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Meshfree Particle Method

Huafeng Liu and Pengcheng Shi

Medical Image Computing Group, Department of Electrical and Electronic Engineering

Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

Abstract

Many of the computer vision algorithms have been posed in

various forms of differential equations, derived from min-

imization of speciﬁc energy functionals, and the ﬁnite el-

ement representation and computation have become the

de facto numerical strategies for solving these problems.

However, for cases where domain mappings between nu-

merical iterations or image frames involve large geometri-

cal shape changes, such as deformable models for object

segmentation and nonrigid motion tracking, these strate-

gies may exhibit considerable loss of accuracy when the

mesh elements become extremely skewed or compressed.

We present a new computational paradigm, the meshfree

particle method, where the object representation and the

numerical calculation are purely based on the nodal points

and do not require the meshing of the analysis domain. This

meshfree strategy can naturally handle large deformation

and domain discontinuity issues and achieve desired numer-

ical accuracy through adaptive node and polynomial shape

function reﬁnement. We discuss in detail the element-free

Galerkin method, including the shape function construc-

tion using the moving least square approximation and the

Galerkin weak form formulation, and we demonstrate its

applications to deformable model based segmentation and

mechanically motivated left ventricular motion analysis.

1 Introduction

1.1 Finite Element Methods

Many of the computer vision algorithms are posed as vari-

ous energy minimization problems, and become partial dif-

ferential equations (PDEs) subject to image data constraints

through natural and essential boundary conditions. Because

the analysis domains in these problems are often spatially

irregular and sampled at discrete points, the ﬁnite element

methods (FEMs) have become the de facto computational

strategy to provide numerical solutions through the dis-

cretization of the analysis domains into meshes with prede-

ﬁned connectivity between nodal points. The main compu-

tational power of these approaches results from the funda-

mental idea of replacing a continuous function f(x) deﬁned

over the entire analysis domain by piecewise polynomial

approximations over a set of ﬁnite number of geometrically

simple sub-domains such as triangles. With suitable regu-

larization constraints (i.e. smoothness, continuum mechan-

ical models, etc.) and proper formulation principles (i.e.

the virtual work), governing differential equations can be

approximated by a set of algebraic equations for all the ele-

ments, and image-derived boundary conditions are enforced

to provide the solutions. An incomplete survey of more

recent works shows a wide range of FEM-based strategies

in computer vision, including object segmentation [5, 23],

shape representation and characterization [11, 20], corre-

spondence and motion estimation [18, 21], image registra-

tion [9], and image guided surgery [4, 7].

Although the idea of domain division has been proven

really ingenious and well-suited for many vision problems,

proper mesh generation from the sampling nodes can some-

times be difﬁcult and time consuming, especially for con-

strained meshing of material and/or kinematics discontinu-

ities and for three-dimensional (3D) cases. Furthermore,

in dynamic formulation which is common for computer vi-

sion problems such as segmentation and motion analysis,

the numerical accuracy and efﬁciency of FEM decreases

drastically whenever the mesh becomes extremely skewed

or compressed, i.e. there is large geometrical shape changes

of the objects between numerical iterations or image frames.

In these situations, adaptive remeshing and/or node reﬁne-

ment must be performed throughout the evolution in order

to prevent the severe distortion of elements [11], to allow

mesh lines to remain coincident with any discontinuities [1],

and to maintain reasonable numerical accuracy. However,

remeshing requires the projection of ﬁeld variables between

meshes in successive stages of the problem, which leads to

huge logistical problems even for medium size problems.

Further, for large three-dimensional problems which are be-

coming more common, the computational cost of remeshing

at each step often becomes prohibitively expensive.

1.2 Meshfree Particle Methods

The emerging meshfree particle methods (MPMs) offer

computationally efﬁcacious alternatives to circumvent the

aforementioned problems encountered by the FEMs. Orig-

Proceedings of the Ninth IEEE International Conference on Computer Vision (ICCV 2003) 2-Volume Set

inated as special numerical methods for computational me-

chanics analysis, with variations such as the smooth particle

hydrodynamics methods, the diffuse element methods, the

element free Galerkin methods, the reproducing kernel par-

ticle methods, and the partitions of unity methods, MPMs

have recently shown their prowess in solving general PDEs

over complex domain [16, 19].

The main motive to adopt MPMs is to eliminate at least

part of the mesh structure by constructing the approxima-

tion of the ﬁeld function f(x) entirely in terms of the nodal

points, whereas no speciﬁc pairwise characterization of the

nodal interrelationship is deﬁned or needed [1]. MPMs rep-

resent the analysis domain with only a set of nodal points

without mesh constraints, and establish a system of alge-

braic equations for the whole problem domain based on the

particle-derived interpolating shape functions. Procedure-

wise, MPMs are actually very similar to FEMs, with the

fundamental differences in the elimination of mesh and the

construction of the shape functions from nodes only. In

FEMs, the shape functions are constructed using the mesh

structure, and they are the same for all the elements of the

same type in the natural coordinates systems. These shape

functions are usually predetermined for different types of

elements before the analysis starts. In MPMs, however,

the shape functions are usually constructed for a particular

point of interest (POI), and they will change as the location

of the POI changes. Hence, the construction of the meshfree

shape function is performed during the analysis.

MPMs are better suited to cope with geometric changes

of the domain of interest, e.g. free s urfaces and large de-

formations, than FEMs [1, 15, 16]. From numerical efﬁ-

ciency and accuracy point of view, the principal attraction

of MPMs is the possibility of simplifying spatial adaptivity

(node addition or elimination) and shape function polyno-

mial order adaptivity (approximation/interpolation types),

and handling moving boundaries and discontinuities. Adap-

tive meshing procedure, which may be needed for a large

variety of problems including deformable models, image

registration, and motion detection, can be effectively treated

in a much simpler manner as a node reﬁnement problem. In

areas where more reﬁnement is needed, nodes can be added

easily to achieve desired numerical accuracy. Since there is

no need to generate the mesh representation and the connec-

tivity between nodes is generated as part of the computation

and can be changed over time, MPMs facilitate the handling

of very large deformations and material/kinematics discon-

tinuity. An illustration of the accuracy advantage of MPM

over FEM (without adaptive remeshing) in a deformation

simulation is shown in Figure 1. Further, several recent

meshfree efforts incorporate the multi-scale concept for

problems involving widely varying scales through wavelet

based basis function enhancement [24], which meshes well

with the scale-space tradition in computer vision research.

Figure 1: Comparison of FEM and MPM computed vertical

strains for a simple elastic deformation s imulation: exper-

iment setup and theoretical strain distribution (left), FEM

mesh and strain map (middle), and MPM representation and

strain map (right). All the strain maps here use the same

color scale.

1.3 Contributions

In this paper, we present a meshfree particle computational

paradigm for certain computer vision problems involving

object deformation. We want to emphasize that we are

not presenting a new deformable model per se, but rather

novel ways for object representation and numerical calcu-

lation. The framework has its roots in the popular ﬁnite

element methods, but is purely based on the nodal points

and does not require the meshing of the analysis domain.

This meshfree strategy can naturally handle large deforma-

tion and domain discontinuity issues and achieve desired

numerical accuracy through adaptive node and polynomial

shape function reﬁnement. We discuss in detail the element-

free Galerkin method, one of the more robust and better

developed meshfree methods, through the construction of

the shape functions using the moving least s quare approx-

imation, the Galerkin weak form formulation, and the im-

position of boundary conditions using t he penalty method.

We demonstrate the computational power of this framework

with applications in object segmentation with one- and two-

dimensional deformable models (contours and annuli), and

multi-frame non-rigid motion analysis of the left ventricle

using mechanical model constrained optimal ﬁltering.

2 EFGM Framework for Vision

Computation

Various meshfree particle methods have emerged in the last

a few years, as summarized in [1, 15], and the most ba-

sic common feature is that a predeﬁned mesh is not nec-

Proceedings of the Ninth IEEE International Conference on Computer Vision (ICCV 2003) 2-Volume Set

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