李纯明的2011年水平集新论文加上试验程序

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 20, NO. 7, JULY 2011 2007

A Level Set Method for Image Segmentation

in the Presence of Intensity Inhomogeneities

With Application to MRI

Chunming Li, Rui Huang, Zhaohua Ding, J. Chris Gatenby, Dimitris N. Metaxas, Member, IEEE, and John C. Gore

Abstract—Intensity inhomogeneity often occurs in real-world

images, which presents a considerable challenge in image segmen-

tation. The most widely used image segmentation algorithms are

region-based and typically rely on the homogeneity of the image

property of the image intensities, and define a local clustering cri-

terion function for the image intensities in a neighborhood of each

point. This local clustering criterion function is then integrated

with respect to the neighborhood center to give a global criterion

of image segmentation. In a level set formulation, this criterion

defines an energy in terms of the level set functions that represent a

partition of the image domain and a bias field that accounts for the

intensity inhomogeneity of the image. Therefore, by minimizing

this energy, our method is able to simultaneously segment the

image and estimate the bias field, and the estimated bias field can

our method has been used for segmentation and bias correction of

magnetic resonance (MR) images with promising results.

Index Terms—Bias correction, image segmentation, intensity in-

homogeneity, level set, MRI.

I. INTRODUCTION

I

NTENSITY inhomogeneity often occurs in real-world im-

ages due to various factors, such as spatial variations in il-

lumination and imperfections of imaging devices, which com-

Manuscript received November 24, 2008; revised June 04, 2009; accepted

lchunming@gmail.com)

R. Huang and D. N. Metaxas are with Department of Computer Science, Rut-

gers University, Piscataway, NJ 08854 USA (e-mail: ruihuang@cs.rutgers.edu;

dnm@cs.rutgers.edu).

Z. Ding and J. C. Gore are with the Institute of Imaging Science, Vanderbilt

University, Nashville, TN 37232 USA (e-mail: zhaohua.ding@vanderbilt.edu;

john.gore@vanderbilt.edu).

J. C. Gatenby was with the Institute of Imaging Science, Vanderbilt Univer-

sity, Nashville, TN 37232 USA. He is now with the Department of Radiology,

University of Washington, Seattle, WA 98195 USA.

laps between the ranges of the intensities in the regions to seg-

mented. This makes it impossible to identify these regions based

on the pixel intensity. Those widely used image segmentation

algorithms [4], [17], [18], [23] usually rely on intensity homo-

geneity, and therefore are not applicable to images with intensity

inhomogeneities. In general, intensity inhomogeneity has been

a challenging difficulty in image segmentation.

for tracking interfaces and shapes [14], has been increasingly

applied to image segmentation in the past decade [2], [4], [5],

matical theories, including calculus of variations and partial dif-

ferential equations (PDE). An advantage of the level set method

is that numerical computations involving curves and surfaces

can be performed on a fixed Cartesian grid without having to

parameterize these objects. Moreover, the level set method is

able to represent contours/surfaces with complex topology and

change their topology in a natural way.

egorized into two major classes: region-based models [4], [10],

[17], [18], [20], [22] and edge-based models [3], [7], [8], [12],

constant (PC) models proposed in [4], [16]–[18]. In [20], [22],

level set methods are proposed based on a general piecewise

smooth (PS) formulation originally proposed by Mumford and

Shah [13]. These methods do not assume homogeneity of image

intensities, and therefore are able to segment images with inten-

sity inhomogeneities. However, these methods are computation-

ally too expensive and are quite sensitive to the initialization of

the contour [10], which greatly limits their utilities. Edge-based

models use edge information for image segmentation. These

models do not assume homogeneity of image intensities, and

ACKGROUND

. This can be formulated as a problem of minimizing the

(1)

is the length of the contour . In the right hand side of

to be close to

, and the second term is the smoothing term, which

LI et al.: A LEVEL SET METHOD FOR IMAGE SEGMENTATION IN THE PRESENCE OF INTENSITY INHOMOGENEITIES 2009

, the constants , and the

bias field

, respectively. The obtained bias field should be

slowly varying and the regions

should satisfy cer-

tain regularity property to avoid spurious segmentation results

caused by image noise. We will define a criterion for seeking

such estimates based on the above image model and assump-

tions A1 and A2. This criterion will be defined in terms of the

regions

Region-based image segmentation methods typically relies

on a specific region descriptor (e.g. intensity mean or a Gaussian

distribution) of the intensities in each region to be segmented.

However, it is difficult to give such a region descriptor for im-

ages with intensity inhomogeneities. Moreover, intensity inho-

mogeneities often lead to overlap between the distributions of

the intensities in the regions

. Therefore, it is impos-

sible to segment these regions directly based on the pixel inten-

ties, which is referred to as a local intensity clustering property

as described and justified below. To be specific, we consider

a circular neighborhood with a radius

centered at each point

, defined by . The partition

of the entire domain induces a partition of the neigh-

borhood

, i.e., forms a partition of . For a

slowly varying bias field

Then, in view of the image model in (3), we have

the intensities in the set

form a cluster with cluster center , which can be

considered as samples drawn from a Gaussian distribution with

mean

. Obviously, the clusters , are well-sepa-

rated, with distinct cluster centers

,

cates that the intensities in the neighborhood

can be classi-

fied into

clusters, with centers , .

This allows us to apply the standard K-means clustering to clas-

sify these local intensities. Specifically, for the intensities

in the neighborhood , the K-means algorithm is an iterative

process to minimize the clustering criterion [19], which can be

bership function of the region

, we can rewrite as

(7)

tion of the cluster center by

, we define a clustering

criterion for classifying the intensities in

as

(8)

(9)

This local clustering criterion function is a basic element in the

formulation of our method.

sification of the intensities in the neighborhood

partition

better the classification. Naturally, we define the optimal parti-