Extract Object Boundaries in Noisy Images using Level Set by: Quming Zhou Literature Survey Submitted to Professor Brian Evans EE381K Multidimensional Digital Signal Processing March 15, 2003 Abstract Finding object contours in noisy images is a challenging task because of the amorphous nature of the object and the lack of sharp boundaries. Classical edge-based segmentation methods have the drawback of not connecting edge segments to form a distinct and meaningful boundary. Many level set approaches, which can deal with changes of topology and the presence of corners, have been developed to extract object boundaries. Previous researchers have used image gradient, edge strength, area minimization and region intensity to define the speed function. However, no paper mentions the edge/gradient direction. Our approach will incorporate direction and magnitude in the speed function.Section 1: Introduction Many of the computer vision applications involve the decomposition of image into region with some homogeneous properties, which are related to the nature of the applications. The boundary of an object is an important feature for the object detection, classification and tracking. Edge based approaches are not suitable for boundary extraction in noisy images [1]. They will detect edges that are not part of an object’s boundary or miss parts of a boundary when the intensity contrast is weak. In general, additional effort is needed to connect the incomplete edges into a distinct and meaningful object boundary. Several approaches have been proposed to extract object boundaries in images using closed curves. Roughly speaking, there are two types of boundary search approaches. One uses a closed contour represented by a parameterized curve. The problem of finding the desirable contour is posed as an energy minimization problem. The classical Euler-Lagrange formulation of the active contour is called ‘snake’ [2]. This kind of method relies on an initial guess of the boundary, image features and parameters. Moreover, its performance suffers from the change of topology and the presence of corners. To overcome these problems, the level set approach has been proposed [3]. The guiding principle of level set methods is to describe a closed curve γin 2R as the zero level set of a higher dimension function ),,( tyxΦ in3R . Instead of propagating the curve γ directly, we consider the evolution of function ),,( tyxΦ with a speed function F and extract the zero level set of points to obtain the boundary curve. Since level set methods represent the curve in an implicit form, they greatly simplify the management of the contour evolution, especially for handling topological changes. Most of the challenges in level set methods result from the need to construct an adequate model for the speed function. This review only focuses on the differential convex function. An algorithm for minimizing a non-differentiable convex function over a convex feasibility has been proposed in [4].Section 2: Background We consider the generation of a family of contours. Let an initial curve0r undergo deformation in a Euclidean plane. Let ),,( tyxr denote the family of curves generated by the propagation of 0r in the outward normal direction Nrwith the speed F. We ignore the tangential velocity because it does not influence the geometry of the deformation, but only its parameterization [5]. The curve velocity ),,( tyxrt is denoted by NFtyxrtr=),,( , (1) where F is a scalar function and Nr is a unit normal vector. According to the level set method, we can express the closed curve )(tr in an implicit form as }0),,(|),{(),,( =Φ= tyxyxtyxr , or 0))),((=Φttr . (2) By the chain rule, 0||=Φ∇Φ∇⋅Φ∇+Φ=⋅Φ∇+Φ=⋅Φ+Φ FNFrtttrtr, yielding the movement equation of curves, 0|| =Φ∇+Φ Ft, with 0)0,,( rtyx==Φ . (3) The above motion equation (3) is a partial differential equation in one higher dimension than the original problem. Given the initial value, it can be solved by means of difference operators in a fixed grid via ))0,min()0,(max(,,,1,−++∇+∇⋅⋅∆−Φ=ΦjijinjinjiFFht , (4) where n is the iterative time, h is the grid step,t∆ is the time step, jiF,is the speed value of pixel (i, j) , nji,Φ is the level value of pixel (i, j) at time n and 5.02222))0,min()0,max()0,min()0,(max(yyxxDDDD+−+−++++=∇ 5.02222))0,min()0,max()0,min()0,(max(yyxxDDDD−+−+−+++=∇jijixD,1, −−Φ−Φ= ,jijixD,,1Φ−Φ=++, 1,, −−Φ−Φ=jijiyD ,jijiyD,1,Φ−Φ=++. This implementation allows the function Φ to automatically follow topological changes and corners during evolution. The speed function F plays a key role in the level set method. Section 3: Prior Work The speed function is essentially a decreasing function of some features. These features should have very high values at the final shape boundary. In general, speed function models can be classified as edge-based, region-based and Motion-based. 3.1 Speed Function Due to Image Gradient Caselle et al. [6] proposed the geometric active contour followed by Malladi et al. [7]. The active contour was extended to extract the surfaces of 3-d objects [8]. The model proposed by Caselles and Malladi was based on the following speed function: |1|/)( IGkaF ∗∇++=σε, (5) where k is the curvature of the curve, a ,εand p are constants and |1| IG ∗∇+σis the edge gradient using a Gaussian filter σG with a known standard deviation σ. Since the stop criterion is the magnitude of the gradient, the speed slows down at strong edges. The drawback of this model is that it only detects objects with edges defined by strong gradients. F is never small enough to stop the curve evolution in a noisy image and the curve may extend beyond the boundary. Moreover the pulling back force is not strong hence it may not be able to pull back the expanding contour if it were to propagate and cross the desired boundary. In paper [9], a curvature profile acts as a boundary regularization term specific to the shape being extracted. This method needed a prior model about the final shape. Yezzi et al. [10] tried to solve the above problems by introducing an extra pull back term. This can be expressed as||/|*1||1|/)(Φ∇Φ⋅∇∇+∇−∗∇++= IGIGkaFσσε (6) where the second term, ||/|*1|Φ∇Φ⋅∇∇+∇ IGσ denotes the projection of an attractive force vector on the normal to the surface. The force is the gradient of a
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