13. A Combined Shape-Newton and Topology Optimization Technique in Real-Time Image Segmentation

Hintermüller, Michael

doi:10.1137/1.9780898718935.ch13

Cited by 4 publications

(9 citation statements)

References 15 publications

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“…We refer the interested reader to [14,16,18,30,5,4,10,11,12] as well as the monograph [32]. Besides the field of shape and topology optimization, topological derivatives are also used in applications from mathematical imaging, such as image segmentation [25] or electric impedance tomography [26,28], or other geometric inverse problems such as the detection of obstacles, of cracks or of impurities of a material, see e.g., [16,23] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Topological derivative for the nonlinear magnetostatic problem

Gangl¹,

Amstutz²

2019

etna

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The topological derivative represents the sensitivity of a domain-dependent functional with respect to a local perturbation of the domain and is a valuable tool in topology optimization. Motivated by an application from electrical engineering, we derive the topological derivative for an optimization problem which is constrained by the quasilinear equation of two-dimensional magnetostatics. Here, the main ingredient is to establish a sufficiently fast decay of the variation of the direct state at scale 1 as |x| → ∞. In order to apply the method in a bi-directional topology optimization algorithm, we derive both the sensitivity for introducing air inside ferromagnetic material and the sensitivity for introducing material inside an air region. We explicitly compute the arising polarization matrices and introduce a way to efficiently evaluate the obtained formulas. Finally, we employ the derived formulas in a level-set based topology optimization algorithm and apply it to the design optimization of an electric motor.

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Section: Introductionmentioning

confidence: 99%

Topological derivative for the nonlinear magnetostatic problem

Gangl¹,

Amstutz²

2019

etna

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“…This has motivated numerous papers in the literature that investigate specific instances of (14) for alternative spaces and manifolds with geometries and inner products that are more appropriate for φ than R n . For instance, some papers use shape sensitivity calculus to solve (14) in a space of geometrical variables or shapes [16]- [18]. Many papers have proposed solving (14) in a Sobolev space because its inner product acts as a smoothing operator inducing favourable regularity properties on the contour [19]- [22].…”

Section: Region-based Level Set Segmentationmentioning

confidence: 99%

“…Many papers have proposed solving (14) in a Sobolev space because its inner product acts as a smoothing operator inducing favourable regularity properties on the contour [19]- [22]. Alternatively, other papers consider Newton-type methods that solve (14) on manifolds whose inner products are related the Hessian matrix of E( y; φ) [1], [16]- [18], [23], [24]. The empirical results reported as described above confirm that solving (14) in an appropriate space, with inner products that induce favourable properties to the gradient flow, can improve the convergence speed dramatically.…”

Section: Region-based Level Set Segmentationmentioning

confidence: 99%

“…In our method this could be achieved by defining position dependent statistical parameters θ 1 = θ 1 (x) and θ 2 = θ 2 (x), which could then be estimated from the neighbourhood of y(x) as in [30]. The resulting FIM would be equal (18) with pixel dependent Bregman divergences evaluated using the values of θ 1 and θ 2 for that pixel; this matrix would still be diagonal and positive definite. Finally, it is also worth mentioning that although in this paper we apply (23) to all the elements of vector φ, one could also consider using a narrow band approach where only the elements of φ that are close to zero are updated (i.e., the pixels within a neighbourhood of C), similarly to [16].…”

Section: Regularisation Of φmentioning

confidence: 99%

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Exploiting Information Geometry to Improve the Convergence of Nonparametric Active Contours

Pereyra

Batatia

McLaughlin

2015

IEEE Trans. on Image Process.

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This paper presents a fast converging Riemannian steepest descent method for nonparametric statistical active contour models, with application to image segmentation. Unlike other fast algorithms, the proposed method is general and can be applied to any statistical active contour model from the exponential family, which comprises most of the models considered in the literature. This is achieved by first identifying the intrinsic statistical manifold associated with this class of active contours, and then constructing a steepest descent on that manifold. A key contribution of this paper is to derive a general and tractable closed-form analytic expression for the manifold's Riemannian metric tensor, which allows computing discrete gradient flows efficiently. The proposed methodology is demonstrated empirically and compared with other state of the art approaches on several standard test images, a phantom positron-emission-tomography scan and a B-mode echography of in-vivo human dermis.

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“…Hence several authors (cf. Allaire, Gournay, Jouve & Toader [3], Burger, Hackl & Ring [16], Hintermüller [28]) tried successfully to combine classical level set methods with the concept of topological derivatives. Their are basically two ideas how to combine these methods.…”

Section: Introductionmentioning

confidence: 99%

Methods for Reliable Topology Changes for Perimeter-Regularized Geometric Inverse Problems

Hackl

2007

SIAM J. Numer. Anal.

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This paper is devoted to the incorporation of topological derivative like expansions of first and second order in volume and perimeter into level set methods for perimeter regularized geometric inverse problems. Based on these expansions we provide a steepest descent type and a Newton-type algorithm to force topology changes in level set methods. Numerous numerical examples are provided that show the strong and also the weak points of these estimates.

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13. A Combined Shape-Newton and Topology Optimization Technique in Real-Time Image Segmentation

Cited by 4 publications

References 15 publications

Topological derivative for the nonlinear magnetostatic problem

Topological derivative for the nonlinear magnetostatic problem

Exploiting Information Geometry to Improve the Convergence of Nonparametric Active Contours

Methods for Reliable Topology Changes for Perimeter-Regularized Geometric Inverse Problems

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