Two-dimensional shape deformation using geometric energy optimization

A good 2D shape deformation system aims to provide intuitive user interface, generate physically plausible results, and perform at interactive frame rates. To achieve these objectives, I propose a new class of 2D shape deformation algorithms based on geometric energy optimization. The proposed geometric energy is an effective metric of the geometric features of the 2D shape. Preserving these features during deformation can produce convincing results. And most importantly, I show that this geometric energy optimization problem can be solved efficiently. Each algorithm corresponds to an interactive 2D shape deformation system that can achieve physically plausible results with simple operations which are difficult to achieve with previous methods. This dissertation consists of three main parts:;(1) 2D shape deformation using nonlinear least squares optimization. The algorithm I propose aims to preserve two local shape properties of the 2D shape: Laplacian coordinates of the boundary curve and local area of the shape interior, which are together represented in a non-quadratic energy function. An iterative Gauss-Newton method is used to minimize this nonlinear energy function. The result is an interactive shape deformation system that can achieve physically plausible deformations that are hard to achieve with previous linear methods.;(2) Keyframe-based video object deformation. Traditional video object editing by modifying each individual frame is time-consuming and tedious. In this work, I propose a keyframe-based video object editing scheme for automatic object shape deformation. Several technologies are developed in the proposed scheme to minimize user interactions and facilitate users' control on deformation with great flexibility and precision. Experimental results fully demonstrate that the proposed scheme can generate natural video shape deformation with few user interactions.;(3) Sketching MLS image deformations on the GPU. I present here an image editing tool that allows the user to deform images using a sketch-based interface. The user simply sketches a set of source curves in the input image and some target curves that the source curves should be deformed to. Then the moving least squares (MLS) deformation technique is adapted to produce realistic deformations while satisfying the curves' positional constraints. I also propose a scheme to reduce image fold-overs in MLS deformations. The resulting system has a very intuitive user interface, generates physically plausible deformations, and can be easily implemented on the GPU for real-time performance.