交替优化新策略：快速多相分割法

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"这篇研究论文探讨了两种用于多相分割的快速交替方向优化方法，即快速交替方向乘子法（Fast ADMM）和快速交替最小化算法（Fast AMA）。这两种方法都是基于分段常数最优近似的基础上，利用多相Chan-Vese模型进行图像分割。" 在图像处理领域，多相分割是一种重要的技术，它能够将图像划分为多个具有不同特征的区域。多相 Chan-Vese 模型是其中的一种经典模型，它通过最小化能量函数来实现图像分割，该能量函数通常包含内部和外部区域的像素差异以及区域间的边缘强度。论文中提出的 Fast ADMM 和 Fast AMA 方法是对传统的交替方向乘子法（ADMM）和交替最小化算法（AMA）的改进。这两种新方法引入了变量拆分策略，以更有效地处理多相分割问题。特别是，它们采用了一个“重置”条件来更新拉格朗日乘子，确保能量函数的值始终保持为正，从而保证优化过程的稳定性。在实际应用中，Osher和Stethian的水平集方法被用来处理连续的水平集函数，而二进制水平集函数则用于表示不同的图像区域。此外，阈值方法和投影公式也在实现过程中发挥了关键作用，它们有助于确定分割边界和区域成员资格。论文通过数值实验展示了这两种方法的快速收敛性，并与其它快速变分方法的结果进行了比较，证明了Fast ADMM和Fast AMA在效率和效果上的优越性。这些结果表明，新提出的方法在处理多相图像分割时，不仅计算速度更快，而且分割精度更高，对于复杂场景的图像处理具有较高的实用价值。这篇论文为多相图像分割提供了新的优化工具，对于计算机视觉和图像处理领域的研究者来说，这两种快速方法的提出无疑为提高图像分割效率和准确性开辟了新的途径。

Two Fast Alternating Direction Optimization Methods for

Multiphase Segmentation

Lu Tan

,WeiboWei

, and Zhenkuan Pan

1,2

College of Information Engineering, Qingdao University, Qingdao, Shandong, China

luna086@163.com, njustwwb@163.com

Abstract - In this paper, two new methods associated with

alternating direction optimization, fast alternating direction

method of multipliers(FastADMM) and fast alternating

minimization algorithm(FastAMA), are proposed for image

segmentation using the multiphase Chan-Vese model, which is

on the basis of piecewise constant optimal approximations.

For these methods, we incorporate the variable splitting

approach and a ‘reset’ condition in order to update the

Lagrange multiplier and make sure the value of energy

functional is always positive. The Osher and Stethian level set

method, binary level set functions, thresholding method and

projection formula are applied in the implementation. Finally,

numerical results with rapid convergence are obtained by our

methods, which are also compared with those of some other

fast variational methods to demonstrate better effectiveness of

our methods.

Keywords: Multiphase segmentation, fast alternating

direction method of multipliers, fast alternating minimization

algorithm, active contours, level sets.

1 Introduction

In image segmentation, major advances were made in

two-phase image segmentation[1-3] in the early days.

Mumford-Shah model proposed by Mumford D and Shah J [4]

is regarded as the most significant region-based model. It has

been extended to a great deal of applications. In 2001, Tony F.

Chan and Luminita A. Vese proposed the Chan-Vese model

[5] for active contours to detect objects in a given image. It is

one of the simplified variants of Mumford-Shah model.

Nevertheless, as the complexity of the images increases, 2-

phase image segmentation is not able to meet the actual needs.

Therefore, multiphase segmentation is applied to satisfy the

demands. On the basis of Potts model [6][7] from statistical

mechanics, Zhao et al. [8] started to study multiphase motion

segmentation by using the level set method and proposed a

model which can represent n different regions by n level set

functions. In order to reduce the number of level set functions,

Chan et al. continued their work and proposed multiphase

segmentation model [9] which is a generalization of Chan-

Vese model. Their scheme can naturally avoid "overlap" and

"leakage" problem. But there are still some problems about

solving the global optimization, accuracy, stability and speed.

Alternating direction method of multipliers(ADMM) was

first described by Glowinski and Marocco [10] and alternating

minimization algorithm(AMA) was presented by Tseng [11].

These techniques are commonly known as the Split Bregman

Method [12], and are known to be an efficient solver for

problems involving the total-variation norm [13]. These

methods can be accelerated using optimal first order methods,

of the type first proposed by Nesterov [14]. And accelerated

variants of ADMM and AMA can be called FastADMM and

FastAMA.

In our paper, we design methods (FastADMM and

FastAMA) that can achieve computational efficiency and own

even faster convergence to solve the functional of multiphase

Chan-Vese model based on binary level sets framework

[15,16]. The gradient descent method (GDM) [17],

Chambolle’s dual method (DM) [18], alternating direction

method of multipliers(ADMM) [19] i.e. the augmented

Lagrangian method (ALM) [20], and alternating minimization

algorithm(AMA) [11] are used to compare with our methods.

But results of these methods which are used in multiphase

segmentation model will be obtained in a slow convergence.

Tom Goldstein et al. [19] introduced FastADMM and applied

it to solve the TV model as an example and some other

strongly convex problems. A predictor-corrector type

acceleration step is used in this method.

The remaining of this paper is organized as follows. In

Section 2, the binary level set formulation of the functional of

multiphase Chan-Vese model used in our paper is reviewed

along with its four traditional solution methods. Our proposed

methods are discussed in Section 3 and its iterative discrete

formulas for implementation will be presented in detail. In

Section 4, some numerical experiments are given to illustrate

the effectiveness of our method by comparing with other

methods. Finally a conclusion is given in section 5.

2 The multiphase Chan-Vese model and

its four traditional methods

2.1 The binary level set based formulation

In order to separate an image domain



into n

subdomains with

i





and

∃

.Veseand

Chan defined up to

n 

phases and

level set functions.

This way makes sure that each pixel

! ∀

y #

will belong

to one, and only one phase.

Int'l Conf. IP, Comp. Vision, and Pattern Reco

nition

IPCV'15

113

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