Keenan Crane
COLUMBIA UNIVERSITY IN THE CITY OF NEW YORK
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Keenan Crane
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About
I am a postdoctoral fellow at Columbia University, working with Eitan Grinspun. I received a bachelor's degree from the University of Illinois at Urbana-Champaign and a PhD from the California Institute of Technology, advised by Peter Schröder and Mathieu Desbrun. My research draws on insights from differential geometry and computer science to develop fundamental algorithms for working with real-world geometric data. My recent work is supported by a Google PhD Fellowship and an NSF Mathematical Sciences Postdoctoral Fellowship. Starting fall 2015, I will be an assistant professor in the School of Computer Science at Carnegie Mellon University.
Research Overview
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I am actively seeking talented, adventurous, and friendly students and postdocs with a passion for geometry, computation, and all things visual to help build a world-class research group at Carnegie Mellon. A strong mathematical background is a definite plus. Interested students should apply this fall through the SCS online application; deadline is December 3, 2014. Feel free to contact me to find out more!
Publications
Robust Fairing via
Conformal Curvature Flow

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Crane, Pinkall, Schröder
ACM Transactions on Graphics 2013

Abstract
This paper presents a formulation of Willmore flow for triangulated surfaces that permits extraordinarily large time steps and naturally preserves the quality of the input mesh. The main insight is that Willmore flow becomes remarkably stable when expressed in curvature space—we develop the precise conditions under which curvature is allowed to evolve. The practical outcome is a highly efficient algorithm that naturally preserves texture and does not require remeshing during the flow. We apply this algorithm to surface fairing, geometric modeling, and construction of constant mean curvature (CMC) surfaces. We also present a new algorithm for length-preserving flow on planar curves, which provides a valuable analogy for the surface case.
BibTeX

@article{Crane:2013:RFC,
author = {Crane, Keenan and Pinkall, Ulrich and Schr\"{o}der, Peter},
title = {Robust Fairing via Conformal Curvature Flow},
journal = {ACM Trans. Graph.},
volume = {32},
issue = {4},
year = {2013},
publisher = {ACM},
address = {New York, NY, USA},
}


Globally Optimal Direction Fields

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Knöppel, Crane, Pinkall, Schröder
ACM Transactions on Graphics 2013

Abstract

This paper presents a method for constructing smooth unit n-direction fields (line fields, cross fields, etc.) on surfaces that is an order of magnitude faster than state-of-the-art methods, while still producing fields of equal or better quality. The method is based on a simple quadratic energy whose minimizers are globally optimal in the sense that they produce the smoothest fields over all possible configurations of singularities (number, location, and index). The method is fully automatic and can optionally produce fields aligned with a given guidance field, for example, principal curvature directions. Computationally the smoothest field is found via a sparse eigenvalue problem involving a matrix similar to the cotan-Laplacian. When a guidance field is present, finding the optimal field amounts to solving a single linear Poisson problem.

BibTeX

@article{Knoppel:2013:GOD,
author = {Kn\"{o}ppel, Felix and Crane, Keenan and Pinkall, Ulrich and Schr\"{o}der, Peter},
title = {Globally optimal direction fields},
journal = {ACM Trans. Graph.},
volume = {32},
number = {4},
year = {2013},
publisher = {ACM},
address = {New York, NY, USA},
}

Geodesics in Heat:
A New Approach to Computing
Distance Based on Heat Flow

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Crane, Weischedel, Wardetzky
ACM Transactions on Graphics 2013

Abstract

We introduce the heat method for computing the geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard linear elliptic problems. The resulting systems can be prefactored once and subsequently solved in near-linear time. In practice, distance is updated an order of magnitude faster than with state-of-the-art methods, while maintaining a comparable level of accuracy. The method requires only standard differential operators and can hence be applied on a wide variety of domains (grids, triangle meshes, point clouds, etc.). We provide numerical evidence that the method converges to the exact distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where greater regularity is required.

BibTeX

@article{Crane:2013:GH,
author = {Keenan Crane and Clarisse Weischedel and Max Wardetzky},
title = {{Geodesics in Heat: A New Approach to Computing Distance Based on Heat Flow}},
journal = {ACM Trans. Graph.},
volume = {32},
issue = {5},
year = {2013},
publisher = {ACM},
address = {New York, NY, USA}
}

Spin Transformations
of Discrete Surfaces

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Crane, Pinkall, Schröder
ACM Transactions on Graphics 2011

Abstract

This paper introduces a new method for computing conformal transformations of triangle meshes in R3. Conformal maps are desirable in digital geometry processing because they do not exhibit shear, and therefore preserve texture fidelity as well as the quality of the mesh itself. Traditional discretizations consider maps into the complex plane, which are useful only for problems such as surface parameterization and planar shape deformation where the target surface is flat. We instead consider maps into the quaternions, which allows us to work directly with surfaces sitting in R3. In particular, we introduce a quaternionic Dirac operator and use it to develop a novel integrability condition on conformal deformations. Our discretization of this condition results in a sparse linear system that is simple to build and can be used to efficiently edit surfaces by manipulating curvature and boundary data, as demonstrated via several mesh processing applications.

BibTeX

@article{Crane:2011:STD,
author = {Crane, Keenan and Pinkall, Ulrich and Schr\"{o}der, Peter},
title = {Spin Transformations of Discrete Surfaces},
journal = {ACM Trans. Graph.},
volume = {30},
issue = {4},
year = {2011},
publisher = {ACM},
address = {New York, NY, USA},
}

Trivial Connections
on Discrete Surfaces

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Crane, Desbrun, Schröder
Computer Graphics Forum /
SGP 2010—Best Paper Award

Abstract

This paper presents a straightforward algorithm for constructing connections on surfaces that are as smooth as possible everywhere but on a prescribed set of isolated singularities with given index. Such connections can be used to design rotationally symmetric direction fields, which are essential in applications such as quadrilateral remeshing and texture synthesis. We compute connections by solving a single sparse linear system built from standard operators. Our solutions are globally optimal in the sense that they describe the trivial connection closest to Levi-Civita among all connections with the prescribed set of singularities. Relative to previous methods our algorithm is surprisingly simple, and can be implemented using standard operations from mesh processing and linear algebra.

BibTeX

@article{Crane:2010:TCD,
author={Keenan Crane and Mathieu Desbrun and Peter Schr\"{o}der},
title={Trivial Connections on Discrete Surfaces},
journal={Computer Graphics Forum (SGP)},
volume={29},
number={5},
pages={1525-1533},
year={2010}
}

Energy-Preserving Integrators
for Fluid Animation

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Mullen, Crane, Pavlov, Tong, Desbrun
ACM Transactions on Graphics 2009

Abstract

Numerical viscosity has long been a problem in fluid animation. Existing methods suffer from intrinsic artificial dissipation and often apply complicated computational mechanisms to combat such effects. Consequently, dissipative behavior cannot be controlled or modeled explicitly in a manner independent of time step size, complicating the use of coarse previews and adaptive time-stepping methods. This paper proposes simple, unconditionally stable, fully Eulerian integration schemes with no numerical viscosity that are capable of maintaining the liveliness of fluid motion without recourse to corrective devices. Pressure and fluxes are solved efficiently and simultaneously in a time-reversible manner on simplicial grids, and the energy is preserved exactly over long time scales in the case of inviscid fluids. These integrators can be viewed as an extension of the classical energy-preserving Harlow-Welch / Crank-Nicolson scheme to simplicial grids.

BibTeX

@article{Mullen:2009:EIF,
author = {Mullen, Patrick and Crane, Keenan and Pavlov, Dmitry and Tong, Yiying and Desbrun, Mathieu},
title = {Energy-preserving integrators for fluid animation},
journal = {ACM Trans. Graph.},
volume = {28},
issue = {3},
month = {July},
year = {2009},
}

Lie Group Integrators for
Animation and Control of Vehicles

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Kobilarov, Crane, Desbrun
ACM Transactions on Graphics 2009

Abstract

This paper is concerned with the animation and control of vehicles with complex dynamics such as helicopters, boats, and cars. Motivated by recent developments in discrete geometric mechanics we develop a general framework for integrating the dynamics of holonomic and nonholonomic vehicles by preserving their state-space geometry and motion invariants. We demonstrate that the resulting integration schemes are superior to standard methods in numerical robustness and efficiency, and can be applied to many types of vehicles. In addition, we show how to use this framework in an optimal control setting to automatically compute accurate and realistic motions for arbitrary user-specified constraints.

BibTeX

@article{Kobilarov:2009:LGI,
author = {Kobilarov, Marin and Crane, Keenan and Desbrun, Mathieu},
title = {Lie group integrators for animation and control of vehicles},
journal = {ACM Trans. Graph.},
volume = {28},
issue = {2},
month = {May},
year = {2009},
}

Capturing and Animating
Occluded Cloth

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White, Crane, Forsyth
ACM Transactions on Graphics 2007

Abstract

We capture the shape of moving cloth using a custom set of color markers printed on the surface of the cloth. The output is a sequence of triangle meshes with static connectivity and with detail at the scale of individual markers in both smooth and folded regions. We compute markers' coordinates in space using marker correspondence across multiple synchronized video cameras. Correspondence is determined from color information in small neighborhoods and refined using a novel strain pruning process. Final correspondence does not require neighborhood information. We use a novel data driven hole-filling technique to fill occluded regions. Our results include several challenging examples: a wrinkled shirt sleeve, a dancing pair of pants, and a rag tossed onto a cup. Finally, we demonstrate that cloth capture is reusable by animating a pair of pants using human motion capture data.

BibTeX

@article{White:2007:CAO,
author = {White, Ryan and Crane, Keenan and Forsyth, D. A.},
title = {Capturing and animating occluded cloth},
journal = {ACM Trans. Graph.},
volume = {26},
issue = {3},
month = {July},
year = {2007},
}

A Two-Color Map of Pluto's
Sub-Charon Hemisphere

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Young, Binzel, Crane
The Astronomical Journal 2001

Abstract

Pluto and its satellite Charon regularly occulted or transited each other's disks from 1985 through 1990. The light curves resulting from these events (collectively called "mutual events") have been used to determine albedo maps of Pluto's sub-Charon hemisphere. We now use a data set of four light curves that were obtained in both B and V Johnson filters to construct a two-color map of Pluto's surface. We are able to resolve the central part of Pluto's sub-Charon hemisphere. We find that the dark albedo feature that forms a band below Pluto's equator is comprised of several distinct color units. We detect ratios of V-filter/B-filter normal reflectances ranging from 1.15 to 1.39 on Pluto's sub-Charon hemisphere.

BibTeX

@article{1538-3881-121-1-552,
author={Eliot F. Young and Richard P. Binzel and Keenan Crane},
title={A Two-Color Map of Pluto's Sub-Charon Hemisphere},
journal={The Astronomical Journal},
volume={121},
number={1},
pages={552},
url={http://stacks.iop.org/1538-3881/121/i=1/a=552},
year={2001}
}

Digital Geometry Processing with
Discrete Exterior Calculus

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Crane, de Goes, Desbrun, Schröder
SIGGRAPH 2013 Course Notes

Abstract

This course provides an introduction to geometry processing using discrete exterior calculus (DEC). DEC provides a simple, flexible, and efficient framework within which one can build a unified platform for geometry processing. The course provides essential mathematical background as well as a large array of real-world examples. It also provides a short survey of the most relevant recent developments in digital geometry processing and discrete differential geometry.

BibTeX

@inproceedings{Crane:2013:DGP,
author = {Keenan Crane, Fernando de Goes, Mathieu Desbrun, Peter Schröder},
title = {Digital Geometry Processing with Discrete Exterior Calculus},
booktitle = {ACM SIGGRAPH 2013 courses},
series = {SIGGRAPH '13},
year = {2013},
location = {Anaheim, California},
numpages = {126},
publisher = {ACM},
address = {New York, NY, USA},
}

Discrete Connections
for Geometry Processing

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Keenan Crane
Caltech Master's Thesis (2010)

Abstract

Connections provide a way to compare local quantities defined at different points of a geometric space. This thesis develops a discrete theory of connections that naturally leads to practical, efficient numerical algorithms for geometry processing, including texture synthesis and quadrilateral remeshing. Our formulation is motivated by real-world applications where meshes may be noisy or coarsely discretized. Further, because our discrete framework closely parallels the smooth theory, we can draw upon a huge wealth of existing knowledge to develop and interpret mesh processing algorithms. The solutions we produce are globally optimal in the sense that they describe the trivial connection closest to Levi-Civita among all solutions with the prescribed set of singularities. Relative to previous methods our algorithm is surprisingly simple, and can be implemented using standard operations from mesh processing and linear algebra.

BibTeX

@mastersthesis{caltechthesis5880,
title = {Discrete connections for geometry processing},
school = {California Institute of Technology},
author = {Keenan Crane},
year = {2010},
url = {http://resolver.caltech.edu/CaltechTHESIS:05282010-102307125}
}

Multiscale 3D Reference Visualization
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Glueck, Crane, Anderson, Rutnik, Khan
i3D 2009 /
Symposium on Interactive Graphics

Abstract

Reference grids are commonly used in design software to help users to judge distances and to understand the orientation of the virtual workspace. However, despite their ubiquity in 3D graphics authoring applications, little research has gone into important design considerations of the 3D reference grids themselves, which directly impact their usefulness. In an effort to resolve some of these outstanding issues, we have developed two new techniques; the multiscale reference grid and position pegs that form a consistent foundation for presenting relative scale and position information to the user.
Our design of a multiscale reference grid consistently subdivides and coalesces gridlines, based on the computation of a closeness metric, while ensuring that there are neither too many nor too few subdivisions. Position pegs extend the grid so that objects that are lying above or below the ground plane can be brought into a common environmental frame of reference without interfering with the grid or object data. We introduce an analytical system, similar to MIP mapping, to provide a stable viewpoint-determined result. Our design solves several depth cue problems in a way that is independent of viewing projection.

BibTeX

@inproceedings{Glueck:2009:MRV,
author = {Glueck, Michael and Crane, Keenan and Anderson, Sean and Rutnik, Andres and Khan, Azam},
title = {Multiscale 3D reference visualization},
booktitle = {Proceedings of the 2009 symposium on Interactive 3D graphics and games},
series = {I3D '09},
year = {2009},
isbn = {978-1-60558-429-4},
location = {Boston, Massachusetts},
pages = {225--232},
numpages = {8},
url = {http://doi.acm.org/10.1145/1507149.1507186},
doi = {http://doi.acm.org/10.1145/1507149.1507186},
acmid = {1507186},
publisher = {ACM},
address = {New York, NY, USA},
}


Data Driven Cloth Animation

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White, Crane, Forsyth
SIGGRAPH 2009 Technical Sketches

Abstract

We present a new method for cloth animation based on data driven synthesis. In contrast to approaches that focus on physical simulation, we animate cloth by manipulating short sequences of existing cloth animation. While our source of data is cloth animation captured using video cameras, the method is equally applicable to simulation data. The approach has benefits in both cases: current cloth capture is limited because small tweaks to the data require filming an entirely new sequence. Likewise, simulation suffers from long computation times and complications such as tangling. In this sketch we create new animations by fitting cloth animation to human motion capture data, i.e., we drive the cloth with a skeleton.

BibTeX

@inproceedings{White:2007:DDC,
author = {White, Ryan and Crane, Keenan and Forsyth, D. A.},
title = {Data driven cloth animation},
booktitle = {ACM SIGGRAPH 2007 sketches},
series = {SIGGRAPH '07},
year = {2007},
location = {San Diego, California},
articleno = {37},
acmid = {1278825},
publisher = {ACM},
address = {New York, NY, USA},
}

Real Time Simulation and
Rendering of 3D Fluids

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Crane, Llamas, Tariq
GPU Gems 3 (2007)

Abstract

Physically based animation of fluids such as smoke, water, and fire provides some of the most stunning visuals in computer graphics, but has historically been the domain of high-quality offline rendering due to great computational cost. In this chapter we not only show how these effects can be simulated and rendered in real time, but also how they can be seamlessly integrated into real time applications.

BibTeX

@InBook{Crane07,
author = "Crane, Keenan and Llamas, Ignacio, and Tariq, Sarah",
title = "Real Time Simulation and Rendering of 3D Fluids",
booktitle = "GPUGems 3",
editor = "Hubert Nguyen",
chapter = "30",
publisher = "Addison-Wesley",
year = "2007"
}

Rectangular Multi-Chart
Geometry Images

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Carr, Hoberock, Crane, Hart
SGP 2006 /
Symposium on Geometry Processing

Abstract

Many mesh parameterization algorithms have focused on minimizing distortion and utilizing texture area, but few have addressed issues related to processing a signal on the mesh surface. We present an algorithm which partitions a mesh into rectangular charts while preserving a one-to-one texel correspondence across chart boundaries. This mapping permits any computation on the mesh surface which is typically carried out on a regular grid, and prevents seams by ensuring resolution continuity along the boundary. These features are also useful for traditional texture applications such as surface painting where continuity is important. Distortion is comparable to other parameterization schemes, and the rectangular charts yield efficient packing into a texture atlas. We apply this parameterization to texture synthesis, fluid simulation, mesh processing and storage, and locating geodesics.

BibTeX

@inproceedings{Carr:2006:RMG,
author = {Carr, Nathan A. and Hoberock, Jared and Crane, Keenan and Hart, John C.},
title = {Rectangular Multi-Chart Geometry Images},
booktitle = {Proceedings of the fourth Eurographics symposium on Geometry processing},
year = {2006},
}

Fast GPU Ray Tracing of Dynamic
Meshes using Geometry Images

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Carr, Hoberock, Crane, Hart
Graphics Interface 2006

Abstract

Using the GPU to accelerate ray tracing may seem like a natural choice due to the highly parallel nature of the problem. However, determining the most versatile GPU data structure for scene storage and traversal is a challenge. In this paper, we introduce a new method for quick intersection of triangular meshes on the GPU. The method uses a threaded bounding volume hierarchy built from a geometry image, which can be efficiently traversed and constructed entirely on the GPU. This acceleration scheme is highly competitive with other GPU ray tracing methods, while allowing for both dynamic geometry and an efficient level of detail scheme at no extra cost.

BibTeX

@inproceedings{Carr:2006:FGR,
author = {Carr, Nathan A. and Hoberock, Jared and Crane, Keenan and Hart, John C.},
title = {Fast GPU ray tracing of dynamic meshes using geometry images},
booktitle = {Proceedings of Graphics Interface 2006},
series = {GI '06},
year = {2006},
isbn = {1-56881-308-2},
location = {Quebec, Canada},
pages = {203--209},
numpages = {7},
url = {http://portal.acm.org/citation.cfm?id=1143079.1143113},
acmid = {1143113},
publisher = {Canadian Information Processing Society},
address = {Toronto, Ont., Canada, Canada},
}

Code
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libgeodesic —highly optimized library for distance transforms, based on the heat method. Coming Soon!
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fieldgen —fully automatic generation of optimal direction fields on surfaces, with optional curvature alignment. Coming Soon!
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Comb —optimal direction field design with user-specified singularities. Based on our trivial connections paper.
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SpinXForm —core solver for conformal (i.e., angle-preserving) geometry processing, based on spin transformations. Naturally preserves texture and mesh quality.
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DGPDEC —unified framework for geometry processing algorithms based on discrete exterior calculus (DEC).
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QJulia —fragment shader for real-time rendering of quaternion Julia sets.
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tilings —generates meshes of all the regular and semi-regular tilings of the plane (in Wavefront OBJ format).
Miscellaneous / Fun
Triangle Areas Cheat Sheet (2010)
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Variational Principles Cheat Sheet (2009)
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Synthesizing the
Sound of Splashing (2007)

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Homotopy, Homology,
and Cut Graphs (2006)

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A Survey of Efficient Structures
for Geometry Processing (2006)

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Bias in Rendering (2006)
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Making Fractals with Christmas Ornaments (2005)
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GraphDraw (2005)

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Poincaré-Hopf, Revisited (2005)

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Ray Tracing Quaternion
Julia Sets on the GPU (2004)

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3D Tashoku Go (2003)

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