LBF模型matlab代码_水平集LBF分割代码资源-CSDN文库

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活动轮廓模型

局部二值拟合模型

CV模型

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局部二值拟合模型（Local Binary Fitting, LBF）是一种在计算机视觉和图像处理领域广泛应用的算法，尤其在图像分割、目标检测和图像分析中占据重要地位。它基于活动轮廓模型，即曲线演化理论，旨在寻找最优的分割边界，以最小化某种能量函数。LBF模型是这一理论的一种实现，其核心思想是通过比较像素点与其邻域内像素点的灰度差异来定义局部二值特征，从而进行图像的精确分割。 LBF模型的构建主要分为以下几个步骤： 1. **特征提取**：LBF首先对每个像素点建立一个局部二值模板，该模板包含了像素点周围一定范围内的邻域像素。通常，这个邻域被选择为一个圆形或正方形，且大小可调。对于邻域中的每个像素，计算其与中心像素的灰度差，然后将其二值化，形成一个二进制编码，这就是LBF特征。 2. **模型构建**：利用这些LBF特征，可以构建一个描述图像局部结构的模型。在模型中，每个像素点的LBF特征与其邻域内像素的LBF特征进行比较，形成一个能量函数，该函数衡量了当前像素点位置与模型的匹配程度。 3. **曲线演化**：活动轮廓模型的核心是曲线演化，即通过迭代更新边界曲线，使其逐渐接近目标物体的边缘。在LBF模型中，这一过程通过最小化能量函数实现，能量函数包括数据项（描述像素点与模型的匹配程度）和光滑项（控制曲线的平滑性），通过梯度下降法或水平集方法更新曲线位置。 4. **水平集方法**：水平集是一种将曲线表示为零等值线的连续函数，可以方便地处理曲线的拓扑变化。在LBF模型中，水平集方法用于表示和更新分割边界，确保分割过程的稳定性。 5. **代码实现**：提供的matlab代码实现了上述算法，用户可以直接运行，进行图像分割。这些代码可能包括数据预处理、特征提取、模型构建、能量最小化和曲线演化等模块，用户可以根据需求调整参数，适应不同的应用场景。 6. **应用范围**：LBF模型由于其对图像局部信息的敏感性和鲁棒性，常用于复杂背景下的目标检测、医学图像分析、纹理分割等领域。结合其他机器学习或深度学习技术，还可以进一步提升分割性能。 LBF模型是一种有效的图像处理工具，通过局部二值拟合实现图像分割，其matlab代码的提供使得研究者和开发者能够快速理解和应用这一技术，解决实际问题。对于计算机视觉领域的初学者和专业人士，理解并掌握LBF模型的原理和实现方法，对于提升图像处理能力大有裨益。

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Implicit Active Contours Driven by Local Binary Fitting Energy

Chunming Li

, Chiu-Yen Kao

, John C. Gore

, and Zhaohua Ding

Institute of Imaging Science

Department of Mathematics

Vanderbilt University The Ohio State University

Nashville, TN 37232-2310, USA Columbus, OH 43210-1174, USA

{chunming.li,john.gore,zhaohua.ding}@vanderbilt.edu kao@math.ohio-state.edu

Abstract

Local image information is crucial for accurate segmen-

tation of images with intensity inhomogeneity. However, im-

age information in local region is not embedded in popular

region-based active contour models, such as the piecewise

constant models. In this paper, we propose a region-based

active contour model that is able to utilize image informa-

tion in local regions. The major contribution of this paper is

the introduction of a local binary ﬁtting energy with a kernel

function, which enables the extraction of accurate local im-

age information. Therefore, our model can be used to seg-

ment images with intensity inhomogeneity, which overcomes

the limitation of piecewise constant models. Comparisons

with other major region-based models, such as the piece-

wise smooth model, show the advantages of our method in

terms of computational efﬁciency and accuracy. In addi-

tion, the proposed method has promising application to im-

age denoising.

1. Introduction

Active contour models have been one of the most suc-

cessful methods for image segmentation [1, 3–7, 12]. The

existing active contour models can be categorized into two

classes: edge-based models [1, 4–6] and region-based mod-

els [2,9,10]. These two types of models both have their pros

and cons, and the choice of them in applications depends on

different characteristics of images.

Edge-based models utilize image gradient to stop the

evolving contours on the object boundaries. Typical edge-

based active contour models have an edge-based stopping

term and a balloon force term to control the motion of the

contour. The edge-based stopping term serves to stop the

contour on the desired object boundary. The balloon force

term is introduced to shrink or expand the active contour

so that the initial contour can be placed far away from the

desired object boundary. However, appropriate choice of

balloon force is sometimes difﬁcult. If the balloon force

is not large enough, the evolving contour may not able to

pass some narrow parts of the object. If the balloon force is

too large, the active contour is likely to pass through weak

object boundary.

Region-based active contour models have the following

advantages over edge-based models. First, region-based

models do not utilize the image gradient and therefore have

better performance for the image with weak object bound-

aries. Second, they are signiﬁcantly less sensitive to the

location of initial contours. One of the most popular region-

based active contour models is Chan-Vese model [2]. This

model has been successful for images with two regions,

each having a distinct mean of pixel intensity. In [11], Vese

and Chan extended their earlier work in [2] by using a mul-

tiphase level set formulation, in which multiple regions can

be represented by multiple level set functions. These mod-

els are called piecewise constant (PC) models, since they

assume that an image consists of statistically homogeneous

regions. However, the regions of interest in images are often

not statistically homogeneous, and therefore the PC models

are not applicable to those types of images.

To handle more general scenario, Vese and Chan [11]

and Tsai et al. [10] proposed two similar region-based ac-

tive contour models, aiming at minimization of Mumford-

Shah functional [8]. In [11], Vese and Chan proposed a

piecewise smooth (PS) model, which overcomes the limita-

tion of their original work [2]. But these methods are com-

putationally inefﬁcient. The technique proposed by Tsai et

al. [10] can address the segmentation of images with in-

tensity inhomogeneity. However, the computation in their

method is also expensive. As proposed in [10], one way to

reduce the computational cost is to use a contour near the

object boundaries as the initial contour. In their method,

such initial contour is obtained by a preliminary segmenta-

tion using other active contour models, such as Chan and

Vese’s PC model. However, for images with intensity inho-

mogeneity, PC model can result in a contour that is still far

away from the object boundary. In this situation, their two-

stage approach still cannot signiﬁcantly reduce the compu-

tational cost.

In fact, intensity inhomogeneity occurs in many real im-

ages of different modalities. In particular, it is often seen

in medical images, such as X-ray radiography/tomography

and magnetic resonance (MR) images, due to technical lim-

itations or artifacts introduced by the object being imaged.

For example, intensity inhomogeneity typically appears in

MR images (see Fig. 4 for example). The inhomogeneity

in MR images arises from non-uniform magnetic ﬁeld pro-

duced by radio-frequency coils as well as from object sus-

ceptibility. The degree of this inhomogeneity is worse for

higher ﬁeld imaging (e.g., 7T MR) which is being increas-

ingly used in research. Therefore, overcoming the difﬁculty

of segmentation due to image inhomogeneity is a main mo-

tivation of this paper.

In this paper, we propose a novel active contour model

that is able to segment images with intensity inhomogene-

ity. The basic idea is to introduce a kernel function to de-

ﬁne a local binary ﬁtting energy in a variational formulation,

so that local intensity information can be embedded into a

region-based active contour model. The local binary ﬁtting

energy functional is further incorporated into a variational

level set formulation without reinitialization proposed by Li

et al. [6]. Therefore, no reinitialization is necessary in the

proposed method.

2. Background

2.1. Mumford-Shah functional for image segmen-

tation

Let Ω ⊂

be the image domain, and I :Ω→be a

given image. In [8], Mumford and Shah formulated image

segmentation as a problem of seeking an optimal contour C

that divides the image domain into disjoint subregions, and

an optimal function u that ﬁts the original image I and that

is smooth within each of the subregions. They proposed the

following energy functional:

(u, C)=



Ω

(I−u)

dxdy+ν



Ω\C

|∇u|

dxdy+ν|C|

(1)

where |C| is the length of contour C. Thus, image segmen-

tation can be performed by minimizing the above functional

over all the contours and ﬁtting function u. However, it is

difﬁcult to minimize the above functional (1), due to differ-

ent nature of the two unknowns: a contour C and a function

u, and the non-convexity of the functional as well. For prac-

tical applications, much works have been done to simpliﬁed

or modiﬁed the above Mumford-Shah functional, including

the two well known approaches that are reviewed below.

Figure 1. Error of thresholding and Chan-Vese model for images

with intensity inhomogeneity. Column 1: Original images; Col-

umn 2: Thresholding results; Column 3: Results of PC model.

2.2. Piecewise constant model using active contours

Chan and Vese [2] proposed an active contour approach

to the Mumford-Shah problem for a special case where the

image u in the functional (1) is a piecewise constant func-

tion. For an image I(x, y) on the image domain Ω,they

proposed to minimize the following energy

(C, c

)=λ



in(C)

|I(x) − c

dx (2)

+ λ



out(C)

|I(x) − c

dx + ν|C|

where in(C) and out(C) represent the region inside and

outside of the contour C, respectively, and c

and c

are two

constants that approximate the image intensity in in(C) and

out(C). We call the ﬁrst two terms in (2)theglobal binary

ﬁtting energy. This energy can be represented by a level set

formulation, and then energy minimization problem can be

converted to solving a level set evolution equation [2].

In the above Chan-Vese model, the constants c

and c

are introduced to ﬁt the image intensities in the regions

in(C) and out(C), respectively. Obviously, such global

ﬁtting will not be accurate if the image intensities in ei-

ther in(C) or out(C) are not homogeneous. This is the

reason why Chan-Vese model [2] cannot handle intensity

inhomogeneity. Similarly, Vese and Chan’s piecewise con-

stant model in a multiphase level set framework [11] still

cannot address intensity inhomogeneity.

For example, the synthetic image and the real vessel im-

age in Fig. 1 are typical examples of images with inten-

sity inhomogeneity. In the vessel image, the intensity in

the background decreases gradually from the top to the bot-

tom. Moreover, part of the background (the upper left cor-

ner) has even higher intensities than part of the vessel (the

lower left branch). The second and the third columns of

Fig. 1 show the segmentation results using thresholding and

the PC model, respectively. It is clearly seen that part of

the background is incorrectly identiﬁed as the foreground,

while part of the foreground is labeled as the background.

2.3. Piecewise smooth model and its difﬁculties

The PS models proposed by Vese and Chan [11] and Tsai

et al. [10] have overcome the difﬁculties of PC models in

the presence of image inhomogeneity. Instead of approx-

imating an image I by a piecewise constant function, the

PS model approximates the image by two smooth func-

tions u

(x) and u

−

(x) in the subregions Ω

= {x ∈

Ω:φ(x) < 0} and Ω

−

= {x ∈ Ω:φ(x) < 0},re-

spectively. By deﬁnition, the functions u

and u

−

are de-

ﬁned on two exclusive subregions Ω

and Ω

−

, respectively.

However, the level set evolution equation derived to mini-

mize the energy functional in the PS model is with respect

to the level set function φ, which is deﬁned on the full do-

main Ω. Therefore, there is an unavoidable step that extends

and u

−

to the full domain Ω. Moreover, the smoothness

of the functions u

and u

−

is enforced by the smoothing

terms µ



|∇u

dx and µ



|∇u

−

dx in the energy func-

tional in the PS model, which leads to two PDEs with the

unknown functions u

and u

−

that have to be solved at each

iteration of the level set function φ.

In addition, it is necessary to periodically reinitialize the

level set function φ to a signed distance function in practi-

cal application of the PS model. In summary, the need of

solving PDEs for u

and u

−

, the extension of u

and u

−

and the reinitialization makes the PS model computation-

ally very expensive. Moreover, these procedures introduce

quite a few parameters to the PS models, which increases

the difﬁculty of selecting parameters in using the PS model,

especially for inexperienced users.

3. Local binary ﬁtting active contour model

and its level set formulation

3.1. Active contours with local binary ﬁtting energy

Consider a given vector valued image I :Ω→

where Ω ⊂

is the image domain, and d ≥ 1 is the

dimension of the vector I(x). For gray level images, d =1,

for color images, d =3.LetC be a contour in the image

domain Ω. We deﬁne for each point x ∈ Ω the following

energy

LBF

(C, f

(x),f

(x))

= λ



in(C)

K(x − y)|I(y) − f

(x)|

+ λ



out(C)

K(x − y)|I(y) − f

(x)|

dy (3)

where λ

and λ

are positive constants, and K is a kernel

function with a localization property that K(u) decreases

and approaches zero as |u| increases, and f

(x) and f

(x)

are two numbers that ﬁt image intensities near the point x.

We call the point x the center point of the above integral,

and the above energy the local binary ﬁtting (LBF) energy

around the center point x.

In this paper, we choose the kernel function K(x) as a

Gaussian kernel

(x)=

(2π)

n/2

−|x|

/2σ

, (4)

with a scale parameter σ>0. It should be emphasized that

the numbers f

and f

that minimize the energy (3)vary

with the center point x. The introduction of the spatially

varying ﬁtting functions f

and f

makes our method es-

sentially different from piecewise constant models.

In the proposed model, the ﬁtting energy in (3) is local

to the center point x in the sense that the values f

and f

only ﬁt the image intensities near each x. This is due to

the kernel function K with the above localization property

that K(x − y ) takes larger values at the points y near the

center point x, and it decreases to 0 as y goes away from

x. Therefore, the image intensities at the points y near the

point x have dominant inﬂuence on the values of f

and f

that minimize E

LBF

(C, f

), whereas the image intensi-

ties at the points y far away from the center point x have

almost no inﬂuence on the values of f

and f

As mentioned above, the values f

and f

that minimize

the LBF energy E

LBF

(C, f

) are functions of the center

point x due to the localization property of the kernel func-

tion K(x − y). Obviously, for each center point x, the local

ﬁtting energy E

LBF

can be minimized when the contour

C is exactly on the object boundary and the ﬁtting values

and f

are chosen optimally. However, the above en-

ergy E

LBF

is deﬁned locally for a center point x ∈ Ω.To

ﬁnd the entire object boundary, we must minimize E

LBF

for all the center points x in the image domain Ω. This can

be achieved by minimize the integral of E

LBF

over all the

center points x in the image domain Ω. So, we deﬁne the

following energy functional

E(C, f



Ω

LBF

(C, f

(x),f

(x))dx (5)

This energy can be converted to an equivalent level set for-

mulation, from which an implicit active contour model will

be obtained to automatically handle topological changes

(see below).

3.2. Variational level set formulation of the model

In level set methods, a contour C ⊂ Ω is represented

by the zero level set of a Lipschitz function φ :Ω→.

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