New Characterizations of Simple Points, Minimal Non-simple Sets and P-Simple Points in 2D, 3D and 4D Discrete Spaces

We propose a purely discrete deformable partition model for segmenting 3D images. Its main ability is to maintain the topology of the partition during the minimization process. To do so, our main contribution is a new definition of multi-label simple points (ML simple point) that is easily computable. An ML simple point can be relabeled without modifying the overall topology of the partition. The definition is based on intervoxel properties, and uses the notion of collapse on cubical complexes. This work is an extension of a former restricted definition [16] that prohibits the move of intersections of boundary surfaces. A deformation process is carried out with a greedy energy minimization algorithm. A discrete area estimator is used to approach at best standard regularizers classically used in continuous energy minimizing methods. We illustrate the potential of our approach with the segmentation of 3D medical images with known expected topology.

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“…Then, we prove that the topology of each surface is preserved. This proof is straightforward using the works in [12].…”

Section: (X R): S(x R) Is Empty and Thus C(x R)mentioning

confidence: 99%

“…Therefore, we use the work of [12] which defines the notion of simple sets for cubical complexes. We recall here the main notions of this paper restricted to the specific case used in this work, called specific cubical complex (SCC).…”

Section: Cubical Complexes and Collapsementioning

confidence: 99%

Fully deformable 3D digital partition model with topological control

Damiand

Dupas

Lachaud

2011

Pattern Recognition Letters

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“…Also, they can only capture centeredness within the bounds of the sampling resolution of O ( Fig. [CB08]. Image surface skeletonization is also discussed in recent surveys [SJT14,SBdB15].…”

Section: Image Surface Skeletons (Iss)mentioning

confidence: 99%

3D Skeletons: A State‐of‐the‐Art Report

Tagliasacchi

Delame

Spagnuolo

et al. 2016

Computer Graphics Forum

214

152

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International audienceGiven a shape, a skeleton is a thin centered structure which jointly describes the topology and the geometry of the shape. Skeletons provide an alternative to classical boundary or volumetric representations, which is especially effective for applications where one needs to reason about, and manipulate, the structure of a shape. These skeleton properties make them powerful tools for many types of shape analysis and processing tasks. For a given shape, several skeleton types can be defined, each having its own properties, advantages, and drawbacks. Similarly, a large number of methods exist to compute a given skeleton type, each having its own requirements, advantages, and limitations. While using skeletons for two-dimensional (2D) shapes is a relatively well covered area, developments in the skeletonization of three-dimensional (3D) shapes make these tasks challenging for both researchers and practitioners. This survey presents an overview of 3D shape skeletonization. We start by presenting the definition and properties of various types of 3D skeletons. We propose a taxonomy of 3D skeletons which allows us to further analyze and compare them with respect to their properties. We next overview methods and techniques used to compute all described 3D skeleton types, and discuss their assumptions, advantages, and limitations. Finally, we describe several applications of 3D skeletons, which illustrate their added value for different shape analysis and processing tasks

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“…We say that a complex Y is simple for X if Y can be "removed" from X without modifying the topology of X, and symmetrically, a complex Y ′ is add-simple for X ′ if Y ′ can be "added" into X ′ without modifying the topology of X ′ . These two notions can be defined thanks to the collapse operation (see [6] for precise definitions, and some examples in Fig. 1).…”

Section: Interpixel Topology and Cubical Complexesmentioning

confidence: 99%

“…1 corresponds to the last condition of Definition 1; we only detail the two conditions add-simple and simple by using the definitions given in [6]. The main principle of these definitions is to test if the given set can be collapsed onto a specific set.…”

Section: Algorithm 1: Issimple(xr)mentioning

confidence: 99%

Combining Topological Maps, Multi-Label Simple Points, and Minimum-Length Polygons for Efficient Digital Partition Model

Damiand

Dupas

Lachaud

2011

Lecture Notes in Computer Science

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Abstract. Deformable models have shown great potential for image segmentation. They include discrete models whose combinatorial formulation leads to efficient and sometimes optimal minimization algorithms. In this paper, we propose a new discrete framework to deform any partition while preserving its topology. We show how to combine the use of multilabel simple points, topological maps and minimum-length polygons in order to implement an efficient digital deformable partition model. Our experimental results illustrate the potential of our framework for segmenting images, since it allows the mixing of region-based, contour-based and regularization energies, while keeping the overall image structure.

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New Characterizations of Simple Points, Minimal Non-simple Sets and P-Simple Points in 2D, 3D and 4D Discrete Spaces

Cited by 5 publications

References 20 publications

Fully deformable 3D digital partition model with topological control

Fully deformable 3D digital partition model with topological control

3D Skeletons: A State‐of‐the‐Art Report

Combining Topological Maps, Multi-Label Simple Points, and Minimum-Length Polygons for Efficient Digital Partition Model

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