A topology-preserving level set method for shape optimization

Alexandrov, Oleg; Santosa, Fadil

doi:10.1016/j.jcp.2004.10.005

Cited by 55 publications

(36 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For an implicitly represented surface, the dynamics of the surface are determined by the evolution of the embedding function and admit a convenient description of topologically complex interfaces (Osher and Fedkiw, 2003). LSM is quite simple to implement in a wide range of applications involving dynamic surfaces and curves such as shape optimization (Alexandrov and Santosa, 2005) and incompressible two-phase flow (Sussman et al, 1998;Sussman and Fatemi, 1999; Kaliakatsos and Tsangaris, 2000). The time-dependent LSM algorithms for externally generated velocity fields are treated as numerical algorithms for Hamilton-Jacobi equations.…”

Section: Introductionmentioning

confidence: 99%

Two-Phase Flows of Droplets in Contractions and Double Bends

Christafakis

Tsangaris

2008

Engineering Applications of Computational Fluid Mechanics

View full text Add to dashboard Cite

Transient two-phase liquid-liquid flows of Newtonian fluid droplets in a primary Newtonian fluid with different viscosities, through constricted channels and double-bend channels, are numerically modelled and studied. The deformation and migration of various arrangements (arrays) of droplets, as they are carried by the primary phase into the low Reynolds number (Re), developing Hagen-Poiseuille laminar flow, is computed and discussed on the basis of the non-dimensional numbers: Capillary number (Ca), Reynolds number (Re), Weber number (We) and viscosity ratio (λ). The observations are well compared with the results of the literature. To predict the contours of droplets, we implement a level-set method (LSM) for capturing interfaces with high order WENO and TVD Runge-Kutta schemes. The LSM implicitly provides the conditions for the revised, nearly second-order, two-phase, time-dependent, 2-D Navier-Stokes momentum equations' solver, incorporating viscous and surface tension terms depending on the level-set function.

show abstract

Section: Introductionmentioning

confidence: 99%

Two-Phase Flows of Droplets in Contractions and Double Bends

Christafakis

Tsangaris

2008

Engineering Applications of Computational Fluid Mechanics

View full text Add to dashboard Cite

show abstract

“…Among the family of topology constrained front propagation methods [35,5,174,105,78,157], the works in [78] and in [157] rely on simple points [13,90,14], that is, points such that their addition or removal to the component will not change the topology of the image. They start from initial seeds distributed in the areas of interest in the space of the image, and then modify these components by adding or removing simple points.…”

Section: Front Propagation and Well-composed Segmentationsmentioning

confidence: 99%

A Tutorial on Well-Composedness

2017

View full text Add to dashboard Cite

Due to digitization, usual discrete signals generally present topological paradoxes, such as the connectivity paradoxes of Rosenfeld. To get rid of those paradoxes, and to restore some topological properties to the objects contained in the image, like manifoldness, Latecki proposed a new class of images, called wellcomposed images, with no topological issues. Furthermore, well-composed images have some other interesting properties: for example, the Euler number is locally computable, boundaries of objects separate background from foreground, the tree of shapes is well-defined, and so on. Last, but not the least, some recent works in mathematical morphology have shown that very nice practical results can be obtained thanks to well-composed images. Believing in its prime importance in digital topology, we then propose this state-of-the-art of well-composedness, summarizing its different flavours, the different methods existing to produce well-composed signals, and the various topics that are related to well-composedness.

show abstract

“…2) Work of Alexandrov and Santosa: Our work is also much motivated by the interesting work [1], which is a curve evolution method based on level sets for shape optimization problems arising in material science. The signed-distance function is used, and their proposed method avoids overlaps of the narrow band of the evolving contour.…”

Section: ) Work Of Han Et Almentioning

confidence: 99%

“…Their method, like ours, is a variational one, but it has not been proposed for image segmentation. Only artificial tests of shape optimization illustrate their point in [1]. In the case of a shape optimization problem, their model minimizes , with , where is a general shape optimization functional and is the topology constraint term defined by where and , are given parameters.…”

Section: ) Work Of Han Et Almentioning

confidence: 99%

“…Finally, the speed on the contour is extended to all level lines by solving an additional PDE, reinitialization to the distance function being needed as in our approach. Note that in all variational relevant approaches in [1], [25]- [27], [36], and [37], the penalty term is a line integral (more difficult to handle in practice), unlike our case where we have a region integral. To conclude this remark, in Rochery's work, the application is different, devoted to the detection of long and thin objects, such as roads in satellite images and also the function is different in our model.…”

Section: ) Description Of the Proposed Topology-constrained Problemmentioning

confidence: 99%

See 1 more Smart Citation

Self-Repelling Snakes for Topology-Preserving Segmentation Models

Guyader

Vese

2008

IEEE Trans. on Image Process.

View full text Add to dashboard Cite

Abstract-The implicit framework of the level-set method has several advantages when tracking propagating fronts. Indeed, the evolving contour is embedded in a higher dimensional level-set function and its evolution can be phrased in terms of a Eulerian formulation. The ability of this intrinsic method to handle topological changes (merging and breaking) makes it useful in a wide range of applications (fluid mechanics, computer vision) and particularly in image segmentation, the main subject of this paper. Nevertheless, in some applications, this topological flexibility turns out to be undesirable: for instance, when the shape to be detected has a known topology, or when the resulting shape must be homeomorphic to the initial one. The necessity of designing topology-preserving processes arises in medical imaging, for example, in the human cortex reconstruction. It is known that the human cortex has a spherical topology so throughout the reconstruction process this topological feature must be preserved. Therefore, we propose in this paper a segmentation model based on an implicit level-set formulation and on the geodesic active contours, in which a topological constraint is enforced.

show abstract

A topology-preserving level set method for shape optimization

Cited by 55 publications

References 10 publications

Two-Phase Flows of Droplets in Contractions and Double Bends

Two-Phase Flows of Droplets in Contractions and Double Bends

A Tutorial on Well-Composedness

Self-Repelling Snakes for Topology-Preserving Segmentation Models

Contact Info

Product

Resources

About