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Chapter 2 Image Segmentation and Limitations Figure 2.8: Segmentation using geodesic active contours. Left: The initial image. Right: Solution of the geodesic active contour method initialized with small circles inside the object. Chan and Vese [31] proposed a method that is independent of gradient information. Instead, they proposed to segment homogeneous regions inside the image. To be more precise, Chan and Vese [31] proposed to minimize the functional ∫ ∫ F(c 1 ,c 2 ,C) = µ · Length(C) + ν · Area(inside(C)) + λ 1 |u 0 − c 1 | 2 dx + λ 2 |u 0 − c 2 | 2 dx . inside(C) The corresponding Euler-Lagrange equation is ( ( ) ) ∇φ φ t = δ(φ) µ∇ · − ν − λ 1 (u 0 − c 1 ) 2 + λ 2 (u 0 − c 2 ) 2 |∇φ| outside(C) (2.39) , (2.40) where δ is the Dirac δ-function [42]. This equation is a parabolic PDE that contains a curvature smoothing term, a term penalizing the segmented area, and two terms penalizing variations from the mean value of the segmented object and the background. Instead of δ, we use a regularized δ-function δ ε for the discretization given by the derivative of the Heaviside approximation H ε = 1 ( 1 + 2 ( z ) ) 2 π arctan . (2.41) ε By using H ε from above, δ ε is 1 δ ε (x) = πε + π . (2.42) ε x2 The mean value of the object and the background can be computed using the Heaviside function: ∫ D c 1 (φ) = u ∫ 0(x)H ε (φ(x))dx ∫ D D H , resp. c 2 (φ) = u 0(x)(1 − H ε (φ(x)))dx ∫ ε(φ(x))dx D (1 − H . (2.43) ε(φ(x)))dx The user chooses λ 1 ,λ 2 , µ,ν. The advantage of the Chan-Vese model is that it does not need edges in the image to segment objects. In fact, the model is independent of gradient information. Instead, it tries to separate homogeneous regions in the image. Fig. 2.9 shows a typical result of Chan-Vese segmentation on an image without edges. This concludes the presentation of classical segmentation algorithms based on PDEs. The presented segmentation algorithms range from interactive, nearly parameter free algorithms, like random walker segmentation, over semi-automatic with a moderate number of variables, like the level set based algorithms, to automatic methods like Mumford-Shah segmentation, where no user interaction is necessary. All these segmentations are able to produce accurate results on a wide range 20
2.5 Why is Classical Image Processing not Enough? Figure 2.9: Segmentation of an object without sharp edges using the Chan-Vese approach. In red, we show the steady-state solution of the Chan-Vese segmentation method initialized with a small circle inside the object. of images and from the perspective of segmentation of single images, there is no need for new concepts. Nevertheless, the approaches presented in this chapter have drawbacks regarding robustness with respect to noise, reproducibility, and error propagation. The next section investigates this. 2.5 Why is Classical Image Processing not Enough? In the last sections, we introduced five segmentation methods and showed that all these segmentation methods perform well on some selected medical images. Besides the segmentation of single images, a segmentation method has to fulfill other features not presented so far: • It is unclear how robust the methods are with respect to image noise. • The robustness of the methods for parameter changes and different initializations is unclear. • Propagating error information through these algorithms is hard, i.e. if information about measurement errors at the image acquisition is available, it is impossible to propagate this information through the segmentation to get segmentation results containing the error information. We organized this section as follows: First, we give an introduction to image noise, show how the image noise influences the image quality for different acquisition modalities and how image noise is modeled mathematically. Then, we investigate the noise robustness of the presented segmentation methods, and finally, we discuss error propagation in classical image segmentation methods. 2.5.1 Image Noise Image noise is a serious problem when dealing with medical images and images from digital cameras. Different noise sources degrade the images. A principal problem of image acquisition devices is the noise due to the random arrival of the photons. Light or X-ray emission is a stochastic process [44]. In addition, the instrumentation noise due to thermal effects in the acquisition device degrades the image quality. Further sources of image noise are the quantization noise due to the conversion from analog to digital signals and the compression process for the images, if any. Physical effects influencing the path of the photons, like blurring, diffraction, and scattering, cause image noise, too. 21
Page 1: Segmentation of Stochastic Images u
Page 5: Abstract The task of segmentation,
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Chapter 9 Summary, Discussion, and
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List of Figures 1.1 Left: CT image
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Appendix A Publications Written Dur
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Bibliography [14] L. Ambrosio and M
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Bibliography [46] B. Echebarria, R.
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