In the past I've worked on projects in other areas, such as human computation and geometric subdivision.

Click the project titles or thumbnails to go to the project page/paper.

Click the project titles or thumbnails to go to the project page/paper.

Diamonds from the Rough: Improving Drawing, Painting, and Singing via CrowdsourcingYotam Gingold, Etienne Vouga, Eitan Grinspun, and Haym Hirsh Proceedings of the AAAI Workshop on Human Computation (HCOMP), 2012 It is well established that in certain domains, noisy inputs can be reliably combined to obtain a better answer than any individual. It is now possible to consider the crowdsourcing of physical actions, commonly used for creative expressions such as drawing, shading, and singing. We provide algorithms for converting low-quality input obtained from the physical actions of a crowd into high-quality output. The inputs take the form of line drawings, shaded images, and songs. We investigate single-individual crowds (multiple inputs from a single human) and multiple-individual crowds. |

On the Smoothness of Real-Valued Functions Generated by Subdivision Schemes Using Nonlinear Binary AveragingRon Goldman, Etienne Vouga, and Scott Schaefer Computer Aided Geometric Design, 2009 Our main result is that two point interpolatory subdivision schemes using C^k nonlinear averaging rules on pairs of real numbers generate real-valued functions that are also C^k. The significance of this result is the following consequence: Suppose that S is a subdivision algorithm operating on sequences of real numbers using linear binary averaging that generates C^m real-valued functions and S is the same subdivision procedure where linear binary averaging is replaced everywhere in the algorithm by a C^n nonlinear binary averaging rule on pairs of real numbers; then the functions generated by the nonlinear subdivision scheme S are C^k , where k = min(m, n). |

Two Blossoming Proofs of the Lane-Riesenfeld AlgorithmEtienne Vouga and Rom Goldman Computing, 2007 The standard proof of the Lane-Riesenfeld algorithm for inserting knots into uniform B-spline curves is based on the continuous convolution formula for the uniform B-spline basis functions. Here we provide two new, elementary, blossoming proofs of the Lane-Riesenfeld algorithm for uniform B-spline curves of arbitrary degree. |

Nonlinear Subdivision Through Nonlinear AveragingScott Schaefer S, Etienne Vouga, and Ron Goldman Computer Aided Geometric Design, 2008 We investigate a general class of nonlinear subdivision algorithms for functions of a real or complex variable built from linear subdivision algorithms by replacing binary linear averages such as the arithmetic mean by binary nonlinear averages such as the geometric mean. Using our method, we can easily create stationary subdivision schemes for Gaussian functions, spiral curves, and circles with uniform parametrizations. More generally, we show that stationary subdivision schemes for e^p(x), cos(p(x)) and sin(p(x)) for any polynomial or piecewise polynomial p(x) can be generated using only addition, subtraction, multiplication, and square roots. The smoothness of our nonlinear subdivision schemes is inherited from the smoothness of the original linear subdivision schemes and the differentiability of the corresponding nonlinear averaging rules. While our results are quite general, our proofs are elementary, based mainly on the observation that generic nonlinear averaging rules on a pair of real or complex numbers can be constructed by conjugating linear averaging rules with locally invertible nonlinear maps. In a forthcoming paper we show that every continuous nonlinear averaging rule on a pair of real or complex numbers can be constructed by conjugating a linear averaging rule with an associated continuous locally invertible nonlinear map. Thus the averaging rules considered in this paper are actually the general case. As an application we show how to apply our nonlinear subdivision algorithms to intersect some common transcendental functions. |