Hybrid Function Representation for Heterogeneous Objects
HH YBRID F UNCTION R EPRESENTATION FOR H ETEROGENEOUS O BJECTS
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A. Tereshin
National Centre for Computer AnimationBournemouth UniversityPoole, United Kingdom [email protected]
A. Pasko
Skolkovo Institute of Science and TechnologySkolkovo, Russia [email protected]
O. Fryazinov
National Centre for Computer AnimationBournemouth UniversityPoole, United Kingdom [email protected]
V. Adzhiev
National Centre for Computer AnimationBournemouth UniversityPoole, United Kingdom [email protected]
January 1, 2021 A BSTRACT
Heterogeneous object modelling is an emerging area where geometric shapes are considered inconcert with their internal physically-based attributes. This paper describes a novel theoreticaland practical framework for modelling volumetric heterogeneous objects on the basis of a novelunifying functionally-based hybrid representation called HFRep. This new representation allows forobtaining a continuous smooth distance field in Euclidean space and preserves the advantages of theconventional representations based on scalar fields of different kinds without their drawbacks. Wesystematically describe the mathematical and algorithmic basics of HFRep. The steps of the basicalgorithm are presented in detail for both geometry and attributes. To solve some problematic issues,we have suggested several practical solutions, including a new algorithm for solving the eikonalequation on hierarchical grids. Finally, we show the practicality of the approach by modelling severalrepresentative heterogeneous objects, including those of a time-variant nature. K eywords hybrid representation · distance fields · eikonal solver · function representation · heterogeneous objects · volumetric modelling. Heterogeneous volumetric object modelling is a rapidly developing field and has a variety of different applications.Volume modelling is concerned with computer representation of object surface geometry as well as its interior.Homogeneous volume modelling, better known as solid modelling, deals with volume interior uniformly filled by asingle material. Heterogeneous object is a volumetric object with interior structure where different physically-basedattributes are defined, e.g. spatial different material compositions, micro-structures, colour, density, etc. [1, 2]. This typeof objects is widely used in applications where the presence of the interior structures is an important part of the model.Additive manufacturing, physical simulation and visual effects are examples of such applications.The most widely used representations for defining heterogeneous objects are boundary representation, distance-basedrepresentations, function representation and voxels. Boundary representation (BRep) [3] maintains its prevailing roledue to its numerous well-known advantages. It works well in solid modelling for objects consisting of a set of polygonalsurface patches stitched together to envelope the uniform and homogeneous structure of its material. However, BRep isnot inherently natural for dealing with heterogeneous objects, especially in the context of additive manufacturing and a r X i v : . [ c s . G R ] D ec PREPRINT - J
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1, 20213D printing [4], where volume-based multi-material properties are paramount as well as in physical simulation wherethe exact representation rather than an approximate one can be important [5].On the contrary, volumetric representations in the form of voxels [6] are more natural for defining such heterogeneousobjects as they are based on volumetric grids. Voxels represent an object as a set of cubic cells at which the geometryalong with the object attributes are defined. However, this representation essentially approximates both the geometrymodel and the material distribution in interior of the object as their definition is limited by the resolution of the voxelgrid.On the other hand, function-based, and more specifically, distance-based representations are able to represent theobject and its interior structure in both continuous and discrete forms [7]. They are exact, embrace a wide range ofgeometric shapes and naturally define many physically-based attributes. There are a lot of well-established operationsfor these representations. Most of them provide distances to the object surface. However, distance functions (DFs)are not essentially continuous, they can have medial gradient discontinuities and are not necessarily smooth. Thispotentially results in non-watertight surfaces, and in artefacts, such as creases, after applying some operations, forinstance, blending and metamorphosis, which are important for many applications. Undesired artefacts (stresses,creases, etc.) can also appear as the result of defining distance-based attribute functions.We consider function-based and distance-based representations as a promising conceptual and practical scheme to dealwith heterogeneous objects, especially in the context of a number of topical application areas concerned with exactvolume-based geometric modelling, animation, simulation and fabrication. However, the existing representationalschemes of that type appear in many variations and the field as a whole exhibits a rather fragmented and not properlyformalised suite of methods. There is an obvious need for a properly substantiated and unifying theoretical and practicalframework. This challenge can be considered in the context of the emergence of new representational paradigmssuitable for the maturing applications, such as modelling of material structures, that was outlined and substantiatedin [8].In this work we propose a novel function-based representational scheme. We introduce a mathematical frameworkcalled hybrid function representation for defining a heterogeneous volumetric object with its attributes in continuousand discrete forms. It is based on hybridisation of several DF-based representations that unifies their advantages andcompensates for their drawbacks. This representational scheme aims at dealing with heterogeneous objects with somespecific time-variant properties important in physical simulations related to both geometry and attributes. The ideawas initially tested in a short paper [9] where the scheme unifying the function representation (FRep) and the signeddistance functions (SDFs) had been sketchily outlined.The contributions of our work can be formulated as follows:• We provide a thorough survey of the relevant representations aiming at their classification and identifying theiradvantages and drawbacks. We formalise the notions of the adaptively sampled distance fields (ADFs) andinterior distance fields (IDFs).• On the basis of an analysis of the well-established FRep and DF-based representations, namely SDFs, ADFsand IDFs, we formulate the requirements for a novel unifying hybrid representation called HFRep.• We propose a mathematically substantiated theoretical description of the HFRep with an emphasis on definingfunctions for HFRep objects’ geometry and attributes.• We describe a basic algorithm for generating HFRep objects in terms of their geometry and attributes, anddevelop its main steps in a detailed step-by-step manner.• We identify the problematic issues associated with several steps of the basic algorithm and propose severalpractical solutions. In particular, we present a novel hierarchical fast iterative method for solving the eikonalequation on hierarchical grids in 2D. The developed algorithm was used for generating HFRep based on FRepand ADF.
There is a huge body of works dealing with different aspects of representational schemes for volumetric heterogeneousobjects. In this section we first concentrate on those works that deal with representations for geometric shapes. Thenwe consider some existing hybrid representations. The basic methods for defining attributes in interior of volumetricobjects are also reviewed. 2
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An overall object geometry can be represented by boundary surfaces or by any other solid representational schemeincluding procedurally defined scalar fields.Boundary representation (BRep) remains the most popular representation. It can be described by a polygonal or othersurface model. Polygonal models can be represented as nested polygonal meshes bounding the regions with differentmaterial density values [3]. This representation scheme has the following problems: the possible presence of holesor gaps in a mesh, normals can be flipped, triangles in the mesh can be intersecting or overlapping with each other,polygonal shells can be noisy. Among many polygon-based approaches applicable to heterogeneous objects we payattention to the diffusion surfaces introduced in [10]. This approach deals with 3D surfaces with colours defined onboth sides, such that the interior colours in the volume are obtained by diffusing colors from nearby surfaces. It wasused for modelling objects with rotational symmetry. It is efficient to compute, but cross-sections of the mesh obtainedwith further triangulation could suffer from discretisation artefacts.Another way to represent the overall object geometry is constructive solid geometry (CSG) [11]. Originally all solidswere homogeneous, but later primitives could carry on some information that can be interpreted as a material index [12].The operations on attributes corresponding to set-theoretic operations were provided.The most widely used method for defining heterogeneous objects is the voxel representation [6, 13]. The object issubdivided into multiple cubic cells with defined geometric and attribute parts in them. However, geometric and attributeproperties are essentially approximated according to the voxel grid resolution.In the context of this work we pay a special attention to defining a heterogeneous object geometry using different typesof scalar fields. The most common schemes of that type are already mentioned FRep [14], SDFs [7], ADFs [15] as wellas the shape aware distance fields which are represented by functions that we call interior distance functions (IDFs). Wewill discuss them in more details in the next section.Another widely used approach for obtaining a continuous distance based definition of the object is to compute thesolution of the optimal mass transportation [16]. This method assumes the numerical solution of a partial differentialequation (PDE) dedicated to the Monge-Kantorovich optimisation problem which can be quite time-consuming. In [17],volumetric objects with multiple internal regions were suggested to define the object-space multiphase implicit functions.These functions preserve sharp features of the object and in some cases provide better results than SDFs.Distance-based objects can be also defined using the level-set method [18] which provides an implicit representation ofa moving front. The main advantage of this method is that it could handle various topological changes of the object thusimplementing the dynamic implicit surfaces. The evolution of the front is controlled by the solution of the level-setequation. The obtained function is transformed into a signed distance function using the solution of some reinitialisationequation. Level-set methods have been used in many applications, such as shape optimisation, computational fluiddynamics, trajectory planning, image processing and others [18].
The main feature of any hybrid representation is that it unifies advantages of several representations and compensatefor their drawbacks. In [19], the concept of hypervolumes was introduced as an extension of the general objectmodel [20] that unifies the advantages of FRep and hybrid volumes. Hypervolume describes a heterogeneous object asn-dimensional point-set with defined attributes, operations and relations over them. Another hybrid approach calledhybrid surface representation was introduced in [21]. It is based on BRep and an implicit surface representation (V-Rep)and was used for heterogeneous volumetric modelling and sculpting.There are approaches when an entire object can be split into disjoint or adjacent components sharing their boundaries.The space partitions can be defined by additional boundary surfaces or scalar fields. In the most general case, thesepartitions are represented by mixed-dimensional cells combined into a cell complex. The combination of a cellularrepresentation and a functionally based constructive representation was proposed in [22]. This model makes it possibleto represent dimensionally non-homogeneous elements and their cellular representations. The authors showed thatattributes may reflect not only material, but any volumetric distribution such as density or temperature.There are some works dedicated to the construction of hybrid representations based on SDFs, ADFs and IDFs. In [23],the authors have introduced hybridisation of meshfree, RBF-based, DF-based and collocating techniques for solvingengineering analysis problems. The proposed technique enables exact treatment of all boundary conditions and canbe used with both structured and unstructured grids. In patent [24], Sullivan has introduced the hybrid ADFs whichrepresented the object by a set of cells. In work [25], the authors introduced a new structure called HybridTree thatis based on an extended CSG tree which unifies advantages of skeletal implicit surfaces and polygonal meshes. The3
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1, 2021hybrid biharmonic distances that are defined similarly to diffusion and commute-time (graph) distances were introducedin [26] for solving some shape analysis tasks. In [27], a concept of the hybrid ADF was introduced for the detailedrepresentation of the dynamically changing liquid-solid mixed surfaces.
A notable early framework called constructive volume geometry (CVG) for modelling heterogeneous objects usingscalar fields was decsribed in [28]. The CVG algebraic representation describes both interior and exterior of the objectthat using regular or hierarchical data-structures. The CVG mathematical framework works with spatial objects definedas a tuple O = ( F O , A , ..., A n ) , where F O is an opacity field that F O : R (cid:55)→ [0 , and A i are attribute fields. Theopacity field defined by the function F O is non-distance based and it is not essentially continuous. Discrete fields alsocan be used in this representation using some interpolation procedure.Multi-material heterogeneous volumetric objects [29] consist of three elements: object geometry, object components(e.g. domains, partitions or cells sharing their boundaries) and material distribution. Material distributions can bedefined using material indexes, piecewise polynomials or continuous scalar fields that provide a resolution independentdistribution.To define material in the interior of the object, a spatial partitioning of the object in several spatial regions should bemade. Perhaps, the most widely used approach is a voxel representation of the object [6]. In [30], Hiller and Lipsonsuggested to use a voxel data structure as a material building-block for layered manufacturing. In another work [31],a bitmap voxel-based method that uses multi-material high-resolution additive manufacturing (AM) was introduced.The material properties are combined in local material compositions that are further fetched in a AM system. In [13] amulti-material voxel-printing method using a high-resolution dithering technique was introduced. The material in thevoxelised object is defined using spatial indexing.Material distribution in interior of the volumetric object can also be defined using DF-based approaches. In [32], DFswere used for parameterisation of the space by distances from the material features either exactly or approximately,taking into account that the defining attribute function should be at least C continuous to avoid creases and stressesin it. In [33], an IDF based method for defining gradient materials was introduced. IDFs are represented as anapproximate Euclidean shortest path and are used for interpolation between sources. In [34], the authors consideredthe decomposition of the geometry using the existing class of material distance-based functions that set up a materialvariation in heterogeneous objects using the medial axes transform. In this section we provide some mathematical background and outline in a formalised manner four functionally-basedrepresentations that will be used to devise the hybrid function representation to be introduced in Section 4. We describein necessary detail the mathematical basics of those representations and propose the formal definitions for two of them,namely ADF and IDF. The advantages and drawbacks of the representations are also systematically outlined.
Let us introduce the mathematical definitions which will be used hereinafter. First we introduce the definition of ametric space and a distance function that follows [35]:
Definition 3.1
Let X be a non-empty point set in a Euclidean vector space R n and let function d : X × X (cid:55)→ R besuch that for points ∀ p i ∈ X ⊂ R n the following conditions are satisfied: d ( p , p ) ≥ ; d ( p , p ) = 0 ⇔ p = p ; d ( p , p ) = d ( p , p ) ; d ( p , p ) ≤ d ( p , p ) + d ( p , p ) . Then thefunction d ( · , · ) is called a metric or a distance function on set X and the pair ( X, d ) is called a metric space. In this work we are focusing on distance-based representations for defining volumetric objects. Let us introduce a moreinstrumental notion for the distance function that satisfies definition 3.1 and that we will use subsequently in the nextsections, as follows [36]:
Definition 3.2
Let X be a point set in a Euclidean vector space R n and let (cid:104) · , · (cid:105) be an inner product defined in R n .Then the Euclidean norm of the point p ∈ X is defined as || p || = (cid:112) (cid:104) p , p (cid:105) . If q ∈ X is another point, the distancebetween these two points is defined as a function: F DF ( p , q ) = || p − q || = (cid:112) (cid:104) p , q (cid:105) (1)4 PREPRINT - J
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1, 2021Figure 1: a) The FRep field of the functionally defined ’bat’ object and b) The SDF field computed for the functionallydefined ’bat’ object. The colours in the pictures correspond to the point membership rule: blue colour corresponds tothe negative values of the defining function, black colour corresponds to the boundary of the object and yellow colourcorresponds to the positive values of the defining function.In this work we deal with functionally defined objects that are specified as closed point subsets G ⊆ X . As we aredealing with the objects defined by the functions, a point membership classification is used to distinguish betweenexterior, boundary and interior of the object. Therefore, let us introduce a formal definition of the boundary ∂G of thesubset G as follows: Definition 3.3
Let G be a subset of the defined metric space ( X, d ) . The boundary ∂G of this subset G is defined as G \ G in , where G = (cid:84) { G C : G C ⊇ G } is a closure of a metric space ( X, d ) , G C is a closed set in X , and interior of G is G in = (cid:83) { G U : G U ⊆ G } , where G U is an open set in G . There are two important properties of the functions that we rely on in the next sections: continuity and smoothness. Thecontinuity of the function is defined as follows [37]:
Definition 3.4
Let X be an open subset of R n . Let C ( X ) be the space of continuous functions X (cid:55)→ R n . Let N n bethe set of all tuples α = ( α , ..., α n ) ∈ N n . Then | α | is the order of α and ∂ α is the partial derivative. For an integer k ≥ C k ( X ) := { f ∈ C k − ( X ) : ∂ α f ∈ C ( X ) , ∀ α, | α | = k } (2) where | α | = N (cid:88) i =1 α i , ∂ α = ∂ | α | ∂x α ...∂x α N N (3)In this work we discuss functions that are either at least C or C continuous. A function f is said to be of class C ifit is continuous on X ⊂ R n . A function f is said to be of class C if it is differentiable and continuous on X ⊂ R n .Formally, smoothness of the function follows from the previous definition and can be defined as in [36]: Definition 3.5
A function f : X (cid:55)→ R n is called smooth if it is n-times differentiable, i.e. if it belongs to a specificclass of functions that can be defined as C n ( X ) for which f ( n ) exists and it is continuous, particularly if it satisfies C ∞ ( X ) = (cid:84) ∞ n =1 C n ( X ) . Let us introduce the definition of FRep [14]: 5
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1, 2021Figure 2: A constructive tree for the FRep object in the form of a ’snow flake’ that was converted to SDF. This treeconsists of objects defined by SDF functions f i stored in the tree leafs and operations applied to them stored in the treenodes. Definition 3.6
Let the geometric shape of the object O F Rep be defined as a closed point subset G of n-dimensionalpoint set X in Euclidean space R n with p = ( x , ..., x n ) ∈ R n using a real-valued defining function F F Rep ( p ) . Thenfunction representation is defined as O F Rep := F F Rep ( p ) ≥ (4)The FRep function (see Fig. 1, (a)) provides the information about point membership: F F Rep ( p ) < p ∈ X \ GF F Rep ( p ) = 0 p ∈ ∂GF F Rep ( p ) > p ∈ G in (5)The major requirement for F F Rep ( p ) is to be at least C continuous.FRep is a high-level and uniform representation of multidimensional geometric objects. The subject of particularinterest is 4D objects with fourth coordinate specified as time. FRep generalises implicit surface modelling and extendsa CSG approach. FRep has a closure property as operations applied to the FRep defining functions produce continuousresulting FRep functions. The FRep object can be defined as a primitive (e.g. sphere, octahedron, cylinder, etc.) or as acomplex object that is defined in the form of a constructive tree. In this case, primitives are stored in the leaves of thetree and operations are stored in its nodes.There exist many well-developed operations, e.g. set-theoretic operations, metamorphosis, blending and boundedblending, offsetting, bijective mapping and others [14]. FRep covers traditional solids [38], scalar fields, heterogeneousobjects including both static and time dependent volumes [39]. Fig. 1 (a) shows the FRep field obtained using 14set-theoretic operations applied to triangles and rectangles to construct the ’bat’. In Fig. 2 we present a constructive treethat describes how a FRep object in the form of a ’snow flake’, that was converted to SDF, was created using union ∪ and intersection ∩ set-theoretic operations. In general case, the FRep field is not distance-based as field isolines do notprecisely follow the object shape. The advantages and drawbacks of the representation can be found in table 1, in thefirst column. 6 PREPRINT - J
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Let us introduce the definition of SDF that relies on definitions 3.1, 3.2 and 3.3 : Definition 3.7
Let ( X, d ) be a metric space. Let the geometric shape G of the object O SDF be specified in ( X, d ) as apoint subset G ⊆ X . Then a signed distance function F SDF ( p ) is defined as: F SDF ( p ) = (cid:26) d ( p , ∂G ) if p ∈ G − d ( p , ∂G ) otherwise (6) where d ( p , ∂G ) ≡ F DF ( p , ∂G ) . Then the SDF representation is defined as follows: O SDF := F SDF ( p ) ≥ (7)The SDF function is at least C continuous as it can be not differentiable at some points of Euclidean space R n and ithas gradient discontinuities on the object’s medial axes. The SDF representation provides the information about pointmembership in the same manner as FRep.The most common operations that are defined for SDF are: offsetting [40], surface interpolation, multiple-objectaveraging, spatially-weighted interpolation, texturing, blending, set-theoretic operations, metamorphosis [41] and others.SDF can be used for a material definition in heterogeneous objects [32], additive manufacturing [42], collision detectionproblems, particle simulations [27] and others.Fig. 1, (b) shows the SDF field generated for the ’bat’ object. As it can be seen, the isolines are spaced equidistantlyand follow the shape of the object. The advantages and drawbacks of SDF can be seen in table 1, second column. Adaptively sampled distance function (ADF) [15] is a distance function that is computed on hierarchical grids, e.g.tree-like data structures. ADF satisfies all the requirements of definitions 3.1, 3.2 and . To our knowledge, there isno well-established formal definition of ADF in the literature. There are several works where ADF is interpreted ina different way compared to [15]. For example, in [6] ADF was defined using T-meshes with different interpolationoperation for restoring the field, in [43] it was suggested to use a hierarchical hp-adaptation for constructing ADF,in [44] it was suggested to construct ADF using estimation of the principal curvatures of the input surface. In this workwe introduce a formal definition of ADF. Let us first give the definition of the hierarchical tree structure:
Definition 3.8
Let a set of nodes and edges ( Q, E ) be an undirected connected graph T that contains no loops andstarts at some particular node of T . Then such a graph T is defined as a tree. Let space R n be subdivided according to the local details using some k-ary tree T := ( Q, E ) with nodes q ∈ Q . Eachnode q is defined as an n-dimensional cell. According to the SDF definition 3.7 we need to compute the distance to theboundary ∂G of the geometric subset G . Taking these preliminaries into account, let us formulate the ADF definition inthe constructive manner: Definition 3.9
Let the geometric shape G ⊆ X of the object O ADF be defined in a metric space ( X, d ) . Let ( X, d ) besubdivided into nodes q ∈ Q with corner vertices p i according to the level of detail using k-ary tree T := ( Q, E ) . Letthe boundary ∂G be subdivided with the maximum tree depth, while X \ G and G in be subdivided with some minimumtree depth. Let the corner vertices of the boundary nodes q be defined as p b i . Then the distance function between thesepoints is d ( p i , p b i ) ≡ F DF ( p i , p b i ) . Thereafter, the ADF distance function F ADF ( p ) on the tree T is restored at eachnode q using some interpolation function F I ( p ) and is defined as follows: F ADF ( p ) = (cid:26) ( F I ◦ F DF )( p ) if p ∈ G − ( F I ◦ F DF )( p ) otherwise (8) The ADF representation is defined in the form of an inequation: O ADF := F ADF ( p ) ≥ (9)The ADF field generated as it was described in [15] has C discontinuities where the cells of different size appear and ithas C discontinuities caused by the bilinear/trilinear interpolation that was used for restoring a DF at each cell. TheADF representation provides the information about point membership in the same manner as FRep. The subset X canbe subdivided using one of the types of k-ary trees: quadtrees or octrees. ADF can be used for an efficient interactive7 PREPRINT - J
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1, 2021Figure 3: The IDF field computed for the ’Stanford Bunny’ 3D mesh using the method described in [46] and an SDFslice to show the difference in nature of these fields. a) the IDF field computed on the boundary of the mesh. Blackisolines show how the field is changing according to the shape of the object; b) the interior slice of the mesh withcomputed IDFs. The yellow point p s in the slice corresponds to the ’source’ point. c) the SDF slice of the same modelwith computed interior and exterior distances. Colour changing reflects how the distances are changing from interior toexterior of the object.real-time modelling, e.g. sculpting, of the heterogeneous objects as the tree data structure provides fast access to object’sgeometry and its specified attributes. ADF is also suitable for solving surface restoration problems [44, 45]. It supportsthe same operations as SDF. ADF are especially suitable for dynamic simulations [43], for example, morphing betweenshapes, as a hierarchical data structure can efficiently be rebuilt at each animation frame [15]. The advantages anddrawbacks of ADF can be seen in table 1, third column. Interior distance function (IDF) is not a well-established notion yet as in literature there is neither a general approachfor generating DFs of this rather broad nature nor one unique name for them. In this work we suggest to use this notionfor a representation with a defining function obtained as follows: the distance function is computed on the boundary ofthe object and then the generated distances are smoothly interpolated in its interior. Let us introduce the definition ofIDF that relies on the definitions specified in subsection 3.1:
Definition 3.10
Let the geometric shape G ⊆ X of the object O IDF be defined in a metric space ( X, d ) . Let points p b i belong to ∂G , and let points p in k belong to G in . Let a distance function d ( p b i , p b j ) ≡ F DF ( p b i , p b j ) = || p b i − p b j || R n between any boundary points p b i and p b j on a curved domain ∂G be recovered. Thereafter, by constructing aninterpolation function F I ( F DF ( p b i , p b j ) , p in k ) that is at least C continuous, boundary distances are extended tointerior of the object O IDF . Therefore, the IDF function can be defined as: F IDF ( p in k ) = F I ( F DF ( p b i , p b j ) , p in k ) (10) where ≤ i, j, < N , N is the number of boundary points, ≤ k < M , M is the number of interior points. The IDFrepresentation is defined in the form of an inequation: O IDF := F IDF ( p ) ≥ (11)IDF is usually obtained by solving a partial differential equation (PDE) or applying some numerical method, e.g., graphapproaches [47] or Markov chains [48]. Among PDE-based methods the following methods can be considered asrepresentative: geodesic distances obtained as the solution of heat equation [49], diffusion maps combined with smoothbarycentric interpolation of the distances in interior of the object [46], the optimal mass transport [16] and some others.IDF is usually used in the tasks related to shape analysis [46], geometry restoration [50], morphing and less commonlyfor an attribute definition in interior of the object [33]. The advantages and drawbacks of IDF can be found in table 1,last column.In Fig. 3 we show how the approach described in [46] can be applied to the polygonal mesh of the ’Stanford Bunny’.The distances are computed on the boundary, as it can be seen in Fig. 3 (a), using the diffusion maps, and thenpropagated in interior of the object, as it can be seen in Fig. 3 (b), using the barycentric interpolation. For convenienceof data visualisation we compute distances from the fixed ’source’ point p s to other points of the mesh. If we comparetwo pictures shown in Fig. 3 (b) and (c), we can see that the distance fields obtained in interior of the bunny are8 PREPRINT - J
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FRep SDF ADF IDF a dv a n t a g e s • FRep generalises implicit sur-face modelling and extendsa constructing modelling ap-proach;• FRep supports point member-ship;• FRep is closed guaranteeing toget an at least C continuous re-sulting function;• FRep covers solids, scalar fields,volumes, time-dependent vol-umes and hypervolumes for het-erogeneous object modelling.• FRep has many well-developedoperations that support multi-dimensional transformations in R n ; • SDF provides distances to the ob-ject surface both inside and out-side it;• SDF defines a watertight object;• SDF is a Lipschitz continuousfunction;• SDF is Fre´chet differentiable al-most everywhere;• SDF satisfies the solution of theeikonal equation;• SDF supports point membership;• SDF is effectively discretised,has a predictable field behaviourand is efficiently rendered. • ADF data structure efficientlysubdivides the Euclidean space R n according to the level of de-tail;• ADF distances are adaptivelysampled;• ADF supports point member-ship;• ADF possesses an efficient mem-ory management: in a smallamount of memory a significantamount of information about theobject can be stored;• ADF hierarchical tree data struc-ture is fast to rebuild that makesit possible to handle time-variantobjects;• ADF can be efficiently renderedin real time. • IDF is shape-aware;• IDF is deformed with the bound-ary;• IDF is smooth;• IDF is suitable for the distance-based attribute definition in inte-rior of the object. d r a w b ac k s • Distances can be obtained for alimited number of FRep objects;• FRep object can have a bound-ary with dangling portions thatare not adjacent to the interior ofthe object;• FRep has an unpredictable non-distance based behaviour of theresulting field and, as a conse-quence, it is sometimes problem-atic to render in 3D. • SDF is not differentiable at somepoints of Euclidean R n space.Loss of SDF differentiabilityhappens when the current pointis sufficiently close to a con-cave singularity (a concave cor-ner/edge);• SDF has discontinuous gradientson the object’s medial axes;• SDF is not smooth;• SDF is not suitable for attributemodelling due to C discontinu-ity. • ADF field has C discontinuitieswhere cells of the different sizeappear as the result of the hierar-chical subdivision;• ADF field has C discontinu-ities that are introduced by the bi-linear/trilinear interpolation [15]during reconstruction of the fieldat each cell;• ADF is not suitable for attributemodelling due to C and C dis-continuities. • IDF can be computationally ex-pensive;• IDF field accuracy for somemethods is highly dependent ona time step and type of the useddiscretisation;• IDF is defined only in interior ofthe object. Table 1: Comparison table of the advantages and drawbacks of FRep, SDFs, ADFs and IDFs.completely different. The IDF field (b) is smooth and continuous while the SDF field (c) is not smooth and has somesharp features in interior of the object.
In the previous subsections we have discussed how geometric shape of objects can be defined using distance-basedmethods. In this subsection we discuss how attributes can be considered in concert with the geometric shape of theobject to represent the heterogeneous object. Let us first introduce a general definition of the heterogeneous object.
Definition 3.11
Let the object O H be defined as a two component tuple: geometric shape G ⊆ X in the form of amultidimensional point-set geometry and attributes A i corresponding to the physical properties of the object O H . Thensuch object O H is a heterogeneous object defined as: O H := ( G, A , ..., A n ) , (12) where n ∈ N is the number of attributes. Attribute distributions specified in heterogeneous objects O H can be uniform or non-uniform. For instance, the simpleexample of the uniform distribution can be a homogeneously coloured object. As to non-uniformity, it can be presentedas porous structures or microstructures with non-linear varying density.In this work we will apply the hypervolume model [19] to define heterogeneous objects O H V using FRep or any otherdistance function-based representation. A hypervolume object is defined as follows: Definition 3.12
Let the geometric shape G of O H V be defined by a real-valued function F G ( p ) , p ∈ R n that is at least C continuous and let attributes be defined by any F A i ( p ) . Then heterogeneous object O H V is defined as: O H V := ( G, A , ..., A n ) : ( F G ( p ) , F A ( p ) , ..., F A n ( p )) , (13) where n ∈ N is the number of attributes. PREPRINT - J
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1, 2021Figure 4: The STB-based metamorphosis operation over the initially FRep ’heart’ converted to HFRep and initiallyBRep ’cube’ converted to SDF ’cube’.In general case, attribute functions F A i ( p ) are not necessarily continuous. However, as it was shown in [32], bettercontrol of the attributes on the surface and in the interior of the distance-based objects can be achieved when the attributedefining functions are parameterised by the distances. The main requirement for the distance function is to be at least C continuous. This requirement prevents the appearing of stress concentrations, creases and other singularities inmodelled attribute distributions.There are several interesting examples discussed in [32]. In particular, the distance-based smooth and differentiableattribute functions were applied to represent a parabolic distribution of the graded refractive index in Y-shaped solidof the waveguide. In this case, it is important that the distribution of the index of refraction is uniform and smooth.Another example is to use such distance-based attribute functions for modelling different types of materials, e.g. siliconcarbide (SiC). It is important to note that the approach introduced in [32] was not applied to such attributes as textures,colours and similar attributes. In this section we introduce and systematically describe a general approach for defining heterogeneous volumetricobjects using a hybrid function representation (HFRep). First, we list the requirement to HFRep, then outline itsmathematical basics, and finally describe its properties with respect to four basic DF-based representations.
Let us give the exact problem statement. Our goal is to propose a hybrid function representation (HFRep) that is suitablefor defining volumetric heterogeneous objects. We assume that the geometric shape G of the given object is defined byFRep, and its defining function is known. To devise the HFRep embracing advantages and circumventing disadvantagesof FRep, SDF, ADF, IDF, it is essential to obtain a real-valued defining function in an n-dimensional Euclidean spacewith the following properties:1. the HFRep function should provide sufficiently accurate distance approximation in Euclidean space R n without C and C discontinuities.2. the HFRep function should be at least C continuous with possibility to enforce it to be at least C continuous.3. the HFRep function should satisfy the point membership test: it should be positive in interior of the geometricshape G , take exact zero values only at the object boundary ∂G and it should be negative in exterior of thegeometric shape X \ G ;4. the HFRep should be a multidimensional object representation; in particular, dealing with 4D objects is ofparamount importance to cover time-variant models with the fourth ’time’ coordinate;5. the HFRep representation should be suitable for the heterogeneous object modelling allowing for definingattribute functions related to the geometry;6. the HFRep attribute functions should depend on evaluation point p ∈ G and be parameterised by distancevalues of the obtained HFRep geometry function.The fulfilment of these conditions guarantees that the generated object will be watertight and such operations as blendingand metamorphosis will not suffer from creases. Overall, the defining HFRep function that is considered in concert with10 PREPRINT - J
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1, 2021Figure 5: The illustration of HFRep based on FRep and ADF with applied PHT-spline (a polynomial spline overhierarchical T-mesh) interpolation to restore the distance field at each cell. ADFs are generated using a numericalsolution of the eikonal equation on the quadtree. a) the FRep field; b) a hierarchical quadtree subdivision; c) UDFcomputed on the quadtree with the applied PHT-spline interpolation for restoring distances at each quadtree cell; d) theHFRep field that was obtained using the generated ADF.attribute functions parameterised by distances will be suitable for dealing with multi-material aspects of heterogeneousobjects including time-variant ones.
First, we provide a mathematical definition for the geometric aspects of HFRep. Then we add the part related toattributes. The geometric shape G of an HFRep object O HF Rep is defined as follows:
Definition 4.1
Let the geometric shape G ⊆ X of the object O HF Rep be defined in a metric space ( X, d ) . Givenat least C or C continuous FRep function F F Rep ( p ) , the distance to the object boundary ∂G is defined as ( F I ◦ F DF )( p , ∂G ) ≡ ( F I ◦ F DF )( p ) , where F I ( · ) is at least C continuous interpolation function and d ( · , · ) ≡ F DF ( · , · ) is a distance-based function, in particular SDF, ADF or IDF. Then the HFRep function is defined as follows: F HF Rep ( p ) = ( F sign ◦ F F Rep )( p ) · ( F I ◦ F DF )( p ) (14) where F sign ( · ) is an at least C continuous function that provides a sign for the computed function ( F I ◦ d )( p ) andsatisfies the FRep point membership test, equation (5). Finally, the HFRep representation is defined as: O HF Rep := F HF Rep ( p ) ≥ (15)The continuity of the HFRep function F HF Rep ( p ) depends on the continuity of the FRep function F F Rep ( p ) . In thecase when we are dealing only with geometric shapes, it is sufficient to have C continuity for the HFRep function.Otherwise, in case of heterogeneous object modelling, the HFRep function should belong to the class of functions thatare at least C continuous. We will give details on how to control the continuity of the HFRep function later in thissection.Now let us show that F HF Rep ( p ) continuity is either C or C . By applying a smoothing interpolation function F I ( · ) that is at least C continuous to the discrete unsigned distance field (UDF) obtained using F DF ( p , ∂G ) ∈ C ,11 PREPRINT - J
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1, 2021
Inherited from FRep Inherited from SDF Inherited from ADF Inherited from IDF• The continuity of the HFRep functiondepends on the continuity of the FRepfunction.• The HFRep object is watertight.• HFRep represents multidimensionalobjects, in particular 4D objects withthe fourth coordinate specified astime. • HFRep provides at least C continu-ous distance function.• the HFRep object is watertight.• the HFRep function is Lipshitz contin-uous and Fre´chet differentiable every-where;• the HFRep function satisfies the solu-tion of the eikonal equation;• the HFRep object can be efficientlydiscretised and rendered. • HFRep provides at least C continu-ous distance function for any FRep ob-ject that was spatially subdivided ac-cording to the local details using a hi-erarchical data structure.• Hierarchical data structure can also beused for defining and storing object’sattributes. • HFRep provides at least a C continu-ous unsigned distance function for anyFRep object in its interior if IDF isused for obtaining distances;• Distances in the interior of the HFRepobject are shape-aware, deformedwith boundaries and are not affectedby the boundary noise.• There is also a potential for modellingattributes in interior of the volumetricobject. Table 2: Properties of the hybrid function representation that depend on the combination of FRep with one of thedistance fields.we enforce the property ( F I ◦ F DF )( p ) ∈ C . The composition of functions ( F sign ◦ F F Rep )( p ) is at least C or C continuous, depending on the continuity of F F Rep ( p ) . The theorem about the continuity of the compositionof two continuous functions was proofed in [51]. Therefore the continuity of the HFRep function is defined as: C HF Rep = min( C mF sign ◦ F FRep , C kF I ◦ F DF ) , where m = 0 or m = 1 , k = 1 , i.e. the minimum class of continuitybetween two function compositions.Now on the basis of definition 3.12 , we can formulate the definition of the heterogeneous HFRep object O H V,HFRep asfollows:
Definition 4.2
Let the geometric shape G of O H V,HFRep be defined by at least C continuous F G ( p ) = F HF Rep ( p ) distance-based function. Let the attribute A i be defined as a real-valued function F A i ( F HF Rep ( p ) , p ) . Then the HFRepheterogeneous object O H V,HFRep is defined as: O H V,HFRep := (cid:26) F G ( p ) := F HF Rep ≥ F A i ( F HF Rep ( p ) , p ) , i = [0 , .., n ] ∈ N (16) where n is a number of attributes. The properties of the introduced hybrid function representation are outlined in table 2. For a particular combinationof FRep with one of the distance fields, namely SDF, ADF or IDF, only one type of properties can be inherited. Weshow some particular properties, mentioned in the table 2, using several examples that will be discussed further in thissubsection.Fig. 4 shows a metamorphosis between two oscillating 4D geometric shapes (’heart’, initially the FRep object thenconverted to HFrep; ’cube’, initially the BRep object then converted to SDF) using the space-time blending (STB)method [52]. The result is a non-distance functionally defined watertight object that is continuous and smooth.In Fig. 5 (d), we demonstrate the restored distance field computed on the hierarchical grid obtained for the initial FRepobject defined as a ’treble clef’, Fig. 5 (a) . There is neither C nor C discontinuities in the field as it can be seen inFig. 5, (c) or (d). All the isolines are smooth and continuous.In Fig. 6, (b) we show a simple example of interior distances computed for the FRep ’star’ object, Fig. 6 (a), that wasconstructed using seven set-theoretic operations. First the boundary of the FRep object was extracted for computingboundary distances. Then the interior of the obtained convex contour was triangulated. Finally, the boundary distanceswere propagated in interior of the shape as it is described in [46]. The black isolines show that the obtained field is atleast C continuous as they are smoothly changing in the object interior. Let us outline in a step-by-step manner the algorithmic solution on generating the HFRep functions. The basic algorithmcovers all paired combinations of FRep with DF representations, namely SDF, ADF and IDF, and allows to generateboth a geometric shape and attributes. Some steps of the basic algorithm will be slightly different depending on theparticular type of the DF paired with FRep. Let us start from the algorithm for generating a geometric shape of theobject O H V,HFRep . Fig. 7 demonstrates the generated function field for each step of the basic algorithm.12
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1, 2021Figure 6: (a) ’Star’ object and its FRep field; (b) the HFRep ’star’ object generated on the basis of the FRep object. Theboundary of the FRep object (a) was extracted and then used for computing boundary distances. The obtained distanceswere interpolated in interior of the HFRep ’star’ object using barycentric interpolation and mean-value coordinates. Theisolines and colour show how the field changes from the source point (white circle) towards the object boundary.
1. According to the definition 4.1 , we start the construction of an HFRep object O HF Rep from defining the FRepfunction F F Rep ( p ) for its geometric shape G . The FRep function F F Rep ( p ) can be defined analytically, withfunction evaluating algorithm or using a point cloud for which it is possible to obtain a real-valued at least C continuous F F Rep ( p ) . It could also be a a complex FRep object that is obtained in the form of a constructivetree.At this step we can also enforce HFRep function F HF Rep ( p ) to be at least C continuous as its continuitydepends on the continuity of F F Rep ( p ) . We have to examine the obtained F F Rep ( p ) for continuity anddifferentiability. The most practically used FRep set-theoretic operations in the form of the following R-function system are [14]: f ∪ ( f ( p ) , f ( p )) = f + f + (cid:113) f + f (17) f ∩ ( f ( p ) , f ( p )) = f + f − (cid:113) f + f These functions have C discontinuity in points where both arguments are equal to zero. Accordingly, theresulting function will only be C continuous. If we need to obtain an at least C continuous resulting function,we can apply another R-function system that is at least C n − continuous [53]: f ∪ ( f ( p ) , f ( p )) = f f ( f n + f n ) − n , ∀ f > , f > f , ∀ f ≤ , f ≥ f , ∀ f ≥ , f ≤ − n +1 ( f n + f n ) n , ∀ f < f < (18) f ∩ ( f ( p ) , f ( p )) = ( f n + f n ) n , ∀ f > , f > f , ∀ f ≤ , f ≥ f , ∀ f ≥ , f ≤ − n +1 f f ( f n + f n ) − n , ∀ f < f < where f ( p ) and f ( p ) are FRep functions. 13 PREPRINT - J
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1, 2021Figure 7: The illustration of the basic algorithm: a) step 1: the computed field of the ’robot’ FRep object; b) steps 2 - 3:the computed unsigned distance field that can be obtained using, e.g., the distance transform or a numerical solution ofthe eikonal equation. The obtained field is smoothed using some spline interpolation; c) step 4: the generated HFRepfield. Fig. 7, (a) shows the FRep field obtained for the ’robot’ object, that was generated using 39 set-theoreticoperations, equation (17), applied to circles and rectangles.2. The values of the function F F Rep ( p ) are used as an input for computing distance functions F DF ( p , ∂G ) thatshould satisfy one of the definitions 3.7, 3.9 or . At this step we obtain an unsigned distance function thatis defined as: F DF ( p ) = d ( p , ∂G ) , ∀ p ∈ X (19)Fig. 7, (b) shows the unsigned distance field that was obtained on the basis of a typical SDF generationalgorithm [54].If the distances are computed using ADF, first, we need to subdivide the space using a hierarchical data-structure, e.g. quadtree, Fig. 5, (b) and during it’s construction we also need to compute basis functions, basisvertices and extraction operators for the hierarchical splines. Then we need to compute the distances at thecorner vertices of each cell. Finally, we restore distances in interior of each cell using at least C continuousspline-based interpolation to obtain a smooth and continuous distance field, e.g. shown in Fig. 5, (c).Specifically for IDFs, the function F DF ( p ) is defined according to equation (10). Distances are computed onthe boundary of the object O F Rep and then interpolated in its interior. In Fig. 6 (a), we can see the field of theFRep-defined ’star’ object that was used for generating HFRep IDF-based field that is shown in Fig. 6, (b).In Fig. 8, (a) we show a possible extrapolation scheme that can be used to obtain distances in exteriorof the object and make an IDF-based field signed at the last step of this algorithm. To do this, weneed to use the boundary distances (Fig. 8, (a), dark blue circles) and an appropriate at least C continu-ous extrapolation operation, that will be used for obtaining distances outside the object (8, Fig.7(a), red circles).3. The distance field obtained at the previous step is unsigned and discrete as it was computed on the finite pointsubset X ⊂ R n . To enforce the continuity and smoothness of the computed field, we need to apply some atleast C continuous interpolation function ( F I ◦ F DF )( p ) to the generated unsigned field, e.g. spline-based: F smDF ( p ) = ( F I ◦ F DF )( p ) (20)We also need to apply a smoothing operation to an IDF field if at the previous step an extrapolation operationwas applied. Otherwise, IDFs are smooth as smoothness is their inherent property. An important requirementfor the interpolation function F I ( · ) is to avoid introducing extra zeros in the distance field generated usingfunction F DF ( p , ∂G ) .4. Finally, as the distance field obtained after previous steps is unsigned, we need to restore the field sign todistinguish between exterior X \ G , boundary ∂G and interior G in of the object O H V,HFRep . We suggest touse some at least C continuous step-function F st ( F F Rep ( p )) with the scope [ − , , that depends on thevalues of the defining FRep function F F Rep ( p ) and approximates its well-defined behaviour ( − in exteriorof the object, on the boundary of the object and +1 in interior of the object). Therefore, the resulting HFRep14 PREPRINT - J
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1, 2021Figure 8: Two cases when extrapolation is important to enforce the continuity of the field: a) when we have DF (lightpink and light orange colours) for two objects that are distantly placed in space. In this case we need to extrapolate thedistance values into the points of the green grid; b) when we computed the IDF and it is essential to obtain distances inexterior of the object.function F HF Rep ( p ) is defined according to definition 4.1 as follows: F HF Rep ( p ) = ( F st ◦ F F Rep )( p ) · F smDF ( p ) (21)The HFRep field generated by this function can be seen in Fig. 7, (c). After a geometric shape of the HFRepobject O HF Rep was generated, we can apply different operations to it provided that they are realised byfunctions which are at least C continuous. The HFRep object is also compatible with other distance-basedobjects. However, to preserve the distance properties for the object obtained after applying multiple operations,we might need to apply the steps of this algorithm again to this object.There is a limited number of operations that preserve the distance property for the geometric shape G of the objectobtained after their application. These operations are rigid (Euclidean) transformations: rotations, translations,reflections or their combination. Another distance preserving operations [41] are affine translations, offsetting, linearsurface interpolation, surface blurring and compression, set-theoretic operations in the form of min( f ( p ) , f ( p ) or max( f ( p ) , f ( p ) [53].In cases of other operations [55] (e.g., scaling, blending, space-time blending, twisting, tapering and sweeping, set-theoretic operations in the form of R-functions [14]) after their application, we have to apply the basic algorithm to theobtained object to restore the distance property.To make the HFRep representation continuous on the whole domain of the Euclidean space R n , we suggest to applysome at least C continuous extrapolation operation to the generated field of the object. To explain this idea in moredetails let us consider the following example shown in Fig. 8, (b). In this figure we have two blue objects definedon their own pink grids and spaced from each other, so their defining grids are not overlapping. If we want to workwith them, e.g. by applying some operation, we need somehow to define the distances in the points of interest of thegreen grid. One can extrapolate and average the distances between two pink grids and avoid full reinitialisation of thedistances for both objects. To set up the attributes in interior of the HFRep object O H V,HFRep , we assume that we have obtained a C continuousdistance function for a geometric shape. Now we can deal with the attributes that are parameterised by the distancesas it was required by definition 4.2 . Object attributes could be of different nature and there is no single algorithm todefine all of them. In this work we consider such attributes as colours, microstructures, and simple 2D and volumetrictextures based on noise functions parameterised by distances.15 PREPRINT - J
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1, 2021Let us formulate the basic algorithm for specifying an attribute component A i of the O H V,HFRep on the basis of alreadydefined geometry:1. Depending on the nature of the attributes and how they are distributed in interior of the object O H V,HFRep ,there are two possible types of object partitioning: single and multiple partitions. At this step we need tosubdivide an object O H V,HFRep according to the chosen partitioning scheme.2. Then we specify and evaluate an attribute function F A i ( F HF Rep ( p ) , p ) for each partition to set up theattributes at the points p ∈ G . These functions depend on the evaluation point coordinate and are parameterisedby the computed distance using F HF Rep ( p ) values.3. In case when we have a multiple partitioned object with several specified attributes, we can obtain a singleattribute function for all subsets A i by applying some interpolation, e.g. transfinite interpolation [56] orspace-time transfinite interpolation [57].The more detailed discussion how to deal with attributes will be provided in section 7. In this section we provide a detailed description of several particular steps of the basic algorithm outlined in the previoussection. We consider a variety of combinations of FRep and SDF, ADF or IDF representations and propose a number oforiginal solutions for solving problematic issues. The first step of the algorithm has been already discussed. In the nextsections we discuss steps 2 - 4 (see Figs. 7, (b) and (c)).
In this subsection we describe the solutions for generating UDFs. We show how some existing techniques can be usedin this context and also introduce a novel method for the ADF generation.
SDF generation.
To compute an approximate UDF, the most widely used class of methods is the distance transform(DT) [7]. DTs are efficiently generated on regular grids. In this work we suggest using the vector DT in which thevector components are propagated across the uniform grid. It provides a sufficiently accurate distance approximation.We follow the typical vector DT algorithm described in [54] for 2D case and [58] for 3D case.A definitive way to obtain an accurate DF for the object is to numerically solve the eikonal equation or the level-setPDEs [59]. The numerical solution of PDE is quite time-consuming unless it is a multi-threading implementation of themethod. The accuracy of the field is also highly dependent on the method. One of the robust methods for solving theeikonal equation is the fast iterative method (FIM) [60]. It numerically solves a nonlinear Hamilton-Jacobi PDE definedon a Cartesian grid with a scalar speed function: H ( p , ∇ φ ) = |∇ φ ( p ) | − f ( p ) = 0 , ∀ p ∈ X ⊂ R n (22) φ ( p ) = 0 , p ∈ Γ ⊂ R n where X is a domain in R n , Γ is the boundary condition, φ ( p ) is a travel time of the distance from the source to thegrid point p , f ( p ) is a positive speed function and H ( p , ∇ φ ) is the Hamiltonian. The computed numerical solution isan unsigned distance on a uniform grid. ADF generation.
To generate UDF on the hierarchical grid we briefly outline an original adaptation of the FIM methodfor solving the eikonal equation that also utilises PHT-splines [6] capability of the accurate geometry restoration. Ouralgorithm partly relies on the algorithm introduced in [60] to inherit its advantages such as independent computation ofeach node and a simple data-structure (an active list L or a doubly linked list) for handling node updates. A detaileddescription of the hierarchical FIM (HFIM) algorithm will be presented elsewhere.The algorithm consists of two parts: (1) initialisation of the grid and (2) iterative updates of the numerical solutionof the eikonal equation. First, we subdivide the space using quadtree/octree according to the values of the FRep field.We need to subdivide the exterior and interior of the FRep object with a small tree depth and its boundary with themaximum tree depth. While executing the hierarchical subdivision of the space, we also need to compute the basisfunctions for the PHT-splines [6] and reconstruct the PHT-spline surface that will be used for restoring distances ininterior of each cell node. 16 PREPRINT - J
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1, 2021Figure 9: The comparison of the field restoration at each subdivided hierarchical cell using bilinear interpolation (a)and PHT-spline interpolation (b). In red circles we can see the C discontinuity in the field isolines where the cells ofdifferent size appear next to each other.The idea of the hierarchical grid initialisation before applying HFIM is similar to the procedure described for FIMon the regular grid. We need to traverse the tree and set to zero those vertices of the cells that store the FRep valuesapproximately equal to zero. The rest of the vertices are set to a relatively huge value. Thereafter, vertices that are equalto zero and the corresponding nodes are stored in the active list. The iterative computation of the solution of the eikonalequation on the hierarchical grid follows the logic of the FIM algorithm [60], but all steps are executed taking intoaccount a hierarchical nature of the grid. The eikonal equation is iteratively solved using the first order upwind Godunovdiscretisation scheme that is modified for computations on the grid with irregular steps. The computed solution is storedand updated in all nodes that share the same vertices. The iterative computation is finished when the active list is empty.After obtaining the solution of the eikonal equation at each corner vertex of each cell of the hierarchical grid, we canrestore the distance field using the already constructed PHT-spline surface.As we have stated in subsection 3.4, the ADF field has C discontinuities that arise after the hierarchical subdivisionwhere cells of different size appear. In Fig. 9, (a), the discontinuities in the white isolines are located in the red circles. C discontinuities are introduced by the bilinear/trilinear interpolation that is used for the field restoration in interior ofeach cell (see Fig. 9, a). As it can be seen in Fig. 9, (b) the field generated by our method with PHT-spline restorationof the field successfully solves these drawbacks. All the isolines are continuous and smooth. IDF generation.
IDFs are usually computed using the solution of some PDE equations or, alternatively, somegraph-based approach. We suggest using the approach described in [46].The generation of IDFs is based on propagation of the distances computed on the boundary of the mesh in its interior.We will use Fig. 10 with the generated IDF field to explain how this method works. We start from triangulating aninput geometric shape G of the FRep object O F Rep to generate the boundary surface ∂G for further computations. Themethod, described in [46] was applied to tetrahedralised meshes and consists of two parts.First, we embed surface vertices p b i in some m-dimensional R m space using a map p b i (cid:55)→ p ∗ b i ∈ R m . This map wassuggested to compute using diffusion maps introduced in [48]. It can be obtained by computing an eigendecomposition { λ k , φ k } nk =1 of a discrete Laplace-Beltrami operator of the mesh. In Fig. 10(a) we show the diffusion map obtainedfor the ’heart’ object. The diffusion distances are computed as a Euclidean distance using obtained eigenvalues andeigenvectors [61].After the diffusion distances were computed on the surface of the mesh as it can be seen in Fig. 10 (b), they are extendedto the interior of the mesh using barycentric interpolation. If point q ∈ G in , then the barycentric representation ofit is q (cid:55)→ q ∗ = (cid:80) i ω i ( q ) ν i , where ω i ( · ) are barycentric coordinates (e.g. mean-value coordinates in 2D [62] or in17 PREPRINT - J
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1, 2021Figure 10: HFRep based on hybridisation of FRep and IDF generated for FRep ’heart’ object using the methodfrom [46]. a) the diffusion map computed on the surface of the object that is used for restoring distances at the shapeboundary; b) the distances obtained at the boundary of the object shape that are shown as black isolines; c) is thetetrahedral slice of the mesh with isolines corresponding to the interior distances. The yellow point p s corresponds tothe ’source’ point defined in the object interior.3D [63]). Finally, the distance in interior of the mesh can be obtained using computed diffusion distances F DF ( p i , p j ) and barycentric interpolation.In Fig. 10 (c) we show a slice of the ’heart’ object. The IDF was was computed between fixed ’source’ point and therest mesh points. One can see that the interior field is continuous, smoothly changing and following the boundary of theobject.At this step we can apply an extrapolation operation (e.g., using a wavenumber based extrapolation [64]) to the obtainedIDF field to propagate the distances to exterior of the object using the already computed boundary distances. Thisoperation will allow us to make the IDF field signed at the last step of the basic algorithm. The resulting distance function F DF ( p ) for SDF, ADF or IDF is unsigned and satisfies the equation (19). Havingobtained UDF, we need to smooth the generated discrete field. To enforce an at least C continuity and essentialsmoothness for the obtained field, we need to use some at least C continuous interpolation function, e.g. B-splines orbicubic/tricubic splines [65]. At the previous step we had obtained a smooth and continuous unsigned distance function F smDF ( p ) defined byequation (20) that we used to compute UDF. Now, at the fourth step of the basic algorithm, we need to define thesign of UDF. To restore the sign we suggest to use a smooth step-function that depends on the values of the FRepfunction F F Rep ( p ) , defined at the first step of the basic algorithm. The step-function F st ( F F Rep ( p )) should satisfy thefollowing requirements:1. it is approximately equal − when it corresponds to the exterior of the FRep object, F F Rep ( p ) < ;2. it should be approximately equal to on the boundary of the FRep object, F F Rep ( p ) = 0 ;3. it should be approximately equal to inside the FRep object, F F Rep ( p ) > ;4. it should be at least C continuous everywhere in a Euclidean space R n ;5. it should barely modify the values of UDF.We have identified two classes of functions which satisfy these requirements. These are sigmoid functions and splinefunctions, particularly cubic splines with Hermite end conditions, to estimate the slopes [65]. In this work we use thehyperbolic tangent sigmoid function [66] (see Fig. 11, a). By controlling slope parameter s l , it is possible to get nearlystep-function behaviour around zero: F sig ( x ) = r exp ( − x/s l ) − r , ∀ x ∈ R (23)18 PREPRINT - J
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1, 2021Figure 11: The illustration of the HFRep function continuity through throught varying the slope controlling parameter s l . a) Plots of hyperbolic tangent sigmoid functions with according slope values s l . Red line: S sig ( x ) , s l = 10 − .Blue line: S sig ( x ) , s l = 10 − ; b) the HFRep ’star’ object that was computed with s l = 0 . for F sig , equation(23); c) the HFRep ’star’ object that was computed with s l = 0 . for the F sig , equation (23); all sharp features aresmooth, i.e. the HFRep function is C continuous.where r controls the range of the F st ( x ) along y -axes. We need to set parameter r = 2 to make the function (23) bedefined in the interval ( − , along the y-axes.The continuity of the HFRep function can be visualised as it is shown in Fig. 11 (b) and (c). In Fig. 11 (b), we show halfof the ’star’ object that was generated with s l = 10 − to follow the step-function shape as close as possible. In Fig. 11(c), we show half of the ’star’ object that was generated with s l = 0 . to smooth the isolines shape. We can see that the C continuity of the generated distance field is preserved and the obtained geometric shape of the object is watertight. In this section we show how we practically work with HFRep heterogeneous objects in terms of their attributes. Insection 5 we have outlined the basic algorithm for generating HFRep attribute functions. However, there is no universalapproach for dealing with HFRep object attributes because of their widely various nature. In this section we showhow the proposed framework works for some representative attributes, namely, microstructures, colour and materialattributes. We show the microstructures (Fig. 12), a heterogeneous model of the COVID-19 virus cell (Fig. 15) and twomodels of metamorphosis dealing with a dynamic (time-variant) smooth transition from one HFRep object to another(Figs. 13 and 16).
In Fig. 12 we demonstrate how microstructures in interior of the O HF Rep object are implemented. The microstructureswere defined as incorporated infinite slabs in interior of the ’sphere’ and ’heart’ objects using set-theoretic operations(18). The infinite slabs were defined according to [67] as follows: S ( p ) = sin( ν (cid:12) p + φ ) + l ; (24)where S ( p ) ≥ is a vector function, with components defined as a set of slabs orthogonal to either X or Y or Z-axes, ν is a frequency vector, with components defined as the distance between parallel slabs along one of the axes, p is a point p ∈ X , φ is a phase vector, with components defined as the position of slabs on one of the axes with respect to theorigin and l , − < l i < is a threshold vector that together with frequency parameters controls the thickness of eachslab. Then the basic algorithm was applied to the obtained function to compute the HFRep objects with microstructures.19 PREPRINT - J
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1, 2021Figure 12: The illustration of the HFRep heterogeneous object based on the FRep and SDF representations withincorporated microstructure. a) the rendered HFRep ’sphere’ object using the sphere-tracing method (left) and itsisolines (right); b) the rendered HFRep ’heart’ object using sphere-tracing method (left) and its isolines (right);Implementation was done using C++ and OpenGL. The HFRep geometric shape was computed as a scalar field whichwas stored in a 3D texture. Then it was passed into a fragment shader for assigning a single colour attribute and renderedusing the sphere-tracing method.
We can specify attributes as simple procedural textures. Fig. 13 shows two heterogeneous HFRep objects O H v ,HF Rep with coloured wooden textures that were obtained using a procedural function f wood ( p ) . This function is constructedusing hash table htab ( p ) allowing for random sampling of the position values p multiplied by the frequency ν . Theprocedural function for the wood can be defined as follows: g ( p ) = htab ( p · ν ) · c ; (25) f wood ( p ) = g ( p ) − int ( g ( p )); where c > is a constant, g ( p ) is a noise function, int ( g ( p )) is an integer part of the function g ( p ) output value. Toparameterise f wood ( p ) by the distance, we assign the distance values to the frequency parameter ν .Then a simple segmentation of the geometric shape of the objects was done (see Fig. 13,(1)). We split the shape into fourregions and assign colours using the obtained HFRep distance function F HF Rep ( p ) and procedural function f wood ( p ) that defines the texture of the wood. The generated objects were used as inputs for 2D heterogeneous metamorphosis onthe basis of the space-time blending (STB) method to handle geometry transformation and the space-time transfiniteinterpolation (STTI) to handle colour transformation [68]. This example was implemented using C++ and OpenCV.In another Fig. 14 we show three textured ’H’ HFRep objects. The textures for these objects were generated using threedifferent parameterisations of the procedural function for the wood by the computed IDFs.20 PREPRINT - J
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1, 2021Figure 13: The illustration of the metamorphosis between two HFRep textured objects using the STB and STTItechniques. The texturing was made using procedural noise functions.Figure 14: The HFRep ’H’ object that is textured using procedural function 25 modelling the ’wood’ texture. Thisfunction was differently parameterised (a), (b), (c) by computed IDF for the given object.
In this subsection we discuss how HFRep objects with voxel-based attributes can be defined. Two following exampleswere implemented in SideFX Houdini using the OpenVDB library.In Fig. 15 we show a 3D model of the COVID-19 cell that was obtained using 207 set-theoretic operations. In Fig. 15,(b), we can see the interior structure of the COVID-19 cell [69]. The central part representing the RNA and N-proteinwas defined using SDF that was further combined with the HFRep spherical shell of the cell. The M-protein was alsodefined as a combination of the SDF arc and two HFRep spheres. The rest of the elements were defined using HFRep.Each element is mono-coloured and colours are assigned per-voxel.Fig. 16 demonstrates a 3D metamorphosis between two heterogeneous objects that are a combination of the HFRep andSDF defined objects [52]. This example served as one of tests for the ’4D Cubism’ project [70]. The input and targetobjects are two SDF cubes spaced from each other. These input shapes were segmented using an octree data-structureto make it possible a local faceting and distortions. Two colours were assigned to them per-voxel. Then differentHFRep and SDF ’cubist’ features were assigned to selected areas of the two basic SDF cubes, which were colouredper-voxel as randomly chosen colours from the specified range. Then we apply the same combination of methods as wehave discussed before for the 2D metamorphosis. The generated colour and geometric shape transformations happen21
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1, 2021Figure 15: The COVID-19 cell model obtained as a combination of the HFRep and SDF functions. a) exterior of thevirus cell; b) interior of the virus cell.simultaneously and interconnectedly. In Fig. 16, 4 (sliced), we show how the interior of the object is transformed duringthe 3D metamorphosis process.
In this work we have introduced a theoretical and practical framework for modelling volumetric heterogeneous objectson the basis of a novel unifying functionally-based hybrid representation called HFRep. First, we have identified fourconventional representational schemes related to scalar fields of different kinds, namely FRep, SDF, ADF and IDF,suggested a formalisation of those approaches and described their advantages and drawbacks. This has allowed usto formulate the requirements for a unifying hybrid representation. The defining functions in the core of HFRep arecontinuous and have a distance property everywhere in a Euclidean space. They also have several other useful properties.We have defined the mathematical basics of the representation and developed an algorithmic procedure allowing togenerate HFRep objects in terms of their geometry and attributes.To make our approach practical, we have provided a detailed description of the main steps of the algorithm and identifiedsome problematic issues associated with them. This has required employing a number of techniques of different nature,separately and in combination. Some of these techniques were already described in literature, others had to be improvedor developed. In particular, a new FIM algorithm for solving the eikonal equation on hierarchical grids has beendeveloped.To show how the proposed framework works, we have illustrated the algorithmic process with a number of implementedexamples, including those that deal with colour, material and microstructure attributes in the interior of functionally-defined shapes in the context of time-variant modelling.While the boundary representation will remain the main and prevailing instrument for geometric modelling, webelieve that the functionally-based representations generalising a well-established implicit modelling approach, arebecoming more important in the context of some modern applications. Hopefully, HFRep that embraces advantages andcircumvents drawbacks of FRep, SDF, ADF, IDF will find its applications.Future work will be concerned with developing operations over HFRep objects in the context of different applications,especially related to physical simulation, additive manufacturing and visual effects. In technical terms, we aim to developa more efficient HFRep field extrapolation procedure beyond the computing domain. One of the interesting directionswill be the introduction of the attribute definition in the interior of the volumetric object using the diffusion-based IDFs.We also consider a further generalisation of the FIM method for 3D hierarchical grids.22
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1, 2021Figure 16: Metamorphosis between multiple coloured objects using the STB and STTI techniques. Colours of theinitial objects are procedurally defined per voxel.
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