Maximal Disjoint Ball Decompositions for shape modeling and analysis

Highlights

• MDBD is defined as a packing of maximal disjoint closed balls inside a d-dimensional domain.

• The proposed decomposition relies on distance computations, which have to be supported by any valid geometric representation.

• MDBD is unique and possesses appealing stability and robustness properties against small boundary perturbations.

• One can construct universal shape descriptors based on MDBD across any valid geometric representation.

• MDBD provides adequate support for key downstream applications for models coming from disparate geometric representations.

Abstract

A large number of geometric representations have been proposed to address the needs of specific engineering applications. This, in turn, exacerbates the inherent challenges associated with system interoperability for downstream engineering applications.

In this paper, we define the Maximal Disjoint Ball Decomposition (MDBD) as the location of the largest d-dimensional closed balls recursively placed in the interior of a d-dimensional domain and show that the proposed decomposition can be used to provide an underlying common analysis framework for geometric models using different representation schemes. Importantly, our decomposition only relies on the ability of an existing geometric representation to compute distances, which must be supported by any valid geometric representation scheme, and does not require an explicit representation conversion. Moreover, MDBD is unique for a given domain up to rigid body transformation, reflection, as well as uniform scaling, and its formulation suggests appealing stability and robustness properties against small boundary modifications.

Furthermore, we show that MDBD can be used as a universal shape descriptor to perform shape similarity of models coming from various geometric representation schemes. A salient attribute of this decomposition is that it provides adequate support for key downstream applications for models coming from disparate geometric representations. For example, MDBD can be naturally used to carry out meshless solutions to boundary value problems; efficient collision detection; and 3D mesh generation of models that use any valid geometric representation scheme. Finally, our hierarchical formulation of the proposed Maximal Disjoint Ball Decomposition allows for a choice of model complexity at run-time to match the available computational resources.

Introduction

Numerous geometric representations have been proposed over the years to address the needs of specific engineering applications. For example, today’s “gold standard” in CAD is the boundary representation with NURBS curves and surfaces, although there are other spline versions that are finding some success as analysis-suitable representations [1]; triangular meshes are the surface representation of choice in graphics-related applications; meshes are being used in solving boundary value problems with Finite Elements [2]; point clouds output by depth cameras are input into mesh reconstruction algorithms [3], which often involves user interaction — thus geometric processing methods operating directly on the point clouds have emerged as an alternative processing approach [4], [5], [6], [7]; cellular representations, such as octrees [8], voxels, and meshes [9], [10], are used to speed up or enable specific solution strategies for a variety of downstream applications; implicit geometric representations have found their niche in many applications involving some form of geometric fit, including fixture design and analysis [11], haptic-assisted assembly [12], [13], [14], and topology optimization [15], [16].

The wealth of geometric representations produced an even larger stream of algorithms designed to take advantage of specific representation schemes. On one hand, all current shape analysis frameworks assume a consistent representation of all models being analyzed, and this assumption requires all representation conversions to be successfully completed prior to undertaking the analysis. On the other hand, converting between existing geometric representations is far from being a trivial or solved problem, and every such conversion generates some information loss that is in general not well understood. It is therefore not surprising that current shape analysis systems have limited ability to deal with models natively existing in different representations having different levels of informational content. In fact, the theoretical and practical challenges of interoperability prompted the development of alternative proposals aimed to control the interfaces between software systems rather than the format of the exchanged data [17].

The ability to measure the similarity between geometric information that exists in different geometric representations is essential for similarity search in geometric database systems [18] containing crowd-sourced geometric models of mechanical components. The current state of the art requires multiple representation conversions prior to measuring similarity with existing algorithms, subject to the limitations discussed above. Furthermore, consider the assembly of large and highly complex systems encountered, for example, in the context described in [19]. In such a case, the user faces the challenge of determining where a given component or subsystem needs to be spatially located to enable its correct assembly. In this context, an Augmented Reality (AR) framework can be used to fuse the information coming from different geometric representations, such as depth cameras and CAD systems, to provide correct assembly instructions to the user. Importantly, the associated computations may have to be performed on models that natively exist in distinct representations. The final example scenario presented here is that of the traditional structural Finite Element Analysis, which requires geometric models to be converted to solid meshes in order to solve the corresponding boundary value problems. However, meshing continues to be an active area of research and remains a demanding geometric pre-processing step in FEA. To circumvent the mesh-related challenges, a variety of “meshfree” boundary value solution methods have been developed, but these operate only on one specific geometric representation, which are reviewed in [20].

It is clear that the representations that dominate a specific application domain may not adequately support other application domains without the use of representation conversion.

We aim to formulate a framework capable of unifying the geometric information that exists in multiple geometric representations without requiring an explicit representation conversion. To this end, we observe that the concept of distance is intrinsic to all valid geometric representations used in 3D geometric modeling, which suggests that distances should play an important role.

Furthermore, the circle is arguably the most studied shape in mathematics. For example, circle packings and their complex (the tangency patterns for circle packings are encoded as abstract simplicial 2-complexes $K$) in 2D, have been shown to bridge combinatorics and geometry [21]; the so-called Ford circles and spheres [22] reveal the approximation of real numbers by rationals; and sphere packing [23] is widely used in crystal chemistry. Our approach presented in this paper is inspired by the extensive research that has been carried out in the mathematics literature on circle packings and their complex $K$. Since our main focus in this paper is on 3D domains, we rely on a unique spherical decomposition of a domain $\Omega$ and on the associated complex $K$ to develop a framework for shape analysis that can (1) interface with different geometric representations; (2) support geometric reasoning and standard geometric algorithms (similarity, segmentation, proximity/collision queries) and (3) support downstream applications including analysis and simulations.

Thus, we seek a geometric representation based on ball packings that is bijective [24], supports the extraction of compact shape descriptors that are invariant to rigid body transformations, and provides support for key downstream applications. In what follows, d-dimensional closed balls, where $d=2,3$, are discs in 2D and spheres in 3D that satisfy $\mathbf{y}\in E^d:d(\mathbf{x},\mathbf{y})\le r$, where $\mathbf{x}$ is the center and $r\in \mathbb{R}$ is the radius.

The standard mathematical model for solids (informally, objects with boundaries both in 2D and 3D) is an r-set [24], and there are a variety of syntactically correct computer representations used to describe such an r-set. Representations schemes can be complete (i.e., unambiguous, that is, a representation corresponds to a single object) and unique (i.e., an object admits a single representation in the representation scheme), although some representations are neither. As discussed in [24], representations that are not complete or unique may still be adequate for specific applications. However, completeness becomes important when the modeling system using the particular representation scheme supports a wide variety of applications. The uniqueness of the shape representation implies the uniqueness of the shape descriptor within a given scheme and is clearly desired in shape similarity computations [25].

Shape analysis is one of the key tasks in 3D modeling, and the associated techniques rely on shape descriptors that should be unique and invariant with respect to rigid body transformations along with possessing several other important properties [25]. A detailed review of existing shape descriptors is beyond the scope of this paper and can be found in [6], [18]. Here we comment on some of the attributes of the shape descriptors that are relevant to our objective. Importantly, the existing shape descriptors are defined for specific geometric representations, which implies that measuring the similarity of objects from distinct representations requires a conversion. At the same time, computing most shape descriptors from given representations requires significant computational resources [18], which could, in turn, impact their utility for real-time analysis of large data sets.

Various shape descriptors were designed for specific geometric representation schemes. For example, moments [7], [26] and spherical harmonics [27] are two global feature-based shape descriptors defined traditionally on voxelized geometric models. However, since high-order moments are sensitive to noise, only a small number of low order moments are used in practice, although these low-order moments tend to only capture global features of the shapes. Furthermore, the computational cost of moments increases exponentially with the size of the domain - a limitation shared by descriptors based on spherical harmonics.

The Heat Kernel Signature (HKS) [28] can capture local characteristics of a mesh or more recently of a point cloud model [6] and is robust against noise, which makes the HKS particularly appealing for measuring the similarity of point cloud models. These signatures are based on the heat diffusion properties of the model, and are stable against boundary noise. One key observation is that each geometric representation that the HKS operates on must have an appropriate definition of the discrete Laplacian, but there are infinitely many ways to define discrete Laplacians for meshes and point clouds. The fact that each such definition results in different convergence properties [29] implies that HKS-based signatures cannot be directly used to measure similarity between, say mesh and implicit models.

There are also many graph-based shape descriptors that capture local features of objects, such as the B-rep graphs [30] and Reeb graphs [31]. However, the B-rep graphs of similar models are not guaranteed to be similar [18], and the Reeb graph-based methods are not applicable to arbitrary shapes.

Spherical decompositions have been shown to be useful geometric constructions in several different application domains. In the context of analytic methods, overlapping spherical decompositions have been used to construct implicit representations of 3D domains. Specifically, a paradigm for efficient computation of analytic correlations based on a grid-free decomposition of potentially overlapping spheres was proposed in [14]. Within this framework, solids are represented as sublevels sets of summations of smooth radial kernels. By utilizing nonequispaced FFTs, one can unify convolution based applications, like holonomic collision constraints, shape complementary metrics, and morphological operations, within an analytic framework. A recent work investigating the quality of the approximations of a 2D domain by finite unions of balls has been presented in [32].

In addition, spherical decompositions have been used in meshing [33], where methods based on circle packings are used to generate high-quality surface elements for polygonal domains. In physics simulations, spherical decompositions offer direct support to meshfree analysis methods, such as Method of Finite Spheres (MFS) [34], [35]. The MFS method, which operates directly on a decomposition of potentially overlapping spheres, uses compactly supported functions defined on a spherical cover of a domain. It has been shown that, even though MFS itself may be less efficient than the Finite Element Method (FEM) in solving the discrete system equations, it can overall be more efficient than FEM because it does not require a mesh of the geometry.

It has also been shown that non-equal spherical decompositions cover a domain more efficiently than uniform spherical decompositions because the former generally need fewer spheres to cover the domain than the latter [36]. Moreover, sphere inner trees (IST) have been proposed as an alternative data structure [37] for standard collision detection and proximity queries. Conceptually, IST are constructed at every level by placing the maximal sphere in the domain left after carving the volume occupied by the spheres placed at all previous steps. The focus in [37] is on keeping the computational cost of collision detection under control, rather than on a careful definition and investigation of the properties of the IST or on using IST for downstream applications. Arguably, the IST presented in [37] is not unique.

In [38], the authors proposed the so-called morphological shape decomposition applied to decompose a 2D object into a union of maximal discs. The morphological shape decomposition proposed in [38] is based on non-uniform circle packing and Minkowski operations, and is shown to be invariant to translation, rotation, and scaling. The morphological shape decomposition is similar with IST in the sense that it recursively seeks the maximal inscribable disk in the domain, and uses Minkowski operations to compute the decomposition as well as the corresponding approximation of the 2D object.

In this paper, we define a new geometric representation, named MDBD, based on packing maximal disjoint closed balls inside a d-dimensional solid domain. We show that this representation is bijective [24], and that computing MDBD only requires a given representation to have the ability to compute distances. We further explore the mathematical properties of this representation, and illustrate its potential to shape analysis by establishing universal shape descriptors that operate directly on the proposed decomposition, and by using them to measure the similarity of models that natively exist in various representations. We also discuss the powerful support that MDBD offers for downstream applications, including various physical simulations. We note that MDBD is used in this paper not as a full-fledged geometric representation, but as a computational framework to unify the geometric information coming from multiple geometric representations without requiring an explicit representation conversion.

Observe that MDBD is a fundamentally different concept from that of the medial axis of the domain. The latter, of course, is the set of points of a domain that have at least two closest points on the boundary of the domain [39] or, in other words, the non-differentiable points of the distance function. On the other hand, computing MDBD does not require the computation of the medial axis. Instead, it is defined as the locations of the largest closed balls recursively placed in the interior of the domain. This difference is explored in more detail in sections 3 Properties, 4 Algorithms.

Section 2 presents the formal definition of MDBD followed by a careful discussion of its key properties in Section 3. Our implementation discussed in 4 is used to generate an ample number of examples illustrating the decomposition and its application to shape similarity in Section 5. We conclude in Section 6 with a summary of the key contributions and some directions for future research.