Using Distance Fields for Object Representation andRendering

Mark W. Jones and Richard SatherleyDepartment of Computer Science, University of Wales Swansea

United Kingdom SA2 8PP

March 12, 2001

Abstract

Distance fields are a two- or three-dimensional array ofvalues, where each value is the minimum distance tothe encoded object. Usually distance fields are signed,such that negative values signify the point is inside theobject, and positive are outside. Distance fields are be-coming a popular research area as more applications arediscovered, and many of these application areas are dis-cussed in this report. More importantly, algorithms forcomputing distance fields quickly are also examined.Both existing algorithms, and those developed by our-selves at Swansea are discussed.

1 INTRODUCTION

The work presented in this paper is closely related tothe field of Volume Graphics [1]. Volume Graphics isan emerging area of Computer Graphics which is con-cerned with the input, storage, construction, modelling,analysis, manipulation, display and animation of spatialobjects in a true three-dimensional form. When com-paring Volume Graphics to traditional surface graph-ics, the relationship has been likened to the relation-ship of that between two-dimensional images and vec-tor graphics [2]. The employment of volume graphicstechniques offers the benefits ofscalable renderingal-gorithms for extremely large scenes,visual effectssuchas fire, fur and roughness (Section 4.3) andconsistencyof rendering(the ability to render objects of differentsource types under the same conditions).

In this paper we shall introduce distance fields in Sec-tion 2 and examine various methods for calculating dis-tance fields in Section 3. We will discuss a naive al-gorithm using binary segmentation in Section 3.1, andimprove this using sub-voxel accuracy (Section 3.2) andoctrees (Section 3.3). Approximate methods are intro-duced in Section 3.4, and improved in Section 3.5. Anew approximate method is presented in Section 3.6,and an analysis of all of these methods is presentedin Section 3.7. Section 4 presents three of our appli-cations of distance fields, namely contour connection(Section 4.1), the morphological closing operator (Sec-tion 4.2) and hypertexture (Section 4.3). Further work

and conclusions are given in Section 5.

2 BACKGROUND

A distance fielddata setD representing a surfaceS isdefined as:D : R3 ! R and forp 2 R3 ,

D(p) = sgn(p) �min fjp� qj : q 2 Sg

sgn(p) =

��1 if p inside+1 if p outside

wherejj is the Euclidean norm

(1)

For each voxel in the three-dimensional grid, the dis-tance to the closest point on the surface is stored. Ad-ditionally, all voxels inside the object are set to be neg-ative, and all voxels outside are set to positive. Thisdistance field has effectively voxelised the object as theoriginal surface can be displayed by rendering the iso-surface of level 0 fromD – i.e. S = fq : D(q) = 0g

whereq 2 R3 . Non-integer grid points can be calcu-

lated from surrounding integer grid points using trilin-ear interpolation or any other appropriate scheme of in-terpolation. Surface normals can be calculated from thegrey-level data [3], and due to the choice of the signfunction point away from the object centre, aswell asbeing perpendicular to the surface due to the distancefield. Figure 1 demonstrates several different forms ofsurface representation.

Figure 1: Binary segmented, original and distance fieldslices from the CThead dataset.

Distance fields have been used for applicationswithin Computer Graphics for morphing [4], virtual en-doscopy or skeletal representation [5, 6] and accelerat-ing ray tracing [7] (amongst other applications). In each

case a chamfer distance transform has been used (Sec-tion 3.4). This paper demonstrates more accurate meth-ods than the basic CDT. A great body of work describesdistance transforms in detail, in particular the interestedreader is directed to Cuisenaire and Macq’s review [8].This paper improves the current best vector distancetransform (in Section 3.6) and proposes a distance shell(Section 3.5) rather than the widely used binary seg-mentation (Section 3.1) for computation. It also de-scribes a practical, efficient algorithm for the computa-tion of an exact solution. Frisken et al. [9] have given aheirarchical computation for distance fields. This givesa compact storage, but trades off accuracy. In this pa-per we are concerned with accuracy. We are also inter-ested in giving practical algorithms for calculation, anddemonstrating new effects (for general objects) such ashypertexture (Section 4.3) and true distance morpho-logical algorithms (Section 4.2). The next section pro-gresses through each algorithm, from simplest to mostcomplex, highlighting our contributions on the way.

3 Distance Field Calculation

3.1 Naive Calculation

A relatively straightforward method of calculation is tofirst apply a segmentation functionf as:

f(v) =

�1 if v is inside the surface0 otherwise

wherev 2 Z3

(2)

Then for each voxelv, we set the distance field atvoxel v to 1 (D(v) = 1). For each voxelw, we setD(v) = min(D(v); jv�wj) wheref(w) = 1, andjj isthe Euclidean distance between two voxels’ positions.Each voxel will be taken in turn, and its distance willbe calculated to all voxels which comprise the object.Themin operator ensures that the minimum distance isstored.

This algorithm produces an accurate distance field inthe sense that each integer grid point in the field hasan accurate distance to the surface encoded by the seg-mentation function. Unfortunately as a discretised seg-mentation function is used, this has the effect that thedistance field represents a discretised surface. Anothernegative aspect of this algorithm is that the complexityof O(n2) results in long computational times. If we takethe CThead dataset for example (256� 256� 113) wemust make around 50 trillion calculations. Even reject-ing voxels immediately where their x, y or z positionis further away than the current minimum distance doesnot bring the rendering time down to a useable level.Figure 2 shows the discretised surface that arises whensampling a sphere using a binary grid. Even when over-sampling is used, the fact that no information is avail-able away from the vicinity of the surface, means thatnormals have a restricted orientation.

Figure 2: Rendering of spheres using jittered (left col-umn) and regular sampling (right column). Samples pervoxel (from top) 1, 8, 27, 125, 9261.

3.2 Sub-Voxel Accuracy

The quality of the distance field representation can beimproved by measuring the distances to a sub-voxel ac-curacy. Instead of employing a binary segmentationfunction, the distances to the actual surface are calcu-lated. In the case of the chess piece we can calculatethe distance to the closest point on each triangle. Thiseffectivelyvoxelisesthe chess piece. Voxelisation is theprocess of converting an object of one source type intovoxel (volume) data. (In fact the binary segmentationfunction mentioned in the previous section is often usedfor voxelisation). We have studied the methods for vox-elising triangular mesh objects in a previous paper [10],and discovered that it is often quicker to voxelise andemploy a volume renderer than to ray trace the originaltriangular mesh. This is mainly because volume ren-dering scales so well. The time to render is based uponthe size of the volume data rather than the complexityof the object. Another benefit of voxelisation is the factthat once all scene objects are encoded by a commondata type, it is possible to apply rendering effects uni-formly across all objects. Also rendering effects such astexture mapping or bump mapping which are tradition-ally carried out in the rendering phase can be carried out

during the voxelisation phase, and therefore view inde-pendent images can be rendered consistently from thevoxelised objects, thus removing the need to performthe texture and bump mapping for each new view.

The distance to a triangle is a non-trivial problem,and is treated in detail in a separate report [11]. De-pending upon the position of the voxel, the closest dis-tance to the triangle may be a vertex, an edge betweenvertices, or the plane of the triangle. We therefore pre-sented an efficient algorithm for projecting a point ontothe triangles plane, performing tests to determine whicharea the point is in, and then calculating the distance ap-propriately.

In the case of triangular mesh objects the complexitybecomes O(nm), where n is the number of voxels and mis the number of triangular facets. For example Figure 3shows volume rendered chess pieces, each voxelised toa grid of60�60�60and originally comprising of about1500 triangles. In that case 324 million calculations arerequired for each piece taking 287 seconds on a 1GHzAthlon (for the pawn).

Figure 3: Distance field encoded chess pieces renderedwith VLib [12].

One of our other contributions to this area is to con-vert CT data (such as the UNC CThead data set) into asub-voxel accurate distance field. A similar algorithmto the naive algorithm can be used, but instead off

encoding a binary segmentation,f encodes those cellswhich are transverse. A cell is a cuboid made up of 8neighbouring voxels. It is transverse when at least onevoxel is inside the surface, and at least one is outsidethe surface. This cell will have a piece of the surfacepassing through it. We triangulate that part of the sur-face using the tiling tetrahedra algorithm [13] as thisremoves the ambiguities of marching cubes. The dis-tance is then calculated to the triangles using the samemethod as for voxelisation. There are alternative meth-ods for constructing a surface from the field, althoughthese are more complex to compute and as we shall seethis is already a computationally intense process.

Using this method an accurate Euclidean distancefield can be created for the UNC CThead. Using thenaive algorithm this would take about 60 days on a1GHz Athlon. We actually implemented it using coarsegrain parallelisation, and executed it on 8 processors (11GHz Athlon, 5 800MHz Athlons and 2 866MHz Pen-

tium IIIs) in just over 8 days. We needed this dataset tocompare the accuracy with the faster methods reportedin the next sections.

3.3 Employing an Octree

We can improve the calculation of distance to O(nlogn)using an octree (in our case a Branch on Need Oc-tree [14] (BONO) as the data size is not a power of 2).We can reject those parts of the octree which are knownnot to contain any transverse voxels. Primarily it is usedto reject portions of the octree (and thus large numbersof voxels) which are further away than our current min-imum distance (D(x,y,z)). This can be improved fur-ther by using a neighbouring voxel’s closet point as agood starting point for the current voxel, thus removinglarge parts of the octree immediately. These two meth-ods improve performance considerably, particularly inthe regions close to the surface. An identical Euclideandistance field to that of the previous section can be com-puted in 126 minutes on a single 1GHz Athlon.

3.4 Approximation Methods – ChamferDistance Transforms

The sheer computational expense of the naive imple-mentation has led to the application ofdistance tran-forms. These have been implemented as multiple di-lations from a binary segmented surface, and more effi-ciently as a 2 pass process. First we begin with an initialdistance field,D (equation 3). Local distance propaga-tion (equation 4) is achieved with a number of passes ofa distance matrix, dM . Each pass loops through eachvoxel in a certain order and direction according to theneeds of the matrix.

D(p) =

�0 if f(p) = 1; 9 q 2 p26; f(q) = 0

1 otherwise

wherep 2 Z3; andp26 is the set of voxelswhich are the 26 neighbours ofp

(3)

D(i; j; k) = min�D(x+ i; y + j; z + k) + dM (i; j; k)

�8 i; j; k 2 dM wherei; j; k 2 Z

(4)Chamfer distance transforms propagate local dis-

tance by addition of known neighbourhood values ob-tained from the distance matrix,dM , (an example ofwhich is in Figure 4). Each value in the matrix repre-sents the local distance value. This matrix is applied intwo passes.

/* Forward Pass */FOR(z = 0; z < fz; z++)

FOR(y = 0; y < fy; y++)FOR(x = 0; x < fx; x++)D(x,y,z) = Eq.4

/* Backward Pass */FOR(z = fz-1; z � 0; z--)

FOR(y = fy-1; y � 0; y--)FOR(x = fx-1; x � 0; x--)D(x,y,z) = Eq.4

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Figure 4: Quasi-Euclidean5 � 5 � 5 chamfer distancematrix.

The forward pass (using the matrix above and to theleft of the bold line, shown initalic font) calculates thedistances moving away from the surface towards thebottom of the dataset, with the backward pass (usingthe matrix below and to the right of the bold line) cal-culating the remaining distances. This method is com-putationally inexpensive since each voxel is only con-sidered twice, and its calculation depends upon the ad-dition of elements of the matrix to its neighbours. It isalso very inaccurate, and hence all of the applicationsthat rely on the method are using inaccurate (althoughin many cases acceptable) data. We desired distancefields to produce hypertextured objects, and we foundthat the best distance transforms did not produce accu-rate enough data for the hypertexture to operate con-vincingly. This led us to investigate vector propagationmethods (Section 3.6).

3.5 Distance Shells

Usually these chamfer (Section 3.4) and vector (Sec-tion 3.6) distance transforms are carried out upon binarysegmented data. Alternatively adistance shellcan becalculated, and use those values as a basis upon whichwe can propogate the surface throughout the volume.This indeed increases the accuracy of these type of dis-tance transforms (both those of the previous and nextsections). To completely encode the surface of an ob-ject, we need to calculate a shell of voxels which areeither side of the surface interface. All those voxelswhich are inside the object and have a neighbour out-side the object should be part of the computed shell,as should all those voxels which are outside the objectand have a neighbour inside the object. In fact a quickvoxelisation algorithm is tosegmentthose voxels, andcalculate the distance for them. The surface can be re-constructed accurately from this shell, but the normalscalculated using the central difference operation samplevoxel values outside of this shell, and therefore as theyare of a uniform background value, it can lead to quan-tization effects (the normals have a restricited numberof unique directions). We can solve this by also includ-ing all those voxels that are used within the grey-levelnormal calculation function.

The whole procedure is to use the segmentation func-tion f (equation 2). Then for each voxel,v, we addvandv26 (the 26 neighbours ofv) to Sv, whenf(v) =

1 and9p such thatf(p) = 0 wherep 2 v26. To includeall of the additional voxels required for central distancecalculation of the normals, we create the shellSn wherefor eachv 2 Sv we addv andv26 to Sn. Sn now con-tains all voxels which are used to display the surface(including valuesused just in normal calculation). Wehave called the voxelsSn thedistance shellof the en-coded object. The distance shell adequately representsthe object, and is a valid method for voxelising objectswhere only the surface needs to be encoded. As a shellvoxelisation it benefits from the advantage of requir-ing less memory to store (if run length encoding is em-ployed). The chess piece takes 11.4 seconds to encodeusing this method, and the UNC CThead takes 124 sec-onds to convert into this form. Such datasets can thenbe used as the basis in the distance transform process(and therefore give a much more accurate result).

3.6 Vector Distance Transforms

Vector methods [15] store a vector to the closest surfacepoint at each voxel. These vectors are propagated toneighbouring voxels in a prescribed way similar to thepropagation of distances via the distance matrix. VDTsgenerally require more passes of the distance matrix.During each pass the vector components are added tothe necessary vector position, the distance calculated, adecision made as to whether any of the new distancesare minimal, and finally the minimal vector is stored.

To allow vector propagation Equations 3 and 4 aremodified as shown in Equations 5 and 6 respectively.

~V (p) =

8><>:(0; 0; 0)

if p 2 S and

9 q 2 p26; q =2 S

(1;1;1) otherwise

(5)

D(p)=min��~V (x+i;y+j;z+k)+dM(i;j;k)

��8i; j; k 2 dM , wheredM = ( ~Mx; ~My; ~Mz)

(6)

The previous best algorithm is Mullikin’s EVDT [16]operating on binary segmented data.

Figure 5 shows the matrix passes employed by ourVector City VDT.

3.7 Comparison

We have compared each of the methods in the previ-ous sections to the correct distance field in terms of theexecution time, the average voxel error, and the mostnegative and positive errors. Table 1 presents these re-sults in detail, and Figure 6 present images for each ofthese methods. If perfect Euclidean distance fields arerequired, we have shown that it is possible to computethem in a reasonable time, by using the octree basedalgorithm running in parallel. It would be possible to

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Figure 5: Eight pass vector-city vector distance trans-form.

Figure 6: 5x5x5 DT on binary, efficient vector DT onbinary, vector-city VDT on sub-voxel accurate data.

calculate each voxel individually from all other vox-els, although in reality we calculate each slice individ-ually. Therefore, the best run-time would arise whenwe have as many processors as slices. For our exper-iments we usually employed 8 processors giving us atime of about 15 minutes for the UNC CThead. We havealso shown that where possible, it is better to reject thebinary segmentation in favour of a sub-voxel accuratesegmentation. We have also demonstrated a procedurefor generating a distance shell, which can then be fedto a distance transform (either vector or chamfer). Theuse of this shell rather than a binary segmentation in-creases accuracy, and further we have shown that ournew VCVDT produces results which are 50% more ac-curate than the previous best algorithm (in a compara-ble time). It is hoped that researchers are able to digestthese results and decide to what accuracy they need dis-tance fields, and use the appropriate method.

4 Applications

4.1 Contour Connection

We originally proposed distance fields as an interme-diate step to create triangular meshes from contourdata [17]. Essentially the algorithm involved calculat-ing the distance from each voxel within the domain, tothe closest point on the set of contours representing theobject of interest. One possible surface that is describedby those contours can be obtained by triangulating theisosurface for the distance of zero. It was found thatthe use of the distance field helped avoid costly and dif-ficult point correspondence problems – particularly in

1 to many and many to many branching cases. Fig-ure 7 demonstrate some particularly difficult situations(for other methods), and the surfaces produced for thosecases.

Figure 7: Some cases of contour connection.

4.2 True 3D Closing Operator

One particularly interesting area for the use of distancefields is that of morphological operations. A closing op-eration can be carried out by usingn dilation operationsfollowed byn erosion operations. This has the effect ofclosing holes in the dataset, smoothing out sharp spikesand filling cavities. It is particularly useful for shapeand object recognition (although we have a further usedescribed in Section 5). The operation ofn dilations in-dicates those voxels which aren voxels away from thesurface, and if displayed will show a shape that has ap-peared to have grown. When followed byn dilations,it will return the original surface, although some ofthe artifacts mentioned above will have been removed.The method is carried out on binary segmented data.Hohne [18] has reported the use of three-dimensionalerosions and dilations as a useful application for theextraction of homogenous regions in body scans. Analternative to erosions and dilations upon binary seg-mented data, is to increase or decrease the grey-levelvoxel values themselves. For example, if the templateis a3�3�3 matrix where each value in the matrix is 1,for a dilation from a binary segmentationf , each voxelv wheref(v) = 1 will be included in the new surfacealong with all of its 26 neighbours. Using the originalvoxel values, a grey-level dilation will set a voxel to thevalue of its maximum neighbour (plus 1).

Intuitively a dilation of degreen should give us theexact surface where each point on the surface isn vox-els away from the original surface. An erosion of de-green should give us the exact surface where each pointon the surface isn voxels inside the original surface.We therefore propose that a true 3D closing operationshould be based on the accurate sub-voxel distance fieldand used in the following manner.

First we calculate the distance fieldD for our surfaceas in equation 1. The surface,S, representing a dila-tion of degreen is equivalent toS = fq : D(q) = ng

whereq 2 R3 . To calculate the erosion of degreen

for this surfaceS, we then calculate a new distance

Table 1: Comparison of Distance Algorithms

Method Time Error Range Average Error

True (octree) 126 minuts 0 0City (binary) 1.280s -2.000000! 76.229546 12.519548City (shell) 1.290s + 124s -2.376824! 73.186974 10.775360EVDT (binary) 6.890s -1.732051! 3.081223 0.261987EVDT (shell) 7.100s + 124s -1.873284! 1.392951 0.015674VCVDT (shell) 7.640s + 124s -1.873284! 1.392951 0.010767

field, D0 based upon distances from surfaceS usingthe techniques described earlier (measuring to the tri-angulation of the isosurface). The surfaceSCn whichhas been closed byn units through a dilation of de-green followed by an erosion of degreen is given bySCn = fq : D0(q) = �ng. Figure 8 demonstrates clos-ing operations of 20, 10 and 5.

Figure 8: True 3D distance closure of 20, 10 and 5 vox-els.

4.3 Hypertexture

A distance field defines the object’s surface, and alsothe distance from and direction to the surface. Thisis similar to the requirements for Perlin and Hoffert’shypertexture [19] effects. We are either inside or onthe surface (D(p) � 0), in thesoft region (D(p) � 0

andD(p) � Æ) or outside of the object completely(D(p) > Æ) (p 2 R

3 ). In the soft region, noise func-tions are used to distort, colour or project inwards toobtain the various effects. Distance fields, as computedin this paper, conform to the data requirements for hy-pertexture, and can be rendered using an extension tothe volume rendering algorithm. Figure 9 demonstratesthe application of hypertexture to distance fields of tri-angular mesh objects and the UNC CThead.

4.4 Other Applications

Other applications includeskeletonization(based uponother morphological operations), morphing by using adistance field representation, general distance opera-tions (such as Voronoi calculation – rather than calcu-lating the distance to a point we can calculate the ac-tual closest point (as demonstrated by our vector shellcode required by the VCVDT algorithm)), shape-based

Figure 9: Hypertexture applied to the distance fields fora chess piece, a dodecahedron and the UNC CTheaddataset.

interpolation schemes, and for accelerating ray-tracingby using the distance fields as proximity clouds.

5 Conclusions and Future Work

In this paper we have shown that various methods arereadily available for the calculation of distance fields.We have demonstrated that a choice can be made be-tween accuracy and speed of computation. Perhapsour more significant results are the fact that true Eu-clidean distance fields can be computed using an ef-ficient O(nlogn) algorithm, and to a far greater de-gree of accuracy by using a sub-voxel distance algo-rithm, rather than a binary segmentation. We have alsointroduced a new vector distance transform function(VCVDT), which has been shown to perform 50% moreaccurately, than the previous best function (EVDT). TheEVDT algorithm was originally proposed for use withbinary segmented data, and to be fair we have com-pared it when applied to sub-voxel accurate data aswell.We have demonstrated how to calculate the initial shellwhich is used for propogation. We have also demon-strated some applications, in particular the new sug-gestion of using the accurate Euclidean distance fieldfor true 3D Closing operations, as demonstrated in Fig-ure 4.2.

Our future work will concentrate on improving ac-curacy – we believe that a hybrid method directed bythe distance transform field will be able to give an ex-act distance field. We also intend using the closed skulldatasets as a basis upon which to create a facial recon-struction of a discovered skull, which will have appli-cation to the field of forensic science.

Acknowledgements

This work has been undertaken with funding from EP-SRC, UK, under grants GR/L88238 and GR/R11186.

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