Shape‐Up: Shaping Discrete Geometry with Projections

Abstract

We introduce a unified optimization framework for geometry processing based on shape constraints. These constraints preserve or prescribe the shape of subsets of the points of a geometric data set, such as polygons, one-ring cells, volume elements, or feature curves. Our method is based on two key concepts: a shape proximity function and shape projection operators. The proximity function encodes the distance of a desired least-squares fitted elementary target shape to the corresponding vertices of the 3D model. Projection operators are employed to minimize the proximity function by relocating vertices in a minimal way to match the imposed shape constraints. We demonstrate that this approach leads to a simple, robust, and efficient algorithm that allows implementing a variety of geometry processing applications, simply by combining suitable projection operators. We show examples for computing planar and circular meshes, shape space exploration, mesh quality improvement, shape-preserving deformation, and conformal parametrization. Our optimization framework provides a systematic way of building new solvers for geometry processing and produces similar or better results than state-of-the-art methods.

Real-Time Deformation of Constrained Meshes Using GPU

“…Besides [6] and [8], other handle-based deformation methods for constrained meshes have been developed in recent years. Zhao et al [26] extended the shape space exploration approach in [25], using curve handles for shape control.…”

“…Poranne et al [19] provided an optimization approach to deform polyhedral meshes, not limited to affine transformations of faces. The deformation is computed through an alternating least-squares approach similar to [6]. However, only face planarity constraints are considered by the method.…”

“…In this way the user only explores panel layouts that satisfy all the requirements, with intuitive feedback about what modifications are possible under the given requirements. Such fabrication-aware shape exploration methods for freeform architecture have been a popular research topic recently [25,6,24,26,19,9].…”

“…Bouaziz et al [6] proposed a general framework for handle-based deformation of meshes subject to soft constraints, formulated as a nonlinear least squares problem. Utilizing projections of individual mesh elements onto their feasible configurations, they propose an iterative solver that alternates between global linear system solving and local mesh element projections.…”

“…Recently, this method was extended in [8] to allow both hard and soft constraints. The proposed numerical solver consists of a series of simple subproblems similar to those in [6], enabling speedup from parallelism. In this paper, we present an implementation of the method in [8] on GPU using CUDA, which provides many more computational cores than CPU.…”