Zohar Levi








School of Engineering and Computer Science

Victoria University of Wellington

New Zealand

I am a lecturer (assistant professor) at Victoria University of Wellington.


I was a postdoc researcher at the Courant Institute of Mathematical Sciences, New York University, working with Prof. Denis Zorin.

I earned my Ph.D. in Computer Science (2013) from the Technion - Israel Institute of Technology, under the supervision of Prof. Craig Gotsman.

I earned my M.Sc. in Computer Science (2007) from the Tel-Aviv University, Israel, under the supervision of Prof. David Levin.

I earned my B.A. in Computer Science (2004) from Tel-Aviv-Yaffo Academic College, Israel.


Research Interests


My main research interest is geometry processing.

Problems in the field include: low distortion mappings with applications of shape deformation and surface parametrization; shape interpolation; shape registration; and surface reconstruction. Algorithms and tools developed to address these problems are useful for shape modeling and animation.



Direct Seamless Parametrization

Zohar Levi

ACM Transactions on Graphics, 2021


Abstract: We present a method for seamless surface parametrization.

Recent popular methods first generate a cross-field, where curvature is concentrated at singular vertices. Next, in a separate step, the surface is laid out in the domain subject to derived seamlessness constraints.

This decoupling of the process into two independent problems, each with its own objective, leads to suboptimal results.

In contrast, our method solves both problems together using domain variables.

The key ingredient to the robustness of our method is a rounding strategy based on local estimation. The insight is that testing a small patch to decide between two likely possibilities is a good estimator.

Most distortion measures can be used with our method, which get minimized consistently throughout the pipeline.

Our method also enables feature alignment, as well as alignment to principle curvatures, and isotropic and anisotropic scaling.


[Paper] [Supplement] [Gallery1] [Gallery2]

Bounded Distortion Parametrization in the Space of Metrics

Edward Chien, Zohar Levi, and Ofir Weber

SIGGRAPH Asia (ACM Transactions on Graphics), 2016


Abstract: We present a framework for global parametrization that utilizes the

edge lengths (squared) of the mesh as variables. Given a mesh with arbitrary topology and prescribed cone singularities, we flatten the original metric of the surface under strict bounds on the metric distortion (various types of conformal and isometric measures are supported). Our key observation is that the space of bounded distortion metrics (given any particular bounds) is convex, and a broad range of useful and well-known distortion energies are convex as well. With the addition of nonlinear Gaussian curvature constraints, the parametrization problem is formulated as a constrained optimization problem, and a solution gives a locally injective map.

Our method is easy to implement. Sequential convex programming (SCP) is utilized to solve this problem effectively. We demonstrate the flexibility of the method and its uncompromised robustness and compare it to state-of-the-art methods.





On the Convexity and Feasibility of the Bounded Distortion Harmonic Mapping Problem

Zohar Levi, Ofir Weber

SIGGRAPH (ACM Transactions on Graphics), 2016


Abstract: Computation of mappings is a central building block in many geometry processing and graphics applications. The pursuit to compute mappings that are injective and have a controllable amount of conformal and isometric distortion is a long endeavor which has received significant attention by the scientific community in recent years. The difficulty of the problem stems from the fact that the space of bounded distortion mappings is nonconvex.

In this paper, we consider the special case of harmonic mappings which have been used extensively in many graphics applications. We show that, somewhat surprisingly, the space of locally injective planar harmonic mappings with bounded conformal and isometric distortion has a convex characterization. We describe several projection operators that, given an arbitrary input mapping, are guaranteed to output a bounded distortion locally injective harmonic mapping that is closest to the input mapping in some special sense. In contrast to alternative approaches, the optimization problems that correspond to our projection operators are shown to be always feasible for any choice of distortion bounds. We use the boundary element method (BEM) to discretize the space of planar harmonic mappings and demonstrate the effectiveness of our approach through the application of planar shape deformation.


[PDF] [Slides]



Smooth Rotation Enhanced As-Rigid-As-Possible Mesh Animation

Zohar Levi, Craig Gotsman

IEEE Transactions on Visualization and Computer Graphics, 2015


Abstract: We improve the As-Rigid-As-Possible (ARAP) animation technique in two aspects. First, we introduce a new ARAP energy, named SR-ARAP, which has a consistent discretization for surfaces (triangle meshes). The quality of our new surface deformation scheme competes with the quality of the volumetric ARAP deformation (for tetrahedral meshes). Second, we propose a new ARAP shape interpolation method that is superior to prior art also based on ARAP energy. This method is compatible with our new SR-ARAP energy, and with the ARAP volume energy.


[PDF] [Movie] [YouTube] [Code]

Strict Minimizers For Geometric Optimization

Zohar Levi, Denis Zorin

SIGGRAPH Asia (ACM Transactions on Graphics), 2014


Abstract: We introduce the idea of strict minimizers for geometric distortion measures used in shape interpolation, deformation, parametrization, and other applications involving geometric mappings. The -norm ensures the tightest possible control on the worst-case distortion. Unfortunately, it does not yield a unique solution and does not distinguish between solutions with high or low distortion below the maximum. The strict minimizer is a minimal -norm solution, which always prioritizes higher distortion reduction. We propose practical algorithms for computing strict minimizers. We also offer an efficient algorithm for  optimization based on the ARAP energy. This algorithm can be used on its own or as a building block for an ARAP strict minimizer. We demonstrate that these algorithms lead to significant improvements in quality.


[PDF] [PDF hi-res] [1-ring optimization] [Linf dual formulation] [Code] [Slides]

Shape Deformation via Interior RBF

Zohar Levi, David Levin

IEEE Transactions on Visualization and Computer Graphics, 2014


Abstract: We present a new framework for real-time shape deformation with local shape preservation and volume control. Given a 3D object, in any form, one would like to manipulate the object using convenient handles, so that the resulting shape is a natural variation of the given object. It is also important that the deformation is controlled, thereby enabling localized changes that do not influence nearby branches. For example, given a horse model, a movement of one of its hooves should not affect the other hooves. Another goal is the minimization of local shape distortion throughout the object. The first ingredient of our method is the use of Interior Radial Basis Functions (IRBF), where the functions are radial with respect to interior distances within the object. The second important ingredient is the reduction of local distortions by minimizing the distortion of a set of spheres placed within the object. Our method achieves the goals of convenient shape manipulation and local influence property, and improves the latest state-of-the-art cage-based methods by replacing the cage with the more flexible IRBF centers. The latter enables extra flexibility and fully automated construction, as well as simpler formulation.


[PDF] [Movie] [YouTube] [Poster] [Slides]

ArtiSketch: A System for Articulated Sketch Modeling

Zohar Levi, Craig Gotsman

Eurographics (Computer Graphics Forum), 2013


Abstract: We present ArtiSketch - a system which allows the conversion of a wealth of existing 2D content into 3D content by users who do not necessarily possess artistic skills. Using ArtiSketch, a novice user may describe a 3D model as a set of articulated 2D sketches of a shape from different viewpoints. ArtiSketch then automatically converts the sketches to an articulated 3D object. Using common interactive tools, the user provides an initial estimate of the 3D skeleton pose for each frame, which ArtiSketch refines to be consistent between frames. This skeleton may then be manipulated independently to generate novel poses of the 3D model.


[PDF] [Movie] [YouTube] [Slides]

D-Snake: Image Registration by As-Similar-As-Possible Template Deformation

Zohar Levi, Craig Gotsman

IEEE Transactions on Visualization and Computer Graphics, 2013


Abstract: We describe a snake-type method for shape registration in 2D and 3D, by fitting a given polygonal template to an acquired image or volume data. The snake aspires to fit itself to the data in a shape which is locally As-Similar-As-Possible (ASAP) to the template. Our ASAP regulating force is based on the Moving Least Squares (MLS) similarity deformation. Combining this force with the traditional internal and external forces associated with a snake leads to a powerful and robust registration algorithm, capable of extracting precise shape information from image data.


[PDF] [Movie] [YouTube]





MLS-Based Remeshing for Improving Surface Approximation

Zohar Levi, David Levin

M.Sc. thesis, Tel-Aviv University, 2007


Abstract: We present a remeshing technique for a better approximation of functions and surfaces. Given a triangulation of a surface, find a triangulation with the same number of vertices, which best approximates the underlying surface. The vertices are redistributed according to a local function error estimation to be close as possible to a MLS surface, which is used to approximate the source surface of the input mesh. A further improvement is achieved by an edge-flip algorithm. Moreover, we show two practical heuristics for choosing parameters for the MLS, and suggest a way to measure the quality of the remeshing.