Geometric Invariants For Bone Fragment Refitting Via Face Matching

Riley O’Neill

Dr. Shakiban

University of St. Thomas

In cooperation with the University of Minnesota:

Katrina Yezzi-Woodley, Jeff Calder, Peter J. Olver, Pedro Angulo-Umaña, Bo Hessburg

September 1st, 2018

Abstract

In an effort to match the faces of bone fragments for reassembly, numerous geometric invariants

have been prepared for discrete implementation on triangulated meshes in MATLAB and

explored for robustness. Four invariants have been examined, ranging from the more elementary

cumulative distance and surface area histograms, which are less robust measures for

triangulations formed from CT scans, to an entirely new method of calculating a volume

invariant, as well as the principal curvature via Principal Component Analysis. The principal

curvature and volume invariants are sufficiently robust for face matching, and algorithms for

face matching shall be developed and synthesized with those for curve matching in the coming

weeks.

Introduction and Objectives

Bones are of crucial importance to understanding life on earth. They are of the few vestiges left

by prehistoric eras and the foremost evidence we have for conjectures regarding the evolutionary

path of terrestrial existence and existence of extinct or unknown species. Were not bones

reposited in ancient mud pits, the notion of a dinosaur would not even permeate modern

consciousness, and many discount the potential existence of bigfoot by the lack of any bones

found. Bones also show where we and others have come from, what we have done, and

potentially foreshadow where we are going. Particularly for the field of anthropology, bones

shed light on ancient average lifespans, hunting patterns (both what hominids hunted and what

hunted them), causes of death (both modern and ancient), and hominid distribution. In a study

published in 2017, a group of researchers unearthed a mastodon in California they claim proves

hominids were in America 130,000 years ago -115,000 years earlier than originally thought

(Zimmer, 2017). The bones appear to have been broken with stone tools (the trademark method

of hominids for getting bone marrow), which were found in the vicinity (Zimmer, 2017). Using

similar tools on elephant bones, the team found the breakages were indeed similar, but the study

remains hotly contested (Zimmer, 2017). As hominids were only thought to have left Africa

40,000 years ago, these bone fragments could fundamentally alter human history.

Bones are breakable - a fact that is both fabulous and troubling for using them to study

ancient life. This fact is fabulous for getting at the conditions of other ages; anthropologists

examine fragment break angles (those on an edge that broke (yes, vague)) to attempt to ascertain

the actor of breakage (who or what broke it: hominid, animal, cave collapse, fall to death, saw,

lava explosion, alien laser, etc.). However, this fact is troubling for both determining break actors

and putting bones back together (to configure species, bone structure, and figure out what broke

entire bone) - sites with bone assemblages can be comprised of hundreds of thousands of

fragments, a fact that can make bone reassembly and study a tedious and time-consuming task.

Anthropologists employ devices hailing from the Victorian era called goniometers (see

Appendix A:1 for picture), or protractors by another name, for both bone reassembly (refitting)

and break angles, but this barely aids the process. To this day, anthropologists describe refitting

as “Painstaking” (Bartram and Marean, 1999), “Time consuming” (Vaquero et al., 2017), and

“Difficult to manage logistically” (Morin et al., 2005). In the age of machine learning and

advanced image processing, how can mathematics be applied to expediate and refine both the

ascertainment of break actors and bone reassembly processes?

As this is a tremendously large and daunting task, we’ve split the project between the break

actors and fragment refitting. For fragment refitting, we have four main end goals: that it

automatically identify potential fits, do so without any beforehand knowledge (not telling the

program 2 fragments go together, so it could function for mass bone assemblages), do so

expediently and reliably (which often can get to be a tradeoff), and not be too difficult or

impractical to implement (that the bone scans don’t have to be ridiculously high resolution).

However, before this can be done, we have to possess data that can be used for matching, which

is the first goal of the project - feature extraction. Numerous means of feature extraction exist, so

we’ve again split the current refitting work into two: curve matching and face matching. My

work has been primarily concerned with face matching, although I’ve broken my fair share of

bones as well (see appendix 1.2 for proof and more information on the bone breaking process).

Background and Methodology

Automated bone refitting has never before been successfully attempted. A brief initiative at

Arizona State University did embark on feature extraction for lithics, ceramics, and bones with

the aim of eventual refitting (Schurmans et al., 2002), but this program seems to have never

come to fruition. However, ceramic and lithic refitting has made substantial progress in

subsequent years. Ceramic and lithic refitting and face differentiation has primarily used

measures of roughness for surface differentiation (Rasheed & Nordin, 2015). However, this

doesn’t work for bones as the majority of the pertinent bone breakages are comparatively smooth

and clean, and thus less discernible in texture (one exception would be for trabecular bone, but

such is not the norm). Others employ predictive methods based off common pottery forms and

distinctive shapes (spheres, cylinders, rectangles, etc.), but we don’t necessarily know what

kind/shape the bone will be as we’re piecing it back together, and bones aren’t such perfect

constructs. Lithic reassembly has further found success with a qualitative approach, using some

probability and several properties like material composition and scar count. Tragically, bones are

not quite as qualitatively unique as rocks.

Thus, we turn to the quantitative, where a number of more advanced mathematical

techniques are employed. Just as done for ceramics and lithics, we employ computerized 3D

models for feature extraction and matching, using DAVID or CT cans (see Appendix 1.3 for

more about DAVID). From these scans, we generate 3D models. As fragment matching must

occur in a 3-dimensional space and it would be frankly ludicrous to attempt to devise concise

equations for each and every elaborate bone fragment surface, we convert the 3D models into

triangulated meshes in MATLAB (see appendix 2.1 for visuals of both). Triangulations serve as

a convenient way of estimating the bone surfaces with triangles for ease of numerical

implementation via discrete differential geometry. The vertices (points) used for these models

are taken factoring into account extrema and averages for a given area, yet the point values vary

slightly from mesh to mesh - even if two fragments perfectly match in real life (no missing

pieces), the triangulations may not match exactly. Furthermore, the chance these triangulations

are in the same orientation is effectively zero without a priori knowledge. As such, we demand

robust geometric invariants, or properties of a surface that will not change with parameterization

or orientation, to match the fragment surfaces.

Curve matching follows from the pioneering work of Grim et. al’s (2016) 2D and 3D puzzle

solving algorithms using the signature curvature of a surface. The signature curvature is a

geometric invariant whereby the curvature is plotted against the derivative of the curvature. Here

we take the curve that best encompasses a break face, take the signature curvature, and match it

to the the corresponding surface’s for matching. So far, this method is very noise sensitive but

has successfully matched some fragments and is still being refined. Face matching, however

encompasses a much larger category of invariants:

1. Cumulative Distance Histograms

Euclidean distance is the most basic of geometric invariants. In aiming to characterize surfaces in

3 dimensions, this is perhaps the readiest go-to. While we could potentially just examine the

maximum and minimum distances across a given face, this doesn’t work for mass bone

assemblages - numerous fragment faces could bear the same maximum and minimum distances,

and one face may not fit to the entirety of another. To remedy the former issue, we employ

cumulative distance histograms. Histograms are charming, simple constructs that plot the

number of an eligible parameter incrementally, usually in the form of a bar graph. For a fixed

region, distance histograms merely plot the number of points at or within a distance parameter

(can be set a single value or at a range) of a given point or points. For for characterizing surfaces,

we like cumulitize this and do every point on a surface, which does often mean double and triple

and sextuple counting the same point. However, these plots tend to say more about the number of

points used than the surface they describe, which is unideal for our triangulations devised from

bone scans as the quantity of points and their distribution can vary tremendously for an unrefined

mesh. Thus, we normalize with cumulative distance histograms, as put forth by Brinkman and

Olver (2012):

(1.1) H(r) = N-2 #{ D(p1,p2) < r }

Where the histogram value H at a given distance r is provided by the number of distances D from

one point to another less than or equal to r (more simply, the number of points within r for each

point) and the total number of points N (Brinkman and Olver, 2012). Graphing this against r

produces a plot that doesn’t alter with regard to the number of points used or the orientation of

the surface. Thus, we have our first invariant (figure 1).

Figure 1: Cumulative Distance histogram. Value of H plotted on y axis, r on x axis. (all figures

can be found on larger scale in Appendix 4, and all code functions are listed in appendix 3).

As the distance is the direct product of the triangulations are devised by the software

(particularly around extrema still), this exhibits the greatest variation of all our methods

presented here (and thus is the least robust); some comparatively substantial range of error would

have to be implemented before potential use in face matching. While this is perhaps a sufficient

metric to test if two known regions of the bone surfaces go together, determining exactly what

portions of the bone fragments those are remains extremely difficult with this method alone, as

plots would need to be generated for every progressively cropped region. Starting from our face

segmentation algorithms with a-priori knowledge on which fragments go together and compiling

distance histograms for each identified face of each fragment would provide a better vantage for

user input matching, but this is not the end solution we are looking for.

2. Surface Area

Averaging and integrating has a smoothing effect (Pottmann et al., 2006), so conceivably the

surface area would be more conducive to face matching. This is where the triangulation begins to

make things wonderful. The area of a triangle is simply half the cross product of two of its legs:

(2.1) |T| = .5( V1 x V2)

V1 and V2 are simply the vectors from one vertex to the other two on the triangle, so we can very

easily approximate the surface area of both the whole fragment surface as well as select regions.

Were the triangulations completely perfect, this could be used for mesh matching. For a 100,000

point triangulation of a sphere of radius 5, the total surface area by this approximation is about

4% off from the actual surface area. Similar to the distance, we can also do cumulative

histograms of this (figure 2), which should be more robust than just a sum of the area in a given

range, here plotting the value of the surface areas within a given radius of a point for every point

(which again is better for verifying face matches than predicting them). However, the the

triangles that make up the triangulations are generally quite similar, so this isn’t always a

fabulous way to go.

Figure 2: cumulative surface area histogram. Surface area within a given radius on y axis

against the radius on the x. Note how similar it is in shape to the volume invariant for the same

bone surface.

3. The Volume Invariant

3.1: Estimating from triangulation

The volume invariant has been by far the most laborious and fruitful development of the

summer. While the total volume of a closed surface is invariant on its own, this tells us next to

nothing about what it can be attached to, so it is not worth calculating. Rather, if we can

configure the volume just on the interior and the exterior within a given distance of the surface,

then we can configure what region goes to what region easily relatively easily and more robustly.

Figure 3: One ball Br shown on the surface of a triangulation. We’re aiming for the volume just

inside the surface, so the portion in the upper half. (See Appendix 2.2 for more representations).

Figure 4: Triangles contained within the radius of this ball.

A method by the name Monte Carlo was devised some years ago for this purpose. For a ball Br

of radius r centered at a given vertex on the mesh, the volume on the mesh’s interior (Figure 3)

is estimated by taking a random sampling of points within r and simply asking “are you in, or are

you out?” (this really should be called the Heidi Klum method). The portion inside can then

merely be taken as the fraction of the points inside times the total volume of the ball. This,

however, is only as strong as the number of points tried, and takes a long time if accuracy is

desired, particularly for large meshes. However, we’ve produced a better way of approximating

that doesn’t require any probability. By the Divergence Theorem, which shows that the surface

area of a closed surface is but the derivative of its volume (just as the area of a sphere is the

integral of its surface area), we can solve for the volume inside the surface by using the triangles

the sphere encompasses (Figure 4). This yields the following equations (as derived by Calder,

2018):

(3.1.1).

Where x is a point in barycentric notation, v is its outward normal vector, and gamma is .5 for

smooth surfaces. A first evaluation of this for a triangulated surface yields the following:

(3.1.2)

(3.1.3)

Where xi yi zi are the coordinates describing a triangle face center within the radius r for a given

vertex, vix , v

iy , v

iz, refer to the components of the outward normal, and |Ti| refers to the surface

area of the triangle (Calder, 2018). For a single ball on a spherical mesh, the portion of the

volume inside the mesh is shown to be an exact quantity (Hulin and Troyanov, 2003):

(3.1.4)

Where Vol(Bp) connotes the volume of the ball of radius t at point p and Bp+ connotes the

volume of the ball inside the sphere of radius R (Hulin and Troyanov, 2003). With this, we can

quite easily test our invariant on a spherical mesh (Figure 5).

Figure 5: a ball on a sphere mesh.

Testing the exact value against the calculated yields the following plots:

Figures 6 & 7: graph of the internal volume of ball of radius r (aka t) on sphere mesh of radius

R. as well as relative error (note axes do not start at 0).

This indicates this first approximation is frankly horrible. While we could just simply add a

constant term, for a bone surface that may not be the case, so we needed to develop something

more accurate. As averaging and integrating have a smoothing effect (Pottmann et al., 2006), we

then try taking (3.1.3) at each vertex of a triangle in range of a given vertex (Calder, 2018):

(3.1.5)

This yields the following plots:

Figures 8 & 9: 3 point approximation for one triangle.

All that’s seemingly changed is the calculation is an overestimate instead of an underestimate by

the same error, which isn’t tremendously encouraging. However, instead of averaging for the

whole triangle, we can subdivide it into 3, average those, and sum for a better approximation of

the whole (Calder, 2018):

(3.1.6)

Testing this yields the following plot:

Figures 10 & 11: 3 triangle subdivisions.

Which is slightly improved, but still poor. Continuing with a similar procedure to (3.6) only for 9

and 12 triangle divisions for each yields the following plots respectively:

Figures 12 & 13: 9 triangle subdivisions.

.

Figures 14 & 15: 12 triangle subdivisions.

Clearly the 12 triangle approximation offers the best correlation yet. As I was preparing for the

lugubrious task of coding 18, 24, 32 triangle subdivisions, I thought I may as well try 4 and 6

triangles:

Figures 16 & 17: four triangle subdivisions.

Figures 18 & 19: 6 triangle subdivisions. Really quite good.

As the six triangle approximation offers the closest and easiest calculation to the actual value,

clearly this is the way to go. While it may at first seem counterintuitive that fewer triangle

divisions would yield a better approximation, the rationale is actually very simple: 6 triangle

subdivisions most effectively reduce the side lengths of each triangle. With the 3, 9, and 12

triangle divisions, l side of each triangle still remained unshortened.

3.2 Analytic Expression for Gamma

Still, a gap remains in the plot. To better try and bridge this gap, we develop an expression for

gamma, which works from the fraction of the adjacent triangles’ (figure 20) surface area inside

the sphere (formal derivation can be found in Calder, 2018).

Figure 20: adjacent triangles for a small ball. Graphic source: (Calder, 2018)

Working in spherical coordinates defined by (Calder, 2018):

(3.2.1)

We use the expression (Calder, 2018):

(3.2.2)

Where:

(3.2.3)

(3.2.4)

(3.2.5)

(3.2.1 – 3.2.5 all from Calder, 2018)

Running gamma and the six-triangle approximation on the sphere mesh produces the following

plots:

Figures 21 & 22: Gamma run with six triangle subdivisions on a sphere.

Note that these are almost identical to the plots received when running just the triangle

subdivisions with .5 for gamma, but that the difference between the simulation and the exact

decreases slightly as the radius increases. As the sphere is as regular a triangulation as possible,

this serves as a testament to it working. Only when run on more irregular bone fragment

triangulations should gamma make a more notable difference.

3.3 Bulk and Adaptive Mesh Refinement

To minimize the number of necessary computations and maximize the accuracy of the volume

invariant, we’ve developed an algorithm for iterative mesh refinement. Working from the error

tolerance of the function (see again Calder, 2018 for formal derivation):

(3.3.1)

Where zi is the coordinate of the triangle face center (in barycentric), vi is still the normal vector

to the triangle, r is still the radius of the ball, epsilon (the curvy E) is the error tolerance between

r and 1 (r is less than 1), Sr is the area of the triangle, and Mi is (Calder, 2018):

(3.3.2)

Where x is the distance from the given center vertex of a ball to a vertex of a given triangle

within the ball (Calder, 2018).

The methods for mesh refinement are still being… refined. Originally we sought a bulk

refinement process for if a triangle would ever need to be refined by producing a large MATLAB

matrix of 1’s and 0’s to determine if a triangle would ever need refining for the whole

triangulation, which was quick and efficient for small things, but for point lists in the order of

28,000 in length (which many fragments even exceed), generating 28,000 by 28,000 arrays of 1’s

and 0’s for refinement exceeded MATLAB’s memory. Ways to do this with only an overarching

loop using shorter cell arrays of points with the range search function are also being explored but

have yet to cumulate in anything fabulous.

Doing adaptive mesh refinement for every single sphere so far has worked for large

triangulations but takes a very long time - eons compare to just implementing the six triangle

approximation. Particularly around sharp corners on the meshes, the program tends to get caught

up over refining, creating hundreds of thousands of triangles if epsilon is too close to the radius

and potentially crashing MATLAB. As the program takes so long to run, we haven’t yet

sufficiently optimized epsilon, but will do so in the coming weeks.

3.4 Spherical Volume Invariant Matching

Figure 23 - How face matching would work with the volume invariant. More spheres

encompassing all valid matches for a face surface would of course be employed for the

orientation. Graphic source: (Hessburg, 2018).

As shown in figure 23, face matching from the volume invariant would follow from trying to

match the interior volume of the sphere for one surface to the exterior volume of another’s on a

point to point basis, similar to outright surface area matching. In an ideal situation of perfect

meshes, for every point on one mesh, the other match would be searched to see where its

spherical volume invariant value matches the other point. From potential matches, the

neighboring points would be examined to see if they match (to both verify the match and obtain

the translation and rotation values for visualizing the match). From the immediate neighbors,

successively more distant neighbors would be examined, and on a sort of “x amount of strike

basis” the match could be verified. Equally doable, the mesh could be traversed from a given

point until the invariant values no longer match, which would also show the extent of the

potential face match on a point to point basis. However, as not all of the vertices of one

triangulation necessarily equate to another by how they are generated by the software, this idyllic

notion is hazier for implementation. While more robust than outright surface area matching, this

still requires a certain range of error and a bout of averaging before potential implementation.

Routes of this are still being explored with averaging the invariant values for a given

neighborhood when traversing the mesh, but have yet to be developed. Finally, cumulative

volume histograms done with respect to both a changing radius and a fixed radius are also

potential routes for verifying a suggested region match, where the operations work in the same

way as for cumulative surface area and distance histograms.

4. Principal Component Analysis and Principal Curvature

The spherical volume invariant is already a robust estimator of mean curvature (Hulin and

Troyanov, 2003); however, it relates no information directly about the individual curvatures,

Gauss curvature, or principal curvatures (Calder, 2018). Just as there are many ways to achieve

the same surface area for a given area around a vertex of a mesh, there are several ways to

achieve the same volume invariant for a mesh, particularly for relatively plain break surfaces.

This is what makes it necessary to obtain an additional invariant. Principal Component Analysis

refers to a broad category of methods for isolating and extracting crucial elements from data to

find stronger and independent correlations from data (Brems, 2017). In the field of discrete

differential geometry, it has found great use in feature extraction and analysis. Here we

implement it to retrieve the principal curvatures, though these can be devised by other means less

robustly. devising the covariance matrix for br, with x again in barycentric coordinates (Calder,

2018):

(4.1)

(4.2)

(Calder 2018) Where:

(4.3)

(4.4)

(4.5)

Where the eigenvalues of MS,r (lamda1 - lamda3) can be used to solve for the curvatures (Calder,

2018):

(4.6)

(4.7)

(4.8)

Thus (Calder, 2018):

(4.9)

Where k1 and k2 are the principal curvatures at a given face center, H is their mean, and O(r7) is

negligible for this computation. These can be estimated in several other ways as well, but this

should stand as the most robust (Calder, 2018). This, paired particularly with the volume

invariant, should stand as enough information to uniquely characterize bone surfaces for face

matching.

Future Work

In the coming weeks, we will begin to implement these invariants for face matching, with the

end goal of creating a suite of measures that can sufficiently characterize and match bone

fragments. For each alone, implementation can quickly become a dance of error ranges to

provide the optimal range of points or plots - not too many as to be effectively meaningless

returns, but not too few (or none) such as to be too stringent ( a very Goldilocks situation).

However, as more and more invariants are applied, this dance should become less a cumbersome

waltz and more a simple shuffle. While we have great confidence in the volume invariant and

principal curvature combination, we do not resist the notion of integrating these method into our

existing curve matching and distance minimization methods.

Bibliography

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Surfaces with MATHEMATICA.” Chapman & Hall/CRC, Taylor and Francis Group.

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nature-study.html.

APPENDIX 1: some more helpful and interesting pictures and quibs:

1. A goniometer, or a protractor by another name. Particularly for small fragments, these bulky

ends yield crude and imprecise measures. Mathematical analyses of break angles is far more

precise; we’re publishing on this imminently as well.

2. Riley O’Neill breaking bones. Bones were broken using four methods: batting, as though the

bone were a baseball bat striking a rock; hammerstone, where a bone is hit with a rock (in some

instances against another rock (these two techniques emanate hominid methods)); rock dropping

(to simulate cave collapses); and fed to hyenas (to simulate, hyenas). Note the meat and marrow

on the bone. In the hot July sun, the smell was quite something. We first did this in a field at the

University of Minnesota near a path to the Mississippi; general passersby were quite revolted at

the bloody mess we were making, but dogs were quite interested.

2.5. Simulating a cave collapse. I would be remiss if I didn’t include this picture.

3. The DAVID laser scanning apparatus. The Database for Annotation, Visualization and

Integrated Discovery has produced an affordable and effective scanning array that functions from

placing objects on a turntable and scanning them with a posable scanning armature.

APPENDIX 2: more helpful plots.

2.1: 3D surface to triangulation:

A. 3D surface

B. Triangulation of 3D surface

2.2: more spherical volume invariant plots:

Note the small ball towards the top. Now imagine one of these corresponding to every point on

the mesh.

APPENDIX 3: MATLAB Code

1. Code I’ve Written:

General Purpose (more or less):

refine_mesh.m - Inputs: T, triangulation, column array k with 0 or 1 for each triangle

corresponding to triangulation’s connectivity list. Refines triangles with 1 into 6 and returns

refined triangulation.

tri_areas.m - returns column array of areas of each triangle of input triangulation.

face_centers2.m - works on triangulations with less than 3 triangles (really petty improvement,

but nice for testing).

face_normal2.m - as refine_mesh.m is completely ignorant towards orienting the points in the

triangle connectivity list in right order (for now), this returns a list of outward face normals

working from a center point, which can be taken as input (handy for extracted, refined

patches), or is automatically calculated from averaging the points. (it seemed far more

ingenious at the time, but I don’t think we’re using refined meshes for patch extractions :/).

Fix_normals.m - takes input triangulation and optional center point (else will determine by

averaging as in face_normal2.m), and returns triangulation with corrected connectivity list.

Should be added to refine_mesh.m. Removes necessity for face_normal2.m.

sphere_mesh.m - generates a triangulation of a sphere with input radius and number of points.

Fabulous for testing.

Distance

Buildhistvec.m -takes point list as input. Plots runs:

cummhist.m - builds

Surface area

surface_area_invariant.m - finds within given range r

surface_area_histogram.m - histogram

Volume

~ SVI = Spherical volume invariant. I got sick of always typing it all out.

svi1.m - using face center for 1

svi2.m - using vertices for 1

svi3.m - divides into 3 triangles, uses vertices

svi4.m - divides into 4 triangles, uses vertices and medians. (pretty ok)

svi6.m - divides into 6 triangles, uses vertices, medians, and face centers. BEST correlation.

svi9.m - divides into 9 triangles using vertices and 4 centers.

svi12.m - divides into 12 triangles.

test.m - tests svi’s (change in text) against exact value. svi6.m won.

svigamma - gamma function for spherical volume invariant. Not all of the above invariant

calculations have had it implemented

svi_ultra - adaptive mesh refinement with gamma. A work in progress.

Principal Curvature

msr.m - generates covariance matrix. Still being tested and developed as eigenvalues are not

lining up.

2. Things I haven’t personally written, but are used by things I’ve written:

General Purpose:

face_centers.m - face centers for things

Principal Curvature:

robust_curvatures.m - alternative way of estimating principal curvatures.

APPENDIX 4: figures on larger scale.

Figure 1: Cumulative Distance histogram. Value of H plotted on y axis, r on x axis.

Figure 2: cumulative surface area histogram. Surface area within a given radius on y axis

against the radius on the x. Note how similar it is in shape to the volume invariant for the same

bone surface.

Figure 3: One ball Br shown on the surface of a triangulation. We’re aiming for the volume just

inside the surface, so the portion in the upper half. (See Appendix 2.2 for more representations).

Figure 4: Triangles contained within the radius of this ball.

Figure 5: a ball on a sphere mesh.

Figure 6: graph of the internal volume of ball of radius r (aka t) on sphere mesh of radius R

using face center approximation.

Figure 7 : relative error of using face centers.

Figure 8: using vertices.

Figure 9: using vertices.

Figure 10: 3 subdivisions.

Figure 11: 3 subdivisions.

Figure 12: 9 subdivisions.

Figure 13: 9 subdivisions.

.

Figure 14: 12 subdivisions.

Figure 15: 12 subdivisions.

Figure 16: 4 subdivisions.

Figure 17: 4 subdivisions.

Figure 18: 6 subdivisions.

Figures 19: 6 subdivisions.

Figure 20: adjacent triangles for a small ball.

Figure 21: Gamma with six subdivisions.

Figures 22: Gamma with 6 subdivisions.

Figure 23: spherical volume invariant matching.