## Piecewise constant level set methods and image segmentation (2005)

Venue: | Scale Space and PDE Methods in Computer Vision: 5th International Conference, Scale-Space 2005 |

Citations: | 3 - 2 self |

### BibTeX

@INPROCEEDINGS{Lie05piecewiseconstant,

author = {Johan Lie and Marius Lysaker and Xue-cheng Tai},

title = {Piecewise constant level set methods and image segmentation},

booktitle = {Scale Space and PDE Methods in Computer Vision: 5th International Conference, Scale-Space 2005},

year = {2005},

pages = {573--584},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. In this work we discuss variants of a PDE based level set method. Traditionally interfaces are represented by the zero level set of continuous level set functions. We instead use piecewise constant level set functions, and let interfaces be represented by discontinuities. Some of the properties of the standard level set function are preserved in the proposed method. Using the methods for interface problems, we minimize a smooth locally convex functional under a constraint. We show numerical results using the methods for image segmentation. 1

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Citation Context ...eed to introduce a constraint K(φ) = 0 (c.f. §3 §4) and solve the constrained optimization problem min c,φ F (c,φ) subject to K(φ) =0. (4) This problem is solved using the augmented Lagrangian method =-=[1, 17]-=-. A minimizer of F corresponds to a saddle-point of the augmented Lagrangian functional � L(c,φ,λ)=F (c,φ)+ λK(φ) dx + r � |K(φ)| 2 2 dx, (5) Ω where λ is a function defined on the same domain as φ ca... |

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Citation Context ...pportunity to measure the lengths of curves surrounding Ωi and the area of each region Ωi by � � |∂Ωi| = |∇ψi|dx, and |Ωi| = ψidx. (2) Ω Here we note that |∂Ωi| is the Total Variation (TV)-norm of ψi =-=[20]-=-. The above framework can be used as a tool for image segmentation. Let u0 be an image to be segmented. We want to construct a piecewise constant function u which approximates u0 in a proper sense. Th... |

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Citation Context ...ize a smooth locally convex functional under a constraint. We show numerical results using the methods for image segmentation. 1 Introduction The level set method was proposed by Osher and Sethian in =-=[19]-=- as a versatile tool for tracing interfaces separating a domain Ω into subdomains. Interfaces are treated as the zero level set of higher dimensional functions. Moving the interfaces can implicitly be... |

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Citation Context ...d the partition of Ω into a set of subdomains Ωi, one can do further modelling on each domain independently and automatically. One general image segmentation model was proposed by Mumford and Shah in =-=[16]-=-. For numerical approximations, see [2]. Using this model, the image u0 is decomposed into Ω = ∪iΩi ∪ Γ , where Γ is a curve separating the different domains. Inside each Ωi, u0 is approximated by a s... |

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Citation Context ...et functions instead of explicitly moving the interfaces. Applications of the level set method include image analysis, reservoir simulation, inverse problems, computer vision and optimal shape design =-=[5, 4, 10, 26, 18, 22, 24]-=-. In this work, we discuss some variants of the level set method. The primary concern for our approach is to remove the connection between the level set functions and the signed distance function and ... |

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Citation Context ...et functions instead of explicitly moving the interfaces. Applications of the level set method include image analysis, reservoir simulation, inverse problems, computer vision and optimal shape design =-=[5, 4, 10, 26, 18, 22, 24]-=-. In this work, we discuss some variants of the level set method. The primary concern for our approach is to remove the connection between the level set functions and the signed distance function and ... |

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Citation Context ...eed to introduce a constraint K(φ) = 0 (c.f. §3 §4) and solve the constrained optimization problem min c,φ F (c,φ) subject to K(φ) =0. (4) This problem is solved using the augmented Lagrangian method =-=[1, 17]-=-. A minimizer of F corresponds to a saddle-point of the augmented Lagrangian functional � L(c,φ,λ)=F (c,φ)+ λK(φ) dx + r � |K(φ)| 2 2 dx, (5) Ω where λ is a function defined on the same domain as φ ca... |

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Citation Context ...ulties associated with the calculation of the Eikonal equation. Another motivation is to avoid numerical problems associated with the Heaviside and Delta functions used in some level set formulations =-=[5, 25]-=-. The third concern of this approach is to develop fast algorithms for level set methods. Due to the fact that the functional and the constraints for this approach are rather smooth, it is possible to... |

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Citation Context ...ch that φ(x) = i. Thus, each point x ∈ Ω can belong to one and only one phase if K(φ) =0. The constraint (13) is used to guarantee that there is no vacuum and overlap between the different phases. In =-=[27]-=- some other constraints for the classical level set methods were used to avoid vacuum and overlap. Following the framework in §2, we will use the basis functions (11), the constraint (12) and the repr... |

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Citation Context ... the characteristic functions ψi. This could for example be accomplished by interchanging the integer values in (10) by the roots of a Chebyshev polynomial of degree n defined on the interval [−1, 1] =-=[12]-=-. � 1 π(i − 2 zi = cos ) � in Ωi, i =1, 2,...,n. (18) n The exact same framework could then be used to construct a set of characteristic functions {ψi} n i=1 by ψi = 1 αi n� (φ − zj) and αi = j=1 j�=i... |

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Citation Context ...rangian method. Numerical experiments indicate that the augmented Lagrangian method is best suited for this minimization problem. At a saddle-point of (5) we must have ∂L ∂L =0, = 0 and ∂φ ∂ci ∂L =0. =-=(6)-=- ∂λ Essentially we minimize L w.r.t c and φ, and maximize L w.r.t λ. In§3 and §4 we introduce iterative algorithms to find the saddle-points in (6) coming from Ω Ωs576 J. Lie, M. Lysaker, and X.-C. Ta... |

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Citation Context ...and numerical results. We conclude with a brief discussion. For a more detailed treatment of the methods, including more numerical results we refer the reader to [14, 15]. We also refer the reader to =-=[23, 21, 9, 11, 7]-=- for some related methods. 2 A Framework for Subdomain Representation In this section a framework for representing subdomains of Ω is developed. Each subdomain Ωi is associated with a basis function ψ... |

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Citation Context ...mains Ωi, one can do further modelling on each domain independently and automatically. One general image segmentation model was proposed by Mumford and Shah in [16]. For numerical approximations, see =-=[2]-=-. Using this model, the image u0 is decomposed into Ω = ∪iΩi ∪ Γ , where Γ is a curve separating the different domains. Inside each Ωi, u0 is approximated by a smooth function. The optimal partition o... |

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Citation Context ...et functions instead of explicitly moving the interfaces. Applications of the level set method include image analysis, reservoir simulation, inverse problems, computer vision and optimal shape design =-=[5, 4, 10, 26, 18, 22, 24]-=-. In this work, we discuss some variants of the level set method. The primary concern for our approach is to remove the connection between the level set functions and the signed distance function and ... |

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Citation Context ...and numerical results. We conclude with a brief discussion. For a more detailed treatment of the methods, including more numerical results we refer the reader to [14, 15]. We also refer the reader to =-=[23, 21, 9, 11, 7]-=- for some related methods. 2 A Framework for Subdomain Representation In this section a framework for representing subdomains of Ω is developed. Each subdomain Ωi is associated with a basis function ψ... |

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Citation Context ...are implicitly represented by the discontinuities of a set of characteristic functions ψi. Note that both the Chan-Vese model and our model can be extended to shape recognition using the framework of =-=[7, 6]-=-. The rest of this article is structured as follows. Our framework and the minimization functional used for image segmentation is formulated in §2. The segmentation problem is formulated as a minimiza... |

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Citation Context ...ero ∂ci if {ci} n i=1 are computed from (8). We use the Uzawa-type Algorithm 1 to find a saddle point of L(c,φ,λ). The algorithm has a linear convergence rate and its convergence has been analyzed in =-=[13]-=- and used in [4, 3]. Algorithm 1. Choose initial values for φ0 and λ0 .Fork =1, 2,..., do: – Find ck from L(c k ,φ k−1 ,λ k−1 ) = min c L(c,φ k−1 ,λ k−1 ). (14) – Use (1) to update u = � n i=1 ck i ψi... |

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Citation Context ...and numerical results. We conclude with a brief discussion. For a more detailed treatment of the methods, including more numerical results we refer the reader to [14, 15]. We also refer the reader to =-=[23, 21, 9, 11, 7]-=- for some related methods. 2 A Framework for Subdomain Representation In this section a framework for representing subdomains of Ω is developed. Each subdomain Ωi is associated with a basis function ψ... |

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Citation Context ...ail. Both sections include algorithms and numerical results. We conclude with a brief discussion. For a more detailed treatment of the methods, including more numerical results we refer the reader to =-=[14, 15]-=-. We also refer the reader to [23, 21, 9, 11, 7] for some related methods. 2 A Framework for Subdomain Representation In this section a framework for representing subdomains of Ω is developed. Each su... |

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Citation Context ...ail. Both sections include algorithms and numerical results. We conclude with a brief discussion. For a more detailed treatment of the methods, including more numerical results we refer the reader to =-=[14, 15]-=-. We also refer the reader to [23, 21, 9, 11, 7] for some related methods. 2 A Framework for Subdomain Representation In this section a framework for representing subdomains of Ω is developed. Each su... |

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Citation Context ... ) = min φ L(c k−1 , φ, λ k−1 ). (25) – Construct u(ck−1 , φ k )by u = �2 N – Update ck by (8), to solve – Update the multiplier by i=1 ck−1 i ψk i . L(c k , φ k , λ k−1 ) = min c L(c, φ k , λ k−1 ). =-=(26)-=- – If not converged: Set k=k+1 and go to step 1. λ k = λ k−1 + rK(φ k ). (27) Remark 4. Remarks 1 and 2 after Algorithm 1 also apply to algorithm 2. (a) (b) (c) Fig. 4. In this example, the image (a) ... |

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Citation Context ... i=1 are computed from (8). We use the Uzawa-type Algorithm 1 to find a saddle point of L(c,φ,λ). The algorithm has a linear convergence rate and its convergence has been analyzed in [13] and used in =-=[4, 3]-=-. Algorithm 1. Choose initial values for φ0 and λ0 .Fork =1, 2,..., do: – Find ck from L(c k ,φ k−1 ,λ k−1 ) = min c L(c,φ k−1 ,λ k−1 ). (14) – Use (1) to update u = � n i=1 ck i ψi(φ k−1 ). – Find φ ... |

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Citation Context ... the image into three different tissue classes in addition to the background. To accelerate the convergence of our method, we first preprocess the image using a simple tresholding, the isodata method =-=[8]-=-. In (b), we show the result of the isodata segmentation of u0 [8]. We here observe that main structures are preserved, but also highly oscillating patterns occur. We use the results from the isodata ... |

3 |
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