Chan_Active-Contours-without-Edges-for-Vector-Valued-Images_Journal-of-Visual-Communication-and-Image-Representation

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### Chan Active Contours without Edges for Vector-Valued Images #### 概述本文献《Active Contours without Edges for Vector-Valued Images》由Tony F. Chan、B. Yezrielev Sandberg 和 Luminita A. Vese 共同撰写，并发表于《Journal of Visual Communication and Image Representation》2000年第11卷第130-141页。文章介绍了一种针对向量值图像（如RGB或多光谱图像）的主动轮廓算法，该算法能够检测图像中的物体边界而无需依赖边缘信息。这种方法是对传统的Chan-Vese模型在向量值图像上的扩展。 #### 主要贡献 ##### 1. 向量值图像分割本研究提出了一种新的主动轮廓模型，用于分割向量值图像。与传统的基于梯度的主动轮廓方法不同，该模型能够在没有明显边缘的情况下检测到物体边界。这使得它在处理诸如遮挡、不同通道中缺失部分等复杂情况下具有独特的优势。 ##### 2. Mumford-Shah 功能性该模型通过最小化Mumford-Shah功能性来确定最佳轮廓位置。此功能性的组成部分包括轮廓长度和图像拟合误差之和。这种设置不仅考虑了轮廓本身的平滑性，还确保了轮廓与图像内容的紧密匹配。 ##### 3. 跨通道检测对于彩色图像或多光谱图像，即使在单个通道内物体不可见，该算法也能检测到这些物体。例如，在RGB图像中，即使某个物体在红色、绿色或蓝色通道中不显著，但整体上仍然可以被有效识别出来。 ##### 4. 对噪声的鲁棒性该模型对噪声具有很高的鲁棒性，这意味着它可以在未经预处理去除噪声的情况下工作。这对于实际应用非常有益，因为许多现实世界的数据集往往包含一定程度的噪声，而预处理步骤通常会增加计算成本。 #### 方法论 ##### 1. 基础概念 - **主动轮廓**：是一种基于曲线演变的技术，通过变形初始轮廓以寻找目标物体的边界。 - **Mumford-Shah功能性**：一种衡量轮廓质量和图像拟合程度的标准。 - **向量值图像**：指具有多个分量的图像，如RGB图像具有红绿蓝三个分量。 ##### 2. 扩展的Chan-Vese模型传统的Chan-Vese模型主要用于灰度图像的分割，通过最小化内部区域和外部区域之间的差异来定位轮廓。本文提出的扩展模型则适用于向量值图像，其关键在于将Mumford-Shah功能性应用于每个图像分量，同时考虑轮廓长度和平滑性。 ##### 3. 算法流程 - 初始化一个闭合的轮廓。 - 计算每个图像分量的拟合误差。 - 更新轮廓位置以最小化Mumford-Shah功能性。 - 迭代执行直到达到收敛标准或最大迭代次数。 #### 实验结果文章提供了多种实验结果来验证所提方法的有效性和优越性： - **遮挡物体的检测**：即使物体在某些通道中部分缺失，也能准确检测。 - **跨通道物体检测**：在多光谱图像中，即使物体在单一通道内不可见，仍能有效检测。 - **噪声环境下的鲁棒性**：无需预处理去除噪声即可进行有效分割。 #### 结论本文提出的向量值图像主动轮廓模型为复杂图像中的物体检测提供了一个强大的工具。通过对Mumford-Shah功能性的扩展应用，该方法不仅能够处理传统意义上的边缘检测问题，还能应对更为复杂的场景，如跨通道检测和噪声鲁棒性。这一成果为图像处理和计算机视觉领域带来了重要的进展。

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Journal of Visual Communication and Image Representation 11, 130–141 (2000)

doi:10.1006/jvci.1999.0442, available online at http://www.idealibrary.com on

Active Contours without Edges

for Vector-Valued Images

Tony F. Chan,

B. Yezrielev Sandberg,

and Luminita A. Vese

Department of Mathematics, University of California, Los Angeles, 405 Hilgard Avenue,

Los Angeles, California 90095-1555

Received November 5, 1999; accepted December 10, 1999

In this paper, we propose an activecontouralgorithmforobjectdetectioninvector-

valued images (such as RGB or multispectral). The model is an extension of the

scalar Chan–Vese algorithm to the vector-valued case [1]. The model minimizes a

Mumford–Shah functional over the length of the contour, plus the sum of the ﬁtting

error over each component of the vector-valued image. Like the Chan–Vese model,

our vector-valued model can detect edges both with or without gradient. We show

examples where our model detects vector-valued objects which are undetectable

in any scalar representation. For instance, objects with different missing parts in

differentchannelsare completely detected (such as occlusion). Also, in color images,

objects which are invisible in each channel or in intensity can be detected by our

algorithm. Finally, the model is robust with respect to noise, requiring no a priori

denoising step.

2000 Academic Press

Key Words: vector-valued images; active contours; levelsets; segmentation;PDEs;

object detection.

0. INTRODUCTION

Active contours are used to detect objects in a given image u

using techniques of curve

evolution. The basic idea is, starting with an initial curve C, to deform the curve to the

boundary of the object, under some constraints from the image u

The classical approaches use the gradient of the image u

to locate the edges. Therefore,

an edge-function is used, strictly positive inside homogeneous regions and zero on the

edges, to stop the evolving curve on the desired boundary. Using this edge-function, the

deformation is usually obtained by minimizing with respect to C a functional, whose (local)

minimizer is given by the boundary of the object [2]. The objective functional is the sum of

two terms: the ﬁrst controls the smoothness of the curve, while the second (depending on the

This work was supported in part by ONR Contract N00014-96-1-0277 and NSF Contract DMS-9626755.

E-mail: chan@math.ucla.edu.

E-mail: bsand@math.ucla.edu.

E-mail: lvese@math.ucla.edu.

130

1047-3203/00 $35.00

2000 by Academic Press

All rights of reproduction in any form reserved.

ACTIVE CONTOURS FOR VECTOR-VALUED IMAGES 131

gradientofu

)attractsthe curvetoward the boundary.Forinstance,thismodelcannothandle

automatic topology changes of the contour, and also it depends on the parameterization of

the curve.

In problems of curve evolution, including snakes and active contours, the levelset method

of Osher and Sethian [3] has been used extensively because it allows for automatic topology

changes, cusps, and corners; moreover, the computations are made on a ﬁxed rectangular

grid. Using this approach, geometric active contour models, using a stopping edge-function,

have been proposed in [4] and also in [5–7]. These models are based on the theory of

curve evolution and geometric ﬂows and in particular on the mean curvature motion of

Osher and Sethian [3]. The evolving curve moves by mean curvature, but with an ex-

tra factor in the speed, by the stopping edge-function. Therefore, the curve stops on the

edges, where the edge-function vanishes. A typical example of edge-function used is given

g(|∇u

|) =

1 +|∇(G

∗u

where g is a positive and decreasing function such that lim

t→∞

g(t) =0. The image u

ﬁrst convolved with the Gaussian G

(x, y) =σ

−1/2

exp

−|x

+y|

|/4σ

, especially for the cases

where u

is noisy. But in practice, g is never zero on the edges, and therefore the evolving

curve may not stop on the desired boundary. To overcome this problem, a new model has

been proposed in [8] as a problem of geodesic computation in a Riemannian space, accord-

ing to the metric g. The associated Euler–Lagrange equation, in level set formulation, has

a new additional gradient term (compared with the previous geometric models [4–7]). This

term increases the attraction of the evolving curve toward the boundary of the object and

is of special help when the boundary has high variations on its gradient values. For another

related approach, see also [9].

These models use the gradient of a smoother version of the image u

to detect edges.

But, if the image is noisy, the smoothing in the edge-function has to be strong, thus blurring

edge features, or a preprocessing has to be implemented, to remove the noise.

In contrast, the Chan–Vese (C-V) active contour model without edges proposed in [1]

does not use the stopping edge-function g to ﬁnd the boundary. Instead, the stopping term

is based on Mumford–Shah segmentation techniques. This model has several advantages:

it detects edges both with and without gradient (see [10] for a discussion on edges without

gradient, called cognitive edges); it automatically detects interior contours; the initial curve

does not necessarily have to start around the objects to be detected and instead can be placed

anywhere in the image; it gives in addition a partition of the image into two regions, the

ﬁrst formed by the set of the detected objects, while the second one gives the background;

ﬁnally, there is no need for an a priori noise removal.

Thepreviously describedmethodshavebeendevelopedtodetectobjectsinasingle image,

but what happens when we have several different registered images of the same object? This

occurs in multispectral images taken at different wavelengths, in medical images taken by

different equipment (i.e., PET, MRI, and CT), in color images, or in textured images. Each

image channel may have signal characteristics that can be combined with other channels to

enhance contour detection.

Severalmodelsofrestoration,edge detection,andalsoactivecontourshavebeenproposed

for vector-valued images. Forrestoration of color images, we mention the works in [11–13].

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