2D角色的拓扑感知MLS变形方法

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本文主要探讨了一种针对2D角色的拓扑感知最小二乘变形方法（Topology-aware Moving Least Square Deformation, T-MLS）。发表在2015年的《计算机动画与虚拟世界》(COMPUTERANIMATIONANDVIRTUALWORLDS)期刊上，该研究论文的在线链接可从Wiley Online Library获取，DOI号为10.1002/cav.1675。这是一项重要的技术革新，因为在实时动画领域，用户能够通过点集或线段对2D角色进行操控，最小二乘法（Moving Least Squares, MLS）是一种常用的方法。然而，传统MLS方法在传播控制点的变形时，只考虑了空间距离的影响，而忽视了形状的拓扑结构，可能导致变形过程中出现扭曲。作者团队，由浙江大学计算机信息工程学院的Xun Wang、Wenwu Yang*、Wangbin Kou、Bailin Yang 和 Guozheng Wang 等成员组成，提出了一种新颖的解决方案。他们首先通过解拉普拉斯方程，计算出一组被称为谐波权重的值。这些权重反映了每个像素或顶点对用户指定控制点的响应，同时考虑了角色形状的拓扑信息。这样，当执行MLS变形时，这些拓扑敏感的权重被用作变形影响力的依据，从而减少了传统方法中的扭曲问题。 T-MLS方法的主要步骤包括：1）通过拉普拉斯方程求解得到的拓扑感知；2）利用获得的谐波权重作为控制点对角色各部分影响的度量；3）根据这些权重调整角色形状，实现更加自然且符合拓扑结构的变形效果。这种技术对于保持2D角色在变形过程中的完整性，尤其是在复杂的动作和变形场景中，具有显著的优势，有助于提高动画质量和用户体验。这篇研究论文对2D角色动画领域的实时操控方法进行了改进，强调了拓扑信息在保持角色形状自然性和真实感方面的重要性。通过引入拓扑感知的最小二乘变形，它不仅解决了传统方法可能导致的变形失真问题，还为艺术家和开发者提供了更精确、更具艺术性的角色动画工具。

COMPUTER ANIMATION AND VIRTUAL WORLDS

Comp. Anim. Virtual Worlds

(2015)

Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cav.1675

RESEARCH ARTICLE

Topology-aware moving least square deformation for

2D characters

XunWang,WenwuYang*, Wangbin Kou, Bailin Yang and Guozheng Wang

School of Computer Information Engineering, Zhejiang Gongshang University, Hangzhou, 310018, China

ABSTRACT

Deformation method based on moving least squares (MLS) allows the user to manipulate 2D characters using either sets

of points or line segments in real time. However, the traditional MLS deformation spreads the deformation of the controls

with respect to the spatial distance, but oblivious to the shape topology, which would possibly lead to distortion. In this

paper, we present a topology-aware MLS deformation approach for 2D characters. First, a Laplace equation is solved to

obtain a set of weights, which are called harmonic weights. Then, the MLS deformation is performed by using the harmonic

weights as the deformation inﬂuence of the user-speciﬁed controls. Finally, the possible distortion in the traditional MLS

deformation can be effectively avoided, as the harmonic weights spread the deformation of the controls in a localized and

topology-aware way. In addition, a simple but effective area-preserving variant of MLS deformation is proposed, which is

KEYWORDS

shape deformation; two-dimensional character animation; moving least squares; harmonic weights

Supporting information may be found in the online version of this article.

*Correspondence

Wenwu Yang, School of Computer Information Engineering, Zhejiang Gongshang University, Hangzhou, 310018, China.

E-mail: wwyang@mail.zjgsu.edu.cn

1. INTRODUCTION

The two-dimensional deformation, which allows the user

to edit and manipulate image or plane shape intuitively, has

been widely used in two-dimensional animation and image

editing to represent moving objects of changing shape.

Nowadays, this tool can be commonly found in commer-

cial software for video editing or 2D character animation

[1], such as Adobe After Effects, Adobe Flash, or Toon

Boom Studio. Recently, many deformation methods that

are based on energy optimization have been introduced

[2–5]. In these methods, the user is typically allowed to

manipulate the shape by placing several handles on the

shape and then moving the handles. By minimizing the

local shape distortion, these approaches generate physi-

cally plausible deformation results, making the user have

a sense of manipulating a real-world object. To achieve

high-quality results, however, these approaches require

solving a nonlinear optimization problem in general. Such

solving is computationally expensive such that the capac-

ity of real-time interactivity of these approaches is reduced

to some extent.

To address the aforementioned problem, Schaefer et al.

[6] propose an image deformation method that is based on

moving least squares (MLS). By enforcing the shape rigid-

ity, this method generates visually pleasing results. Instead

of solving a global nonlinear system, the method, however,

solves a set of much smaller local linear systems, each of

which is deﬁned at a shape vertex. Furthermore, each of

the involved local linear systems has an analytical solution

for a 2D case. Consequently, the method is able to generate

deformations for the shape consisting of tens of thousands

of vertices in real time. The disadvantage of this method

is that it spreads the deformation of the user-speciﬁed con-

trols with respect to the Euclidean distance, oblivious to the

shape topology, which would possibly lead to inappropriate

distortion (Figure 1(b)).

In this paper, we present a 2D MLS deformation

approach, which takes into account the shape topology. The

key idea is to adopt the so-called harmonic weights as the

inﬂuence weights of the user-speciﬁed controls to the shape

points. The harmonic weights spread over the shape of

the deformation inﬂuence of the controls in a shape-aware

and localized manner. Finally, the new approach generates

visually pleasing deformation effects, where the possible

distortion in the traditional MLS deformation is effectively

avoided (Figure 1(d)).

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