mls.rar_MLS_变形_图像变形_图像变形 MLS

Figure 1: Deformation using Moving Least Squares. Original image with control points shown in blue (a). Moving Least Squares deforma-

tions using affine transformations (b), similarity transformations (c) and rigid transformations (d).

Abstract

We provide an image deformation method based on Moving Least

techniques, we provide simple closed-form solutions that yield fast

deformations, which can be performed in real-time.

CR Categories: I.3.5 [Computer Graphics]: Computational Ge-

ometry and Object Modeling—Boundary representations; Curve,

surface, solid, and object representations; Geometric algorithms,

languages, and systems

Keywords: Deformations, moving least squares, rigid transforma-

tions

1 Introduction

Image deformation has a number of uses from animation, to mor-

Figure 2: Deformation of the test shape from figure 1 using thin-

alized this technique to surface deformations.

mations that are desirable for performing deformation. One notable

exception is worked based on thin-plate splines [Bookstein 1989]

that attempts to minimize the amount of bending in the deforma-

tion. Bookstein presents a deformation algorithm using the sim-

plest deformation handle, a point, that uses radial basis functions

with thin-plate splines. Figure 2 (left) shows an example of the

deformation created with thin-plate splines for our example in fig-

ure 1. The deformation appears very similar to the affine-method in

property that amount of local scaling and shearing is minimized.

(The concept of rigid-as-possible transformations was itself first in-

troduced in Alexa [Alexa et al. 2000].)

gulate the input image and solve a linear system of equations whose

size is equal to the number of vertices in the triangulation. In con-

trast, our method creates deformations by solving a small linear

tails). Since, we solve much smaller systems of equations, we can

create very fast deformations of grids consisting of tens of thou-

Contributions

linear Moving Least Squares. To construct deformations that min-

imize the amount of local scaling and shear, we restrict the classes

of transformations used in Moving Least Squares to similarity and

rigid-body transformations. By using MLS, we avoid the need to

triangulate the input image (as done in Igarashi et al.) and produce

deformations that are globally smooth.)

rigid MLS deformations. These formula are simple, easy to imple-

ment and provide real-time deformations. This derivation relies on

a surprising and little-known relationship between similarity trans-

where ˆp

Squares is very general in that the matrix M does not have to be

a fully affine transformation. In fact, this framework allows us to

deformation, much of equation 5 can be precomputed yielding very

fast deformations. In particular, we can rewrite equation 5 in the

form

where A

Notice that, given a point v, everything in A

yielding a simple, weighted sum. Table 1 provides timing results

for the examples in this paper, which shows that these deformations

may be performed over 500 times per second in our examples.

mation applied to our test image. Unfortunately, the deformation

class of deformations l

tion matrix M from being fully linear to similarity and rigid-body

transformations.

2.2 Similarity Deformations

While affine transformations include effects such as non-uniform

scaling and shear, many objects in reality do not undergo even these

simple transformations. Similarity transformations are a special

subset of affine transformations that only include translation, ro-

tation and uniform scaling.