresolution isosurface extraction, isosurfaces in four dimensions, interval volumes, and contour trees. The book also describes data structures for faster isosurface extraction as well as methods for selecting significant isovalues. For more information, including the table of contents, see: http://www.crcpress.com/product/isbn/9781466570979 Chapter 2, Sections 1-3: A sample preview of the text.

## Journal and Conference Articles:

Constructing Isosurfaces with Sharp Edges and Corners using Cube Merging

A, Bhattacharya and R. Wenger.   Computer Graphics Forum, 32, 2013, 11-20.

A number of papers present algorithms to construct isosurfaces with sharp edges and corners from hermite data, i.e. the exact surface normals at the exact intersection
of the surface and grid edges. We discuss some fundamental problems with the previous algorithms and describe a new approach, based on merging grid cubes near sharp edges, that produces significantly better results. Our algorithm requires only gradients at the grid vertices, not at each surface-edge intersection point. We also give a method for measuring the correctness of the resulting sharp edges and corners in the isosurface.

Paper (pdf format)

Related tech report: Experimental Results on MergeSharp (pdf format)

On the  Fractal Dimension of Isosurfaces

M. Khoury and R. Wenger.   IEEE Transactions on Visualization and Computer Graphics,  16, 2010, 1198-1205.

The fractal dimension of an isosurface represents the growth in the isosurface as the number of grid cubes increases. We define and discuss the fractal isosurface dimension, present statistics on the average fractal dimension of 60 publicly available benchmark data sets, show the fractal dimension is highly correlated with topological noise in the benchmark data sets, and present a formula predicting the fractal dimension as a function of noise.

Paper(pdf format)
A Randomized O(m log m) Time Algorithm for Computing Reeb Graphs of Arbitrary Simplicial Complexes

W. Harvey, Y. Wang. and R. Wenger.   Proc. of the ACM Symposium on Computational Geometry (SOCG) 2010.

We present the first sub-quadratic algorithm to compute the Reeb graph for a function on an arbitrary simplicial complex K.  Our algorithm is randomized with an expected running time O(m log n) where m is the size of the 2-skeleton of K and n is the number of vertices.  Our algorithm is very simple to implement.

Paper(pdf format)
Isotopic Reconstruction of Surfaces with Boundaries

T. K. Dey, K. Li., E. A. Ramos, and R. Wenger.   Proc.  Sympos. Geom. Processing.(SGP09), special issue of Computer Graphics Forum, Vol. 28, No. 5 (2009), 1371-1382. [Web-page] [Software]

We present an algorithm for the reconstruction of a surface with boundaries (including a non-orientable one) in three dimensions from a sufficiently dense sample. It is guaranteed that the output is isotopic to the unknown sampled surface. No previously known algorithm guarantees isotopic or homeomorphic reconstruction of surfaces with boundaries. Our algorithm is surprisingly simple. It `peels' slivers greedily from an alpha-complex of a sample of the surface. No other post-processing is necessary. We provide several experimental results from an implementation of our basic algorithm and also a modified version of it.

Paper (pdf format)

#### Quality Isosurface Mesh Generation Using an Extended Marching Cubes Lookup Table

Sundaresan Raman and Rephael Wenger.
Computer Graphics Forum, 27, 2008, 791-798.

Abstract: The Marching Cubes Algorithm may return degenerate, zero area isosurface triangles, and often returns isosurface triangles with small areas, edges or angles. We show how to avoid both problems using an extended Marching Cubes lookup table. As opposed to the conventional Marching Cubes lookup table, the extended lookup table differentiates scalar values equal to the isovalue from scalar values greater than the isovalue. The lookup table has 38 = 6561 entries, based on three possible labels, '-' or '=' or '+', of each cube vertex. We present an algorithm based on this lookup table which returns an isosurface close to the Marching Cubes isosurface, but without any degenerate triangles or any small areas, edges or angles.

Paper (pdf format)

#### Isosurface Construction in Any Dimension Using Convex Hulls

Praveen Bhaniramka, Rephael Wenger and Roger Crawfis
IEEE Trans. on Visualization and Computer Graphics, 10, 2004, 353-400.

Abstract: We present an algorithm for constructing isosurfaces in any dimension. The input to the algorithm is a set of scalar values in a d-dimensional regular grid of (topological) hypercubes. The output is a set of (d-1)-dimensional simplices forming a piecewise linear approximation to the isosurface.  The algorithm constructs the isosurface piecewise within each hypercube in the grid using the convex hull of an appropriate set of points.  We prove that our algorithm correctly produces a triangulation of a (d-1)-manifold with boundary. In dimensions three and four, lookup tables with 28 and 216 entries, respectively, can be used to speed the algorithm’s running time. In three dimensions this gives the popular Marching Cubes algorithm.  We discuss applications of four dimensional isosurface construction to time varying isosurfaces, interval volumes and morphing.

Paper (pdf format)

#### Stability of Critical Points with Interval Persistence

Tamal K. Dey and Rephael Wenger
Discrete and Computational Geometry, 38, 2007, 479-512.

Abstract: Scalar functions defined on a topological space W are at the core of many applications such as shape matching, visualization and physical simulations. Topological persistence is an approach to characterizing these functions. It measures how long topological structures in the sub-level sets {x in W: f(x) <= c} persist as c changes. Recently it was shown that the critical values defining a topological structure with relatively large persistence remain almost unaffected by small perturbations.  This result suggests that topological persistence is a good measure for matching and comparing scalar functions. We extend these results to critical points in the domain by redefining persistence and critical points and replacing sub-level sets {x in W: f(x) <= c} with interval sets {x in W: a <= f(x) < b}. With these modifications we establish a stability result for critical points. This result is strengthened for maxima that can be used for matching two scalar functions.

Paper (pdf format)

#### Contour Area Filtering of 2-Dimensional Electrophoresis Images

Ramakrishnan-Kazhiyur-Mannar, Dominic J Smiraglia, Christoph Plass and Rephael Wenger
Medical Image Analysis, 10, 2006, 353-365.

Abstract: We describe an algorithm, Contour Area Filtering, for separating background from foreground in gray scale images.  The algorithm is based on the area contained within gray scale contour lines.  It can be viewed as a form of local thresholding, or as a seed growing algorithm, or as a type of watershed segmentation.  The most important feature of the algorithm is that it uses object area to determine the segmentation.  Thus it is relatively impervious to brightness and contrast variations across an image or between different images.

Contour Area Filtering was designed specifically for image analysis of 2D electrophoresis gels, although it can be applied to other gray scale images...

Paper (pdf format)

#### Restriction Landmark Genomic Scanning (RLGS) spot identification by second generation virtual RLGS in multiple genomes with multiple enzyme combinations

Authors: Dominic Smiraglia, Ramakrishnan Kazhiyur-Mannar, Christopher Oakes, Yue-Zhong Wu, Ping Liang, Tahmina Ansari, Jian Su, Laura Rush, Laura Smith, Li Yu, Chunhui Liu, Zunyan Dai, Shih-Shih Chen, Shu-Huei Wang, Joseph Costello, Ilya Ioshikhes, David Dawson, Jason Hong, Michael Teitell, Angela Szafranek, Marta Camoriano, Fei Song, Rosemary Elliott, William Held, Jacquetta Trasler, Christoph Plass and Rephael Wenger

BMC Genomics, 8:446, November 2007

Abstract ... We report the development of a virtual RLGS method (vRLGS) that allows for RLGS spot identification in any sequenced genome and with any enzyme combination.  We report significant improvements in predicting DNA fragment migration patterns by incorporating sequence information into the migration models, and demonstrate a median Euclidian distance between actual and predicted spot migration of 0.18 centimeters for the most complex human RLGS pattern. We report the confirmed identification of 795 human and 530 mouse RLGS spots for the most commonly used enzyme combinations.  We also developed a method to filter the virtual spots to reduce the number of extra spots seen on a virtual profile for both the mouse and human genomes.  We demonstrate use of this filter to simplify spot cloning and to assist in the identification of spots exhibiting tissue-specific methylation...

Paper (pdf format)

Other papers (selected):