PDE Surface-Represented Facial Blendshapes

Haibin Fu,Shaojun Bian,Lihua You,Shuangbu Wang,Ehtzaz Chaudhry,Jian Jun Zhang

doi:10.3390/math9222905

Abstract

Partial differential equation (PDE)-based geometric modelling and computer animation has been extensively investigated in the last three decades. However, the PDE surface-represented facial blendshapes have not been investigated. In this paper, we propose a new method of facial blendshapes by using curve-defined and Fourier series-represented PDE surfaces. In order to develop this new method, first, we design a curve template and use it to extract curves from polygon facial models. Then, we propose a second-order partial differential equation and combine it with the constraints of the extracted curves as boundary curves to develop a mathematical model of curve-defined PDE surfaces. After that, we introduce a generalized Fourier series representation to solve the second-order partial differential equation subjected to the constraints of the extracted boundary curves and obtain an analytical mathematical expression of curve-defined and Fourier series-represented PDE surfaces. The mathematical expression is used to develop a new PDE surface-based interpolation method of creating new facial models from one source facial model and one target facial model and a new PDE surface-based blending method of creating more new facial models from one source facial model and many target facial models. Some examples are presented to demonstrate the effectiveness and applications of the proposed method in 3D facial blendshapes.

Highlights

Partial differential equation (PDE)-based geometric modelling and computer animation is physics-based
We have developed a new method of facial blendshapes for creating new facial blendshapes and generating face animation
We have proposed a curve template to extract curves from polygon facial models, used extracted curves to define the boundaries of PDE surface patches, and solved a second-order partial differential equation subjected to the constraints of the extracted boundary curves represented with Fourier series to obtain the mathematical expression of curve-defined and Fourier series-represented PDE surfaces

Summary

Introduction

Partial differential equation (PDE)-based geometric modelling and computer animation is physics-based. Existing methods of facial blendshapes, such as Autodesk Maya blendshapes or Blender shape keys, interpolate or blend vertices of facial models to create new facial models and animation. Compared to Autodesk Maya blendshape or Blender shape keys, which blend polygon models together, facial blendshapes represented with PDE surfaces have the following advantages: (1) The difficulty of interpolating or blending the models with different topologies and/or vertices is overcome by transforming the correspondence between the vertices of polygon models into the correspondence between the boundary curves defining PDE surfaces. Our method is to build curve-defined PDE surfaces to represent the model, so that shape blending between 3D models with different topologies, vertices, and faces could be converted into morphing between PDE-based geometric models. This paper will tackle this problem by transforming polygon models into PDE surfaces for interpolating and blending facial models to create facial blendshapes

Differential Equation Based Modelling

I Dmax

Conclusions