6egmentation of Color Image Bad on Partial Differential Equations
Chun Yuan
Division of Information Technology Graduate School at Shenzhen, Tsinghua University
Shenzhen, China
yuanc@sz.tsinghua.edu
Shangli Liang
Division of Information Technology Graduate School at Shenzhen, Tsinghua University
Shenzhen, China
beat it 歌词
Abstract—Image gmentation is an important task in image processing. A lot of image gmentation methods or models have been propod. But most of the methods cannot work well with color images, which actually contain more uful information about the objects inside. In this paper, a gme
ntation model for color images is propod. The new model is bad on the GAC gmentation model and extends the concept of gradient from one channel to three channels. Experiments show that the new model has better performance than the GAC model, especially in the gmentation of color images.
Keywords- image gmentation; image processing; PDE; GAC;
I.I NTRODUCTION
Image gmentation is one of the most important tasks in image processing. The purpo of image gmentation is to parate the objects from the background of the image for further processing such as Object Recognition, Object Tracking and so on. A lot of gmentation methods have been propod in the traditional image processing aspect [1] [2] [3] [4]. According to the information ud, the methods can be mainly classified into three categories: method bad on region/threshold, method bad on edges and methods bad on texture. And in the recent decades, the theory of partial differential equations – PDEs has been well developed and introduced into the image processing aspect. Many new approaches of image gmentation bad on PDEs have been propod ever since. However, most of the gmentation methods mentioned above just focus on th
e processing of gray images and do not work well with color images. Since color images contain much more information about the objects than gray images, gmentation bad directly on color images can achieve more accurate results. In this paper, a gmentation model of color image bad on PDEs is propod.
II.R ELATED W ORK
Image gmentation bad on PDEs has been well developed in the recent years. In 1987, M. Kass, A. Witkin and D. Terzopoulos came up with the Active Contour model or Snake model which is the first image gmentation model bad on PDEs[5]. The key idea of the model is to translate the problem of parating the objects into minimizing an energy function of a clo curve:
E[C(p)] = The first two parts of the function are the inner energy of the curve which is ud to shorten and smooth the curve. The last part is the energy from the image which is ud to hold the curve onto the edges of the objects. But the problem of the Snake model is that the energy function depends on not only the position and shape of the curve, but also the parameter of the curve. And the value of the energy function changes arbitrarily according to different types of the parameter.
To overcome the shortness of Snake model, V. Calles, R. Kimmel and G. Sapiro propod the Geodesic Active Contour Model which did not contain any free parameter[6]. The GAC model is also an energy function of the curve which consists of the internal power of the curve and the external power from the image. Since our new gmentation model for color image mainly bad of the GAC model, we will discuss more about the behavior and other details of the GAC model later. Cohen L. and Kimmel R. propod an interactive gmentation model which can get accurate even if the background has many nois[7]. T. Chan and L. Ve propod a gmentation model bad on PDEs which can work well with images which do not contain strong edges[8].
During the evaluation of the curve, there may me topological changes. To cope with this situation, S. Osher and J.
A. Sethian propod the level t gmentation method[9]. The basic idea of level t is to embed the curve into a 2D function, which is actually a 3D model. Since the curve corresponds to the zero level t of the embed function, the evaluation of the
embed function actually reprents the evaluation of the curve.
Figure 1. Level Set
consciousnessIII.G EODESIC A CTIVE C ONTOUR M ODEL
In the Geodesic Active Contour model, the problem of finding the best contour of the object is translated into the problem of minimizing the following energy function:
2011 Fourth International Symposium on Computational Intelligence and Design
L R (C) = (3.1)再别康桥英文译稿
where L(C) is the arc length of the curve C and L R (C) is the weighted arc length. And the corresponding gradient descent
疲惫的英文flow is:
(3.2)
where g is any monotonically decreasing non-negative function, κ is the curvature of the curve.
In the equation 3.2, the first part actually acts like an MCM equation, which means the curve will shrink at the area where the curvature is positive and expand at the area where the curvature is negative. As a result, the total arc length of the curve will become shorter. But this transformation is controlled by the scalar function g(C). So in the flat area of the image, the gradient |∇I| is small and the corresponding g(C) is big, the curve will completely transforms according to the MCM equation and quickly pass the flat area. However, when the curve is near the edges of the object, the gradient |∇I| become bigger and so the g(C) smaller, this part will has less effect on the transformation of the
naive什么意思curve. When g(C) is near zero, which means the curve C is on the edges of the object, this part will completely become uless and the curve will stay on the edges. This part is the internal power of the curve. The cond part of equation 3.2 acts more complexly. In the flat area, since g does not change, the ∇g ≈ 0 and this part will has no effect on the transformation. When the curve is near the edges, g varies rapidly. Since g gets its local minimum at the edges and ∇g always reprent the direction where g will increa, ∇g points at the direction leaving the edges. As a result, if the curve is outside the object, ∇g will point outside the object. And at the same time, the normal vector N points inside the curve, so ∇g ∙N is negative and -(∇g ∙N)N points at the same direction of N, which is inside the curve. So the curve will shrink and get clor to the edges. On the contrary, when the curve is inside the object, ∇g will also point inside the object. So ∇g ∙N is positive and -(∇g ∙N)N points at the opposite direction of N. So the curve will expand and also get clor to the edges. This is the external power from the image. So when the curve evaluates according to the GAC model, it is actually under control of two types of power. One is from the deformation of the curve --- MCM and is called the internal power. The other is from the gradient of g which will hold the curve on the edges. This is called the external power.
Figure 2. Segmentation of GAC Model IV. C OLOR I MAGE S EGMENTATION M ODEL
Since the color images contain more uful information about the objects, gmentation directly bad on the color images can achieve better results. The traditional way to deal with color images is to split the components of the color image into veral single-channel images and process them independently. However, in image gmentation, this method will not work well becau different gmentation results can be achieved from different single-channel images and then the collision occurs. So the best way is to develop a gmentation model which regards the color image as a uniform image.
As discusd in the previous ction, the GAC model mainly ba on the edges of the objects in the images. And edges are actually reprented by the gradient of the gray level in the image. So to expand the GAC model into a color image gmentation model, the most important task is to find out a suitable reprentation of gradient of color image.
Suppo the gradient of color image is ∇I color , the GAC model can be easily expanded in to a color image gmentation model by replacing ∇I with ∇I color :
L R (C) =underneath
(4.1)
And the corresponding gradient descent flow:
(4.2)
So the problem now is to construct the gradient of color image is ∇I color . A natural implement of ∇I color is the weighted
sum of all the gradient of different channels, that is:
(4.3)
(4.4)
where
chancesis the ith channel of the color image. The weights
and can be fix, namely 1/m, or lf-adaptive. For
example, the weights may be in the following form:
(4.5)
Which means the channel with larger gradient will contribute more to the total gradient of the color image. Once the gradient ∇I color and the corresponding are obtained, the gradient descent flow 4.2 can be ud to instruct the evaluation of the curve. And this color image gmentation model, or color-GAC model, can work well with color images
while the traditional GAC model fails.
V.
C ONCLUSION
As an important part of image processing, image gmentation is getting more and more attention. A lot of gmentation methods and models have been propod, such as the traditional gmentation method bad on edges and the GAC gmentation model. But most of the methods and models only work well on gray images. Since color images have more information about the object inside, gmentation bad on color images can achieve more accurate results. So in this paper, we propo a gmentation model for color images. The new color image gmentation model is actually an extension of the GAC gmentation model and is called the color-GAC model. And from the experiments, we can e that the color-GAC model can work well with color images while the traditional GAC model fails. The key component of the color-GAC model is the expression of the gradient of the color images. So it is easy to be improved or extended by modifying the expression of the gradient.
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Figure 3. Experiment between GAC and Color-GAC
