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Possible cuts for part-based segmentation. Suboptimal cuts (a). Cuts created by our method (b). Medial surface colored by its importance metric (c).
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![Fig. 5. Part-based segmentations of our method vs Reniers et al. [23]...](profile/Cong-Feng-2/publication/284887483/figure/fig4/AS:614045438730270@1523411249495/Part-based-segmentations-of-our-method-vs-Reniers-et-al-23-Sec-4_Q320.jpg)
We present a new method for part-based segmentation of voxel shapes that uses medial surfaces to define a segmenting cut at each medial voxel. The cut has several desirable properties–smoothness, tightness, and orientation with respect to the shape’s local symmetry axis, making it a good segmentation tool. We next analyze the space of all cuts crea...
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... method has a simple intuition: Say we want to cut the shape in Fig. 1 a close to points A . . . E. Which properties should these cuts have to yield a 'natural' PBS? In other words: How would a human draw such cuts? Figure 1 a shows five undesirable cuts: A is noisy, although it crosses a perfectly smooth surface zone; B is self-intersecting; C and D are too loose (long); and E is unnaturally slanted -a ...
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... other words: How would a human draw such cuts? Figure 1 a shows five undesirable cuts: A is noisy, although it crosses a perfectly smooth surface zone; B is self-intersecting; C and D are too loose (long); and E is unnaturally slanted -a human asked to cut the shape at that point would arguably do it so across the finger's symmetry axis. Figure 1 b shows five cuts for the same points, computed with the method in this paper. ...
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... 1 a shows five undesirable cuts: A is noisy, although it crosses a perfectly smooth surface zone; B is self-intersecting; C and D are too loose (long); and E is unnaturally slanted -a human asked to cut the shape at that point would arguably do it so across the finger's symmetry axis. Figure 1 b shows five cuts for the same points, computed with the method in this paper. We argue that these cuts are more suitable for PBS than those in Fig. 1 a, as they are (1) tight, (2) locally smooth, (3) self-intersection free, (4) and locally orthogonal to the shape's symmetry axis. ...
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... zone; B is self-intersecting; C and D are too loose (long); and E is unnaturally slanted -a human asked to cut the shape at that point would arguably do it so across the finger's symmetry axis. Figure 1 b shows five cuts for the same points, computed with the method in this paper. We argue that these cuts are more suitable for PBS than those in Fig. 1 a, as they are (1) tight, (2) locally smooth, (3) self-intersection free, (4) and locally orthogonal to the shape's symmetry axis. An additional property that cuts should satisfy is (5) being closed curves, so that they divide the shape's surface into different parts. We construct such cuts as follows: First, we compute a simplified ...
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... the MGF monotonically increases from the medial surface boundary to its center, upper thresholding it yields connected and noise-free simplified medial surfaces (though tunnel preservation requires additional work) [25]. Figure 1 c shows a regularized medial surface using the MGF method in [25]. ...
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... This segmentation only results in surface parametrization (Bénière et al., 2013;Gauthier et al., 2017;Vidal et al., 2014) so that the topology itself is not covered quite well. Therefore so-called curve (middle line) or surface skeleton (middle surface) can be used for part segmentation (Agathos et al., 2007;Feng et al., 2015;Reniers and Telea, 2008), which can lead to a more beamline representation (Bremicker et al., 1991;Nana et al., 2017;Stangl and Wartzack, 2015). These curve skeletons serve as a reasonable shape descriptor for tube-like organic geometries (Tagliasacchi et al., 2016). ...
Polygon meshes and particularly triangulated meshes can be used to describe the shape of different types of geometry such as bicycles, bridges, or runways. In engineering, such polygon meshes can be supplied as finite element meshes, resulting from topology optimization or from laser scanning. Especially from topology optimization, frame-like polygon meshes with slender parts are typical and often have to be converted into a CAD (Computer-Aided Design) format, e.g., for further geometrical detailing or performing additional shape optimization. Especially for such frame-like geometries, CAD designs are constructed as beams with cross-sections and beam-lines, whereby the cross-section is extruded along the beam-lines or beam skeleton. One major task in the recognition of beams is the classification of the cross-section type such as I, U, or T, which is addressed in this article. Therefore, a dataset consisting of different cross-sections represented as binary images is created. Noisy dilatation, the distance transformation, and main axis rotation are applied to these images to increase the robustness and reduce the necessary amount of samples. The resulting images are applied to a convolutional neuronal network.
... This segmentation only results in surface parametrization (Bénière et al., 2013;Gauthier et al., 2017;Vidal et al., 2014), so that topological properties are not covered. Therefore, the so-known curve (middle line) or surface skeleton (middle surface) can be used for part segmentation (Agathos et al., 2007;Feng et al., 2015;Reniers and Telea, 2008a), which can lead to a representation in beamline shape (Bremicker et al., 1991;Nana et al., 2017;Stangl and Wartzack, 2015). While curve skeleton can be used for part segmentation of organic shapes (Nana et al., 2017;Reniers and Telea, 2008a;Stangl and Wartzack, 2015), the surface skeleton can be used for patch segmentation (Reniers and Telea, 2008b) and part segmentation (Feng et al., 2015), or a hybrid form using surface skeleton for patch-part segmentation (Koehoorn et al., 2017). ...
... Therefore, the so-known curve (middle line) or surface skeleton (middle surface) can be used for part segmentation (Agathos et al., 2007;Feng et al., 2015;Reniers and Telea, 2008a), which can lead to a representation in beamline shape (Bremicker et al., 1991;Nana et al., 2017;Stangl and Wartzack, 2015). While curve skeleton can be used for part segmentation of organic shapes (Nana et al., 2017;Reniers and Telea, 2008a;Stangl and Wartzack, 2015), the surface skeleton can be used for patch segmentation (Reniers and Telea, 2008b) and part segmentation (Feng et al., 2015), or a hybrid form using surface skeleton for patch-part segmentation (Koehoorn et al., 2017). While points on a curve skeleton can be classified in view cases such as junction-, end-and skeletal point, the surface skeleton extends the number of cases for parametrization dramatically (Hisada et al., 2001;Saha et al., 2000;Svensson et al., 2002). ...
Polygon meshes and particularly triangulated meshes can be used to describe the shape of different types of geometry such as bicycles, bridges, or runways. In engineering, such polygon meshes can occur as finite element meshes, resulting from topology optimization or laser scanning. This article presents an automated parameterization of polygon meshes into a parametric representation using subdivision surfaces, especially in topology optimization. Therefore, we perform surface skeletonization on a volumetric grid supported by the Euclidian distance transformation and topology preserving and shape-preserving criterion. Based on that surface skeleton, an automated conversation into a Subdivision Surface Control grid is established. The final mid-surface-like parametrization is quite flexible and can be changed by variating the control gird or the local thickness.
... In particular organic shapes resulting from topology optimization, a descriptor such as medial axes (skeletons) (Blum, 1967) shows desirable concepts for recent reverse engineering approaches in topology optimization (Mayer and Wartzack, 2020;Nana et al., 2017a;Stangl and Wartzack, 2015;Yin et al., 2020). These skeleton-based methods are capable of reasonable part segmentation (Agathos et al., 2007;Feng et al., 2015;Reniers and Telea, 2008), so that a hand can be segmented into five fingers and the palm of the hand (Wu et al., 2016). Such parametrization has been implemented in various applications for the reverse engineering task in topology optimization (Mayer and Wartzack, 2020;Nana et al., 2017a;Stangl and Wartzack, 2015;Yin et al., 2020) for predefined optimized shapes. ...
The automated reverse engineering of wireframes is a common task in topology optimization, fast concept design, bionic and point cloud reconstruction. This article deals with the usage of skeleton-based reconstruction of sketches in 2D images. The result leads to a flexible at least C1 continuous shape description.
... This segmentation only results in surface parametrization [1]- [3] so that the topology itself is not covered quite well. Therefore so-called curve (middle line) or surface skeleton (middle surface) can be used for part segmentation [8]- [10], which can lead to a more beam-line representation [7], [11], [12]. These curve skeletons serve as a reasonable shape descriptor for tube-like organic geometries [13]. ...
... Another option is to determine first the cross-section and then to construct the skeleton [24], which is not further discussed in this article. For other concepts, we refer to the summary of segmentation methods in [8] or more recent publications [2], [10], [22], [23], [25]. ...
Polygon meshes and particularly triangulated meshes can be used to describe the shape of different types of geometry such as bicycles, bridges, or runways. In engineering, such polygon meshes can be supplied as finite element meshes, resulting from topology optimization or from laser scanning. Especially from topology optimization with low member size settings, frame-like polygon meshes with slender parts are typical and often have to be converted into a CAD (Computer-Aided Design) format, e.g., for further geometrical detailing or performing additional shape optimization. Especially for such frame-like geometries, CAD designs are constructed as beams with cross-sections and beam-lines, whereby the cross-section is extruded along the beam-lines or beam skeleton. In our research, automatic parameterization of polygon meshes into a subdivision surface representation is tried out. For this purpose, the beam-lines are approximated by computation of curved skeletons, which are determined by a homotopic thinning method. These skeleton lines are transformed into a subdivision surface control grid by using the Euclidean distance transformation.
... In contrast, SEG-MAT overcomes this limitation by using the MAT which is rigorously defined as a complete representation for an arbitrary 3D shape. Feng et al. [33] analyze the cut distribution based on the medial geodesic function [34] (MGF) computed with the medial surfaces to identify different parts for shape segmentation. The method solely relies on the single descriptor MGF without considering the shape structure; while SEG-MAT leverages various properties of the MAT, leading to higherquality and more structure-aware segmentation results. ...
... In this section, we evaluate our method with comparisons to two representative segmentation methods based on skeletal representations, i.e., generalized cylinder decomposition (GCD) [50] and skeleton cut space analysis (Skel-Cut) [33]. ...
... Skel-Cut SEG-MAT Fig. 19. Qualitative comparison with representative skeleton-based methods, i.e., Generalized cylinder decomposition [50] (GCD) and Skel-Cut [33]. ...
Segmenting arbitrary 3D objects into constituent parts that are structurally meaningful is a fundamental problem encountered in a wide range of computer graphics applications. Existing methods for 3D shape segmentation suffer from complex geometry processing and heavy computation caused by using low-level features and fragmented segmentation results due to the lack of global consideration. We present an efficient method, called SEG-MAT, based on the medial axis transform (MAT) of the input shape. Specifically, with the rich geometrical and structural information encoded in the MAT, we are able to develop a simple and principled approach to effectively identify the various types of junctions between different parts of a 3D shape. Extensive evaluations and comparisons show that our method outperforms the state-of-the-art methods in terms of segmentation quality and is also one order of magnitude faster.
... Curve skeletons Cornea et al. (2007); Tagliasacchi et al. (2016), compared with the boundary representation, provide a more compact way to encode a shape, and easier to manipulate in practice. In spite of compactness, the hints behind are enough for many geometry processing and analysis tasks, such as animation Angelidis and Cani (2002); Baran and Popović (2007), deformation Schaefer and Yuksel (2007); Jacobson et al. (2011), segmentation Lien et al. (2006); Shapira et al. (2008); Feng et al. (2015) and shape matching/retrieval Hilaga et al. (2001); Sundar et al. (2003); Au et al. (2010). Although skeleton extraction of a 2D shape has been well studied, skeletonization of a 3D object remains fascinating but still challenging, to our best knowledge. ...
... In fact, it is observed that skeletonization is potentially related to the segmentation problem. On the one hand, due to the fact that skeletons are able to encode the overall structure, one can use a skeleton to guide segmentation of a shape Lien et al. (2006); Attene et al. (2006); Tierny et al. (2007); Reniers et al. (2008); Feng et al. (2015). On the other hand, a semantic shape segmentation can also reflect structural configuration of a shape. ...
Curve skeletons of 3D objects are central to many geometry analysis tasks in the field of computer graphics. A desirable skeleton has to meet at least four requirements: (1) topologically homotopic to the primitive shape, (2) truly well-centred, (3) feature preserving and (4) has a reasonable degree of smoothness. There are at least a couple of difficulties with skeletonization. On the one hand, finding the “best” skeleton is related to visual perception, to some extent, and thus hard to be completely solved by a pure geometric technique. On the other hand, how to exactly characterize the centredness of a skeleton, without a pre-computed medial axis surface, still remains challenging.
Due to the fact that skeletons are able to encode the overall structure, a skeleton has been used to guide segmentation of a shape, which implies that there exists a dual relationship between segmentation and skeletonization. Based on the underlying duality, we propose to generate skeletons from a reliable segmentation result that is more easily available by deep learning or alternative techniques. In implementation, we first extract a collection of samples and then compute the Voronoi diagram restricted in the volume w.r.t. those samples, followed by transforming the clipped Voronoi diagram into a graph G. We further equip each edge in G with a centredness score. The user-specific segmentation result is then used to decompose G into a set of subgraphs Gi=1k. The next task is to compute the Steiner tree for each subgraph while requiring that the Steiner trees of two adjacent parts Gi and Gj must be linked together. The global structure of the final skeleton inherits the proximity configuration of the user-specific segmentation, and thus is topologically homotopic to the primitive shape. At the same time, the centredness of the final skeleton is taken into full consideration by maximizing the overall centredness score. We also integrate the other two requirements carefully into our algorithmic framework. We conduct extensive experiments to evaluate the new approach in terms of the above-mentioned aspects. The experimental results show that our approach has an obvious advantage over the state-of-the-arts. As a by-product of our algorithm, users can obtain skeletons with different levels of details by editing the segmentation configurations.
... Other segmentation methods are based on topological information, like Reeb graphs [24] but depend on the function chosen for defining the graph [25]. Skeletons have been applied for segmentation on point clouds, where either the resulting skeletons are not connected [26,27] or do not provide a consistent segmentation [28], and on surface meshes [29], using the skeleton to define a dense set of possible segmentation cuts from which a subset will be chosen based on boundary geometry [30]. Very few of these segmentation methods explicitly profit from the innate parentchild relation between sub-parts, or measure for the relative saliency over the parts they obtain. ...
This paper introduces a measure of significance on a curve skeleton of a 3D piecewise linear shape mesh, allowing the computation of both the shape's parts and their saliency. We begin by reformulating three existing pruning measures into a non-linear PCA along the skeleton. From this PCA, we then derive a volume-based salience measure, the 3D WEDF, that determines the relative importance to the global shape of the shape part associated to a point of the skeleton. First, we provide robust algorithms for computing the 3D WEDF on a curve skeleton, independent on the number of skeleton branches. Then, we cluster the WEDF values to partition the curve skeleton, and coherently map the decomposition to the associated surface mesh. Thus, we develop an unsupervised hierarchical decomposition of the mesh faces into visually meaningful shape regions that are ordered according to their degree of perceptual salience. The shape analysis tools introduced in this paper are important for many applications including shape comparison, editing, and compression.
... Recently, surface skeletons have been used to produce part-based segmentations of 3D shapes [17,16]. A key to this idea is the construction of a so-called cut space containing a large number of well-designed cuts that partition the shape in ways similar to how a human would cut it. ...
... Recently, surface skeletons of voxel models have also been used for part-based segmentation [17]. The key idea is to construct a cut space CS = {c i } that contains a large set of cuts that have suitable properties to act as segment boundaries. ...
... Cut spaces are constructed by building closed loops formed by shortest paths on ∂ between the two feature points of each surface skeleton point. Next, cuts that represent suitable segment boundaries are found by analyzing a histogram of the cut lengths [17] or, alternatively, clustering cuts in terms of length [16]. Related methods of analyzing cut spaces for segmenting shapes have also been proposed, though not using skeletons to construct the cuts; see, e.g., [25,20]. ...
Curve skeletons are well known for their ability to support part-based segmentation of 3D shapes. In contrast, surface skeletons have been only sparsely used for this task and, to our knowledge, only for voxel representations. We present here a method to use such surface skeletons to segment 3D meshes. For this, we extend a recent surface-skeleton-based method for part-based segmentation of voxel shapes [16] to efficiently handle high-resolution mesh shapes, on the one hand, and to compute part-based, patch-based, and hybrid part-and-patch segmentations, a result we refer to as unified segmentation. Our method can handle high-resolution 3D meshes with low memory and computational costs and produces segmentations that compare favorably with those delivered by other state-of-the-art methods. We demonstrate our method on a wide collection of both natural (articulated) and man-made (faceted) shapes.
... The distance-to-boundary directly delivers a way to define and estimate local shape thickness, or wall thickness [Geo10], used to estimate e.g. the printability and mechanical resistance of 3D shapes [DK09,JKT13], anatomic tissue resistance [YP03], characterize anatomic shapes [NSK * 97], and find tubular shape parts [MPS * 04]. Finding similar-thickness shape parts enables high-quality shape segmentation [SSCO08,FJT15]. Separately, finding input-surface points that correspond to the boundaries of the medial surface via the feature transform allows robustly finding edges even for complex noisy shapes, which serves shape classification [RJT08,KJT15]. ...
Given 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.
... In this paper, we extend the related segmentation framework proposed in [13] with the following main contributions: ...
... -We present a clustering-based segmentation technique using the shape cut-space, which works more robustly, and is easier to use, than the earlier histogram-based technique in [13]; -We present a detailed analysis of the parameter space of the entire pipeline, which allows us to nd good preset values for the method's free parameters, and also gives detailed insight in the method's behavior; -We present a new application of our cut-space segmentation technique for the interactive, user-driven, segmentation of 3D shapes. ...
... A rst way to do this is to use the histogram of cut lengths over S, as described in [13]. This works as follows. ...
We present a refined method for part-based segmentation of voxel shapes by constructing partitioning
cuts from every voxel of the shape’s medial surface. Our cuts have several desirable properties –
smoothness, tightness, and orientation with respect to the shape’s local symmetry axis, making it a good
segmentation tool.We analyze the space of all cuts created for a given shape and detect cuts which are good
segment borders. We present a detailed analysis of the parameter space of our method, which yields good
preset values for all its parameters. Our method is robust to noise, pose invariant, independent on the shape
geometry and genus, and is simple to implement.We demonstrate our method for both automatic and interactive
segmentation on a wide selection of 3D shapes.
