Implicit Representation of Inscribed Volumes

Implicit Representation of Inscribed VolumesParto Sahbaei

University of [email protected]

David MouldCarleton University

[email protected]

Brian WyvillUniversity of Victoria

[email protected]

ABSTRACTWe present an implicit approach for constructing smooth 3-D in-scribed volumes intended for modeling porous structures, suchas volcanic rocks, foam, radiolarians, and Swiss cheese. Positiveinscribed volumes can model natural pebbles, and negative vol-umes can model porous structures. We introduce two techniquesfor blending and creating interconnections between these inscribedvolumes to adapt our approach to both regular and irregular. Webegin with a set of convex polytopes such as 3-D Voronoi diagramcells and compute inscribed volumes bounded by the cells. The cellscan be irregular in shape, scale, and topology, and this irregular-ity transfers to the inscribed volumes, producing natural-lookingspongy structures. Describing the inscribed volumes with implicitfunctions gives us the freedom to exploit volumetric surface combi-nations and deformation operations effortlessly.

CCS CONCEPTS• Computing methodologies→ Shape modeling;

KEYWORDSvolumetric modeling, implicit curves and surfaces modeling, in-scribed volume, spongy structures, radiolarian, computer aestheticACM Reference Format:Parto Sahbaei, David Mould, and Brian Wyvill. 2018. Implicit Represen-tation of Inscribed Volumes. In Expressive ’18: The Joint Symposium onComputational Aesthetics and Sketch Based Interfaces and Modeling and Non-Photorealistic Animation and Rendering, August 17–19, 2018, Victoria, BC,Canada. ACM, New York, NY, USA, 7 pages. https://doi.org/10.1145/3229147.3229164

1 INTRODUCTIONModeling porous objects including radiolarians, foams, volcanicrocks, or similar volumes is a challenging task. We introduce aconsistent approach to modeling such natural or man-made formswhile conserving their smoothness and natural aesthetics, as wellas preserving the complex geometry of interconnected structurescontaining inscribed volumes. The problem is described by Bearet al. [2]: it is difficult to formulate a general approach, as thereare wide morphological and structural distinctions between theabove examples. Porous materials – such as volcanic rocks, sponges,or Swiss cheese – are volumes with pores that may or may not

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interconnect. To achieve a general representation of such objects,volumetric modeling is required.

We aim to model a general form of porous structures that arenot only natural looking but aesthetically satisfying, with smoothsurfaces inside the inscribed volumes. We use an implicit volumemodeling approach [17], as this technique produces smooth and aes-thetic shapes that can be seamlessly modified at the local or globallevel while maintaining the consistency of the model to producethe desired target results [15, 21]. In addition, their function defini-tion is compact and usually requires fewer high-level primitives toconstruct a model compared with other methods [3].

The contributions of this work are summarized as follows:(1) A general approach that can represent many types of spongy

structures.(2) A base method that produces convex blobby volumes with

aesthetic curvature, representing either holes or solids.(3) Two techniques for making connections between the cham-

bers of porous substances, achieving different sets of proper-ties.

2 RELATEDWORKFor surface modeling and rendering of objects with porous or meso-scopic details, texture synthesis approaches have been suggested.Tong et al. [19] proposed a bidirectional texture function synthesisconsidering both surface geometry and surface details to achieverealistic porous materials. The synthesis is done by examining dis-tances between 2-D surface textons derived from the 3-D textonmethod [8]. The drawback with textons is that the clustering intro-duces discretization errors, and the distance metric requires costlyaccess to a large inner-product matrix which makes this methodcomputationally slow [6]. Tong et al proposed Quasi-Homogeneousmaterials [18] which produced more realistic surface materials witha complex capture and rendering process. These methods are basi-cally focused on representing porous surface materials with 2-Dtexture synthesis and rendering techniques for objects such as theouter surface of bread showing the mesoscopic scale pores.

Chen et al. [4] used 3-D texture mapping employing complexmicrostructures tomodel porous volumes for the purpose of productdesign and manufacturing. The final structures made with regulargeometries could be warped to some extent and are more suited forindustrial purposes and less beneficial for natural spongy structuresdue to a lack of the required stochastic properties.

Magda and Kriegman [10] described a method for acquiring vol-umetric surface textures which separated geometric informationfrom reflectance data. Since their algorithm offers high compres-sion of the original data, it is especially suitable for the textureswith complex, opaque microstructure, shadowing, and unknownreflectance properties that are difficult to compress. Their methodalso offers fully rendered textures at any orientation to the viewerat extra memory and computation cost.

https://doi.org/10.1145/3229147.3229164

https://doi.org/10.1145/3229147.3229164

https://doi.org/10.1145/3229147.3229164

Expressive ’18, August 17–19, 2018, Victoria, BC, Canada Parto Sahbaei, David Mould, and Brian Wyvill

Baravalle et al. [1] presented an interesting method for modelingporous structures by a particle system generation of 3-D texturesusing a dynamical system. They introduced a 2-D sliced basedtexture mapping method, applying a 3-D texture to a specific sliceof an object; making this method computationally efficient. Thismethod is not suitable for modeling of smooth or round macro-pores as is found in some structures, however it is able to representmesostructure details of bread or similar textures on the outer shellof a surface.

Various techniques have been used for modeling specific struc-tures, but are not intended as generic methods. These include atechnique by Martinez et al. for open-celled foam, representing dryfoams with hollowed internal structures [12]. Their technique usesa 3D Voronoi approach and was used for studying properties offoam microstructure in the context of industrial manufacturing ofsynthetic foams. In other methods, a microstructure model basedon the combination of 3-D image analysis and random Laguerretessellations is presented which captures the basic features of realsolid foams [9, 14]. A similar cell foam modeling method based onVoronoi tessellations is also presented by Maliaris et al. [11].

Since we are using an implicit volume modeling approach, wediscuss the most relevant background work in this field. Implicitmodeling gained attention due to its intuitive mathematical rep-resentation of surfaces. Implicit volume modeling [17] producessmooth and aesthetic shapes that can be seamlessly modified atthe local or global level while maintaining a consistent modelingstructure [15, 20]. In addition, their function definition is compactand they require few high-level primitives to construct a model [3].

Some 3-D implicit modeling and design tools software packageswere developed by employing skeletal-based modeling, such asBlobTree.NET [5] and ShapeShop [16].

The inscribed curves method by Wyvill et al. [21] is the 2-Dfoundation of the work introduced in this paper. This earlier methodis based on 2-D implicit models, and we extend the technique to3-D and generalize to include connected inscribed volumes crucialfor 3-D porous volume construction.

In many applications (such as Swiss cheese) interconnectionsoccur between inscribed volumes. The style of these connections,and whether they are regular or chaotic, can be captured using twodifferent techniques. We demonstrate that the techniques can beadapted to many different types of porous structures.

The inscribed volumes (IVs) in their positive form can be usedto model natural pebbles, liquid foams, or similar objects, however,to achieve complex models, such as spongy structure models, theyshould be defined in a negative form by hollowing out other solidprimitives using Constructive Solid Geometry (CSG) or a negativeblending operation.

We use 3-D Voronoi cells to produce convex, stochastic, andnatural looking geometry with the ability to control the topologyof the inscribed volumes and infer the adjacency of neighboringcells. The Voronoi diagram is a robust tool for representing manygeometric structures for partitioning space into convex polytopes.The distributions and patterns of the input seeds of the Voronoitessellation can affect the overall properties of the IVs: e.g., size,roundness, and regularity vs. stochastic aspect. Choosing the rightpoint distribution algorithm depends on the modeling application.Our approach can be used regardless of the input pattern.

3 BACKGROUNDWyvill et al. [21] conducted a user survey to determine whetherparametric or implicit curves are more appealing. The user studyshowed a strong preference for implicit curves over B-spline curves.Figure 1 shows an example of the favored curve type.

Figure 1: The inscribed curves method. This image receivedthe most votes from users for containing the most aestheti-cally pleasing curves.

The idea behind inscribed curves (IC) in 2-D [21] is to define aset of aesthetically pleasing curves in the interior of each boundedconvex polygon of a 2-D Voronoi diagram such that the curvesapproximate their bounding region. In this context, the aesthetic as-pect refers to the smoothness of the closed curves. For each convexpolygon P the following data is available: a vertex set consisting ofp0,p1, ...,pk−1; ni as the inward unit normal for li , where li is theedge connecting pi and pi+1.

The signed distance of an arbitrary point x ∈ ℜ2 to the edge liis

di (x) =(x − pi

)· ni , (1)

where (·) denotes the dot product.For each point located on the edge li we have

li ={x ∈ ℜ2 | di (x) = 0

}. (2)

Wyvill et al. sought to define a function F that was positive insideP and increased with distance from the boundary. Such a functioncan be obtained by taking the product of the distances:

F (x) =∏

0≤i<kdi (x), (3)

where k is the total number of vertices of P. Any point positionedon any edge of P has F (x) = 0.

In practice, the product of distances can become unmanageableby creating enormous values. Instead, consider the field f , where

f (x) = log F (x), x ∈ P. (4)

Combining 3 and 4, we have

f (x) = log∏

0≤i<kdi (x) =

∑0≤i<k

log di (x), x ∈ P. (5)

Given the field f , we can extract a curve: let Sa be the pointson the surface defined by Sa =

{x | f (x) = a

}. That is, Sa is the

implicit surface in f with isovalue a.The above method for inscribed curves has two drawbacks. First,

it is defined only in 2D; we need 3D volumes. Second, the curves

Implicit Representation of Inscribed Volumes Expressive ’18, August 17–19, 2018, Victoria, BC, Canada

within cells are isolated; we want the cells to communicate, withfields that represent interconnections between the cells allowingus to create sponge-like structures. We address both of these in thefollowing section.

4 INSCRIBED VOLUMESWe start by describing a basic 3D method which considers eachcell independently, defining a distance field function for every cell.Subsequently, we introduce two methods for creating interconnec-tions between neighboring cells. Defining precise shapes in theinterconnection volume between regions gives us the flexibility tosupport a great variety of porous structures.

4.1 Isolated Inscribed VolumesThe idea is to have a distance function to generate a set of smoothinscribed volumes where each volume is approximated from thewalls of a bounded convex region and must be defined in the inte-rior of the region. The smoothness and roundness of the InscribedVolumes (IVs), helps us fulfill our aesthetic goals: smooth curvatureis important for architectural or industrial decorative designs, crea-tures like radiolarians and some living sponges, and even objectslike pebbles and Swiss cheese. Our method is similar to the 2-Dimplicit method [21], but the 2-D geometry elements become 3Delements: bounding polygons become polytopes and edges becomefaces.

Figure 2: A single convex polytope C with one face high-lighted.

Consider a bounded convex polytope C (Figure 2) with a finitenumber of faces, wherepi is the i’th polygon ofC having Li verticesVpi = (v0, ...vLi ). Let ni be the inward unit normal for pi . Thenfor a point x ∈ ℜ3, the signed distance of the polygon pi to thepoint x is

di (x) =(x − vmi

)· ni , (6)

where vmi is them’th vertex of the i’th face of C and can be anyvertex of the face or even the face centroid. For each point locatedon the face itself, the distance is zero.

Just as in 2D, the field function f can be defined as

f (x) = log∏

0≤i<kdi (x) =

∑0≤i<k

log di (x), x ∈ C. (7)

where k is the total number of faces of C.The smoothness and size of inscribed iso-surfaces depend on the

choice of the value of the iso-value a. If a approaches zero then Saapproximates the boundary of the enclosing polytope C. The largerthe value of a, the rounder and smoother the resulting iso-surface.

As written, the field value of f is derived from its distance toeach face of C, with each face weighted equally. However, equalweighting yields undesirable results. Consider polytopes CA andCB with the same geometry. Now, add a face of infinitesimal sizeto CB . The isosurfaces within CB and CA will be quite different,even though the geometries are almost identical. We can addressthis in a manner analogous to the approach of Wyvill et al., addinga coefficient for each face. Let

f (x) =∑

0≤i<kλi logdi (x), (8)

where λi is given by λi = Ai/AC whereAi is the area of the i’th faceand AC is the total area of the polytope. Thus the faces with biggerarea have more effect on the inscribed surface than the smallerones. For better control over the influence of the face areas, λi canbe modified to:

λi = Aci /AC, 0 < c ≤ 1. (9)The value of c can affect the overall size of the inscribed surfacesas well. Therefore, depending on the chosen modeling object, thechoice of c can be adjusted to obtain the desired smoothness andapproximation of C. For the results we show, we typically used avalue of c around 0.8.

4.2 Interconnected Inscribed VolumesIn order to construct a wider range of natural substances, we need tocreate interconnections between the generated volumes within cells.We distinguish between fully-connected and partially-connectedstructures. Fully connected porous structures are entirely intercon-nected; an example is the spongy structure of the Cancellous bone,shown in Figure 11. Partially connected structures have sporadicconnectivity; an example is the bubbles found in some types ofcheese.

4.2.1 The Fully-Connected Method. The fully-connected methodforms links across every shared face. The aim of this algorithm isto produce a network of IVs. For an internal cell C of a 3-D Voronoitessellation with i faces, there will be i passageways through thefaces, one for each face. We achieve this by placing additionalconnection fields across each face. The connection fields are newprimitives that should smoothly blend with the initial inscribedvolume withinC. Our connection fields will pass through the centreof each face; we constrain them to neither break the shared faceedges nor pass through other faces.

To construct interconnections, we first divide the cell C into ksectors where k is the number of faces of C. Each sector is con-structed from the vertices of the shared face and the centroid ofthe cell it lies in. We then create a cage by linking the two adjacentsectors: i.e., the cage is the polyhedron created by linking the ver-tices of the face with the centroids of the two neighbouring cells.


Figure 3: Iso-contours within three Voronoi cells. As a ap-proaches zero (from left to right), Sa approaches the bound-ary shape. The greylevels represent the gradient field con-tours.

This new cage is a bounded region in which we introduce a newfield primitive. The size of this new primitive and its roundness aregiven by the function f in the same way as the initial IV. Finally,the new primitive field blends with the original IVs. We chose thesimple Ricci blending operator to control the smoothness of thefinal shape. However, using other blending operators, e.g., that ofGourmel et al. [7], with better control of the blending surfaces, canbe considered as well.

4.2.2 The Partially-Connected Method. The partially-connectedtechnique is used to develop a random and natural link betweenthe IVs, not necessarily connecting all cells. In this method, a spher-ical booster field is generated around each cell’s centroid. Thesebooster fields then blend with the initial IVs within the cells. Theresulting shape becomes connected where centroids are close to-gether, but the IVs remain disconnected for more distant centroids;the irregular pattern of connectivity gives the method the namepartially-connected.

Each cell’s booster field blends with all its neighbors’ boosterfields. Finally, we blend the result with the initial inscribed field.For this variation, it would be helpful to have an accurate blendingoperator such as gradient-based blending by Gourmel et al. [7]to avoid unwanted bulging artifacts. Incorporating this blendingmethod is future work.

Adjusting the radius for the booster fields is purely optional anddepends on what type of object is being modeled. In general, if thesizes and shapes of the bounded cells are irregular, some geometricrestriction can be arbitrarily applied to the radius of the boosterfields to prevent the final shape (i.e., after blending with initialIVs) from going beyond the cell boundaries. Moreover, restrictingthe radius of the booster fields naturally helps to produce a morerandom and partially connected network of IVs. This is mainlybecause when two cell centroids are far from each other, boosterfields may not blend with each other which results in unconnectedneighboring cells as well.

Another consideration about choosing the sizes of the boosterfields can be obtained by the actual cell sizes and their irregularitiesof shape. In general, if the cells’ tessellations are randomly selected,it is possible to have thin cells along with fairly huge cells. Thisrequires us to pick a smaller radius for a smaller cell so as not tooverwhelm the initial IV shape and also to select a bigger radiusfor larger cells to avoid the booster field getting masked or beingignored by its corresponding initial IV size. In each cell, we usedthe distance of the closest face of the cell to the centroid point asour guidance for determining the radius of the booster field.

5 RESULTSThe techniques described in this paper provide control over the vol-ume’s size, smoothness, and blending operator. The first decision tomake is determining the distribution and density pattern of IVs; forexample, uniform patterns could be used for synthetic foams or nat-ural honeycombs, while random distributions are suitable for manynatural porous structures. The proper point distribution varies fromone usage to another; our approach can be used regardless of thedistribution.

The size and smoothness of IVs can be adjusted by changing theisovalues (see Figure 3), or by changing the weights of function fas explained in the previous section. In general, as the IVs approachthe boundaries their shapes appear less smooth and rounded andtherefore less aesthetic.

Figure 4: Fully connected: a regular interconnection betweenInscribed Volumes.

Another factor is the blending operator, which not only affectsthe smoothness of the chambers in a porous structure, but also hasa crucial role when the IVs blend with the augmenting fields ineither the fully- or partially-connected method. Figure 4 shows a2-D slice of IV field values using the fully-connected method. In thisfigure, we adjusted the size of the cages by moving the centroidvertex of the bigger cell toward the shared face to better balance thecontribution of the interconnection primitives to each neighboringcell.

Figure 5: Partially connected Inscribed Volumes.

Variations of the partially-connected method depend chiefly onthe size of the booster fields before and after blending, as well asthe method used to blend between them and the initial IVs. As thesize of the booster fields increases, the likelihood that neighboringcells will connect increases as well.

We have primarily used IVs as negative space, subtracting themfrom other geometry to produce porous structures. We can alsointerpret them as positive space, and when we do so, the resultingshapes resemble pebbles, as seen in Figure 6. Figure 7 shows poroustriple torus shells using negative IVs. To create this figure, we first


Figure 6: Inscribed volumes in positive; the resulting shapesresemble pebbles smoothed by the action of water.

Figure 7: An aesthetic object: porous triple torus shells.

subtracted a smaller torus from a bigger one to produce a thin shell,and then subtracted a set of inscribed blobs from each shell. Both ofthese figures show the capability of this method for creating objectsof aesthetic interest.

Figure 8: A type of radiolarian called a polycystine, with itsmain spines and round pores on the outer and central shells.Real-life radiolarian (left), image credit Jane K. Dolven; ourmodel (right).

Figure 8 shows a simple model inspired by a type of radiolarianscalled polycystines. Fossils of these marine plankton can be foundin most oceans. They usually have hollowed skeletons made ofopaline silica, spines on the outer shells, and a central capsulewhich separates their inner and outer portions.

For those structures whose pores are not completely smooth orround, or which have short tunnels connecting pore chambers, theregular or partial connection techniques described in section 4.2are used. These algorithms are flexible enough to construct naturallooking sponges such as dry foams, spongy bones,aquatic sponges,some kinds of cheese, and volcanic rocks including pumice.

Figure 9 shows a piece of cheese constructed from a sector of acylinder and cut from the top and one side (right) compared to areal-life cheese (left). The IVs are naturally connected with eachother using the partial-connecting algorithm; the sizes of holes andthe amount of blending power between the booster fields insideeach cell and the IVS, were manually chosen to give a plausibleeffect. These parameters can be easily adjusted in the user interfaceimplementation. Figure 10 shows a real-life rock (left), and twoporous rocks, both created using the partial connecting algorithm(right). The left rock in the right-side figure is a sphere modified by3-D Perlin noise [13], where the value of noise is clamped between[0,1] and then combined with the original sphere distance field. Dueto the implicit definition, other methods of surface displacementcould be employed instead if desired.

Figure 9: A real-life piece of cheese (left), image credit GettyImages-Creativ StudioHeinemann; synthetic cheese createdwith partial interconnections between holes, rendered withsubsurface scattering (right).

Figure 10: A real-life volcanic rock (left), and two porousrocks generated by our partially-connected method (right).

Figure 11 depicts a cross-section of a normal Cancellous bone,modeled inside a solid outer layer of Cortical bone. We generated


this model using the fully-connected method, as the bone structurehas smoothed chambers connected with each other through wideopenings.

Figure 11: A cross-section of Cancellous (spongy) bone in-side a simplified Cortical bone (right); a real-life spongybone (left), image credit Steve Gschmeissner and Getty Im-ages.

5.1 DiscussionNatural modeling of porous or spongy substances is a combinationof art and different sciences including biology, geology, and com-puter graphics. In computer graphics, we use several techniqueswhere each of them helps to mimic one or more characteristics ofporous structures to achieve similar results. Our approaches are ableto construct several types of porous structures, giving the flexibilityof choosing and adjusting the right settings for each model.

The mathematical definition of implicit inscribed volumes of-fers remarkable flexibility and accuracy when it comes to surfacealteration, to customize the modeling construction process. Thisis particularly important when a natural and irregular surface dis-placement or a seamless surface combination is required. To guar-antee that the resulting shapes have smooth curvature and are C2

continuous, an appropriate fall-off filter function (Wyvill function)is performed which is also useful to get aesthetic results.

Using the Voronoi diagram not only helps produce naturallyshaped IVs, but the topological distribution of its seeds assists withcustomized or random topological patterns of IVs as well. Never-theless, for a photo-realistic visualization of natural materials anappropriate texture application and different rendering techniquesseems necessary as even the best volumetric modeling techniquescannot adequately represent all mesoscopic details of some porousmaterials.

6 CONCLUSIONS AND FUTUREWORKIn this work, we presented a procedural approach to creating porousstructures using inscribed volumes. The first step was constructing3-D IVs approximated by a set of bounded and convex regionssuch as a Voronoi diagram. The IVs in their negative (inverted)form can be used for generating porous structures and in theirpositive (normal) form are smooth rounded shapes resemblingpebbles. Some modern furniture designs, exhibiting smooth andaesthetic curved surfaces, might also lend themselves to the designframework presented here.

We introduced two simple, flexible interconnection techniques(fully-connected and partially-connected) to embrace morphologi-cally different spongy structures. The flexibility of our frameworkgives the freedom of adjusting size and smoothness of the poresas well as choosing a number of properties for modeling a specifictype of sponges where connecting IVs are required. We demon-strated that various natural spongy structures such as Cancellousbone, cheese, rocks, and radiolarians can be constructed using ourtechnique. Synthetic porous materials with uniform pore distribu-tions can also be generated from our techniques with ease, as theyusually do not have the complex and unpredictable characteristicsof natural substances. The results can be deformed or effortlesslyblended or combined, using BlobTree [20] operations such as CSGand blending, with other skeletal primitives to construct more com-plex objects. These models can be used in different applicationssuch as animations, and medical visualizations.

The definition of IVs was limited to a set of boundaries that wereconvex and were made of polygonal walls. Nevertheless, the IVs canbe defined over any bounded convex region as long as the region isrepresented by its faces and vertices. This means that in a polytopeC curved faces can replace polygonal faces. The function must bechanged and we have not tested it in our framework yet, but thiswould be a nice feature which can produce interesting results inthe future.

Although our implementation can deploy both random and uni-form control point distributions, as well as create distributions withspecific conditions, it does not use any data from real-world porousmaterials. Generating the mesoscopic details of quasi-homogeneousmaterials can be time-consuming for the end user. Integrating asimple and fast texture synthesis procedure or generating varioustextures and applying them to the surface will reduce the pressureon the user, who can benefit from the combination of our methodsand texture synthesis and rendering techniques to attain beautifuland intricate images.

The Voronoi diagram has been used because of the natural andvaried structures it produces. In the future, we hope to investigatealternative tessellation methods so that we can further increase thevariety of shapes.

ACKNOWLEDGMENTSThis work was supported by the Natural Science and EngineeringResearch Council (NSERC), Canada. Thanks go to Herbert Gras-berger for his help on the implementation. We thank the BellairsResearch Institute for hosting the discussions that ultimately led tothis work.

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Implicit Representation of Inscribed Volumes