The Journal of the Society for Art and Science, Vol. 21, No. 3, pp. 136 – 145 (2022)

Evaluation Method for Shape Similarities between Archaeological
Unorganized Point Cloud
Amartuvshin Renchin-ochir1) , Zorig Badarch1) , Kouichi Konno2)
1)

School of Engineering and Applied Science,
National University of Mongolia, Ulaanbaatar, Mongolia
2)
Graduate School of Science and Engineering, Iwate University, Iwate, Japan
amartuvshin@seas.num.edu.mn

Abstract
Researchers assume that the bodies of several Tara statues may have been molded from the same
template. Therefore, researchers are eager to study the shape similarities between the Tara statues.
Motivated by the archaeologist’s requirements, this study presents an efficient approach to evaluate
shape similarities between point clouds of two different Tara statues right-arm models. The evaluation
approach consists of two main stages. The first is the reconstruction of individual body parts without
decoration using a B-spline surface approximation technique. In previous reconstruction process, thesurrounding surface selection parameter was manually specified. If the manually applied parameteris inadequate, it reduces the accuracy of the surface approximation and hole-filling, resulting in
reduced accuracy of the evaluation result. Therefore, in this study, the surrounding surface selection
parameter is automatically determined based on the Golden-section-search algorithm. The second
stage evaluates the shape similarities between two different point clouds. The evaluation method
considers a simple point-to-surface distance metric method with overlapping-surface detection andwhen compared with generic point-to-point distance metric method, the evaluation results proved thatthe proposed methodology was reliable and effective for this application. The approximation accuracy
affects the evaluation result when the reconstruction method fills the gap by adding new points throughthe approximated surface. Therefore, in this study, the shape similarities were investigated between
the two arm models with holes, immediately after separating the decoration parts.

1 Introduction
Mongolian government maintains remarkablecultural heritages of Tara statues.
The
great artist Zanabazar created those statues
in the 17th-century, known as “White Tara”,“Akshobhya”, and “Vairochana” are shown in
Figure 1.
Many artisans and archaeologists are keen to
study sculptural techniques. They concurredunder the assumption that the several parts of
the Tara statues might be molded from the same
template. To verify this assumption, they want
to examine shape similarity between the body

shapes of the Tara statues. However, this is adifficult with traditional archaeological devices.
Furthermore, working on the Tara statues’
original model is prohibited as the statues areunder government protection. Fortunately, theMongolian government has created a digitallyscanned model of the Tara statues.
Those
digital models were constructed by the method
presented in [1]. The authors measured statues
from various angles to cover the whole surface
with the “Konica Minolta vivid 9i” laser scannerin [1]. The scanning accuracy was 0.2 mm.
Totally 235 scans were required to construct acomplete White tara model with approximately– 136 –

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2 Related works

(a)

(b)

(c)

Figure 1: (a) White Tara, (b) Akshobhya and (c)Vairochana (Gilt, bronze, 68 × 45 × 45cm).
(Photographs from “www.zanabazarfam.mn”
and “www.zanabazar.com”)

29 million points. In the case of Akshobhya,
the total number of scans was 194, and 225
scans were for Vairochana. The complete modelof the Akshobhya contains approximately 19
million points and the Vairochana model contains
approximately 20 million points. The scanning
process took about two days per statues. Using
those scanned models, it is possible to evaluate
the shape similarities between body parts of thetwo statues.
As the statues contains various decorations on
its body surface, it is difficult to evaluate theshape similarities. Therefore, in the previous
study [2], we suggested the two methods:
complex hole-filling and the point-cloud-based
shape similarity evaluation method.
In thecomplex hole-filling method, the surrounding
surface selection parameter is specified
manually. If the manually selected parameteris inadequate, it reduces the accuracy of thesurface approximation and the hole-filling.
Finding a sufficient surrounding surface selection
parameter manually is very time consuming
process.
Therefore, the surrounding surface
selection parameter is automatically determinedbased on the golden-section-search algorithm.
The two arm parts that contain artificiallygenerated points as hole-filling results were usedin the previous evaluation. The evaluation result
include the approximation error.
Therefore,
in this study, the shape similarity of two arm
models were examined with the holes on thesurface instead of artificially generated points tofill the holes.

Reconstructionoftheindividual,
undecorated body parts:
The first step
was to detect decoration from the body using
the point cloud model. The segmentation of
decorated surfaces has rarely been studied [3, 4],
and these methods cannot be applied to finecrafted decorative surfaces.
The method ofsegmenting the decoration was developed in aprevious study [5] with the decorations obtained
from the shoulder and upper arm. This isfollowed by an unusual hole in the surface of
the Tara statue’s right hand. Several hole-fillingalgorithms have been suggested, including
Guo Xiaoyuan’s algorithm was for simple
hole-filling [6] and Wang Jianning’s algorithm [7]for complex holes with an island, covering only
half of our problem, which consists of holeas well as gaps. Another study of hole-filling
was suggested by Amitish [8], which can only
fill simple holes on highly variable surfaces.
A review of these related studies [6, 7, 8]
showed that their methods need modification.
Our suggested method is a uniform B-spline
surface [9]. It is important to determine theexact area surrounding the hole. An improved
version of the method [9] which automaticallydetermines sufficient surrounding area, was used
to reconstruct the holes and gaps of the model,
in this article.
Evaluation of the shape similarity :Measuring similarity between 3D objects is
widely used to measure the global similarity
between two complete objects. However, only
a few methods are proposed for measuring thesimilarity between 3D point clouds [10]. In
point-cloud-based methods, accurate registration
methods are widely used to create a complete
model using separate point clouds captured byscanning devices from different points of view
[11, 12, 13]. Existing registration algorithms can
be classified into two types: pairwise registrationand multi-view registration.
Each of thesemethod attempts to evaluate the optimal hardtransformation, which maps the source point
cloud to the target point cloud [13].
Themost famous of them is the Iterative Closest– 137 –

The Journal of the Society for Art and Science, Vol. 21, No. 3, pp. 136 – 145 (2022)

Point (ICP) algorithm and its variants basedon iterative minimization of error metrics toestimate point correspondences between two or
more point clouds [14]. The study [14] divided the
ICP algorithm process into four steps: Selection,
Matching, Rejection, and Alignment.
Whenpoint clouds were aligned, the similarity between
the two surfaces can be evaluated. In otherwords, the calculated distances between thecorresponding regions of the surface may indicate
the similarity of the shape between the cloudsof points, as in the case of considering thesimilarities in the parts of the body of thestatues of Tara. For example, a point-to-point
error metric method was introduced, which uses
the distance between the source point and theplane described by the target point and its local
surface normal [11, 14, 15, 16]. Therefore, thegeneralized-ICP algorithm, which generalizes themetric of the point-to-point and point-to-planeerrors, was introduced in [12].
With the advantage of the generalized-ICP
method, our method implemented a simple andeffective method for evaluating the similarity ofshapes between two arm models of the Tarastatues. However, another study [17] evaluatedthe point-to-point distance information, in whichan attempt was made to assess the similarity
between the two types of objects obtained bydifferent data acquisition systems. Therefore,
as part of this study, we also compared our
implemented method with the point-to-point
distance-based estimation method. However,
the evaluation results include the approximation
error. Therefore, in this study, we examined theshape similarity of the two arm models with holeson the surface instead of artificially generated
points to fill the holes. The methods presented
in this paper do not consider very complex parts,
such as the crown on the head, earnings or faces,
since they are completely different on all thestatues.

3 Methodology
Archaeologists have focus on the arm modelsof the Tara statues because the arms lookvery similar. The experimental arm models

have a bracelet and chain-shaped accessories on
their surfaces, as shown in Figure 1.
The
general process flow of the reconstruction of
the individual, undecorated body parts, andshape similarity evaluation process is summarized
in Figure 2. The accessories in the sub-areaof the target object have been referred to as“decoration” illustrated by the green shape in
Figure 2, and the other body surfaces havebeen referred to as “background”. The input

Figure 2: The process flow of the evaluation method.

point cloud of the arm model is separated into
decoration and background point clouds using
the segmentation method. After decoration andbackground parts were separated, large holes andgaps were appeared on the background; thus, thecomplex hole-filling method [9] based on B-spline
surface fitting was used to restore a contiguous
background surface. The proposed hole-filling
methods were suitable to apply for three types ofholes: a simple hole, a hole with an island and agap separating the background into two surfaces.
A uniform B-spline surface with two
topological parameters, u and v, in Figure
3, was used. First, regularly shaped either“rectangular ” or “cylindrical ” surfaces Si,j which
encircled the hole boundary, were defined as
represented by the green and blue surfaces in
Figure 3. A rectangular surface is shown in this– 138 –

The Journal of the Society for Art and Science, Vol. 21, No. 3, pp. 136 – 145 (2022)

figure as an example.
When the outer area is defined, the following
steps are completed in the method [9].
1. The outer area size was expanded bymanually initialized parameter to obtainenough points around a target hole. The
expanded outer area is illustrated in thegreen, blue, and red surfaces in Figure 3.
p4

inside pointsof outer area
p3

8

7,3

p2
p1

u
process 2

process 3

process 4

Figure 3: Process diagram of obtaining the inside
points of the outer area.

2. The inside points of the outer area, as
illustrated by the yellow points in Figure 3,
were obtained from the background pointsset.
3. According to the obtained inside points, a
cubic B-spline curve approximation, withknot insertion was then applied for every fourboundaries, according to the instructions
given in chapter 9 in [18].
4. The boundary curves were approximatedagain using the merged knot vectors, andthen the new control points of the boundary
curves were used as the outer control pointsof the B-spline surface.
5. Since the outer control points and knot
vectors are known, the B-spline surface
can be easily approximated using theleast-squares approximation technique,which minimizes the average distance fderived by Eq. (1). Newton’s method wasused to calculate the normal vector between
the B-spline surface and the corresponding
points. The average distance f is referred toas precision. Where, Qs (0 ≤ s ≤ t) denotes
the inside points of the outer area, obtained

by “Step 2 ”. The number of the inside
points (t + 1) depends on the outer area size.
S(us , v s ) is the corresponding points on theapproximated B-spline surface.
Pt|Qs − S(us , v s )|2
f = s=0(1)
t+1
When the outer areas are successfully
approximated, locally uniform points inside
the hole can be generated to complete thebody surface. The standard deviation of the
background point cloud was used to generate thelocal uniform points.
To determine a sufficient expanding
parameter , the parameter used to expand theouter area influences the accuracy of the surface
approximation result. If the outer area is too
small, the approximated B-spline surface might
not efficiently be fitted on the surface arounda hole. In contrast, if the outer area is toolarge, the number of the control points may
be inadequate for approximation. Therefore, asufficient expansion value should be considered
for each hole. Hence, an improved version of themethod [9] is presented and used in this study.
Let j be the expanding parameter of the outerarea. In the example shown in Figure 3, theexpanding parameter j is equal to 1. The
golden-section search algorithm [19] is used tofind the sufficient expanding value, according
to the minimum precision of f , which variancedepends on j. To find the local minimum,
the golden-section search algorithm requires thesearching interval, which is 1 ≤ j ≤ 20 in ourcase. That interval was sufficiently wide to findthe sufficient expanding parameter. For any jvalue, the precision f is calculated by steps 1 to
5 described above.
Once the arm parts from the different statues
were available in an undecorated form, the bodypart shape similarity evaluation process could
proceed on the complete body surface.
Evaluationofshapesimilarity :Archaeologists have been eager to examine
similarities between Tara statues parts, asmentioned earlier, therefore, we developed asimple and efficient evaluation method in theprevious study [2], which displays a color insteadof the distance between two surfaces.

– 139 –

The Journal of the Society for Art and Science, Vol. 21, No. 3, pp. 136 – 145 (2022)

p1

q1
and

o The source and target clouds were aligned
using the generalized-ICP algorithm
proposed in [12]. The median of the twopoint clouds was used as the initial position
for alignment.
o The distance between two surfaces wasestimated using the point-to-plane distancemetric in an overlapping region, which wasderived using the method proposed in thissection.
o The distance between two overlappedsurfaces was visualized using the color
according to the color mapping function.
The distance between two aligned surfaces of
the arm models was derived using the improved
point-to-plane error metric in the study [2]. The
distance between two surfaces was derived fromthe distance between the source point, pk , andthe plane described by target point qi and its
local surface normal. The points pk and qi
are the corresponding nearest neighbors. The
two surfaces matched locally because the arm
models were separated from different statues;
therefore, some regions near the armpits have low
density and might have been missed. For thisreason, overlapping surfaces must be included
when estimating shape similarity.
Let li be a line defined by the point qi andits normal vector to the local surface normal.
Then, the distance si between point pk and li
can be easily calculated. For example, distance s1
between point p1 and the line l1 is shown in Figure
4, where if the distance si ≥ 3 × sP , point qi

f (di ) =0, 255, 255 · (1 − L −L ) ,
2 1

d −L
255 · Li −L2 , 255, 0 ,
3
2


d −L
255, 255 · (1 − Li −L3 ), 0 ,4

3

if L1 ≤ di < L2
if L2 ≤ di < L3(2)

if L3 ≤ di ≤ L4

Where the di (1 ≤ i ≤ n) is the distance, L1 =min(di ) is the minimum, L4 = max(di ) is the
maximum, and L2 = d̄i is the mean of the di .
Furthermore, another distance level, L3 , is equal
to L2 + (L2 − L1 ). The color mapping function
and RGB vectors are shown in Figure 5.
The comparison between our implemented
evaluation method and the distance between two
surfaces estimated by the point-to-point distancemetric is presented in the following Section.
1.5
1.4
1.3
1.2

Steps of the golden-section search

1.1
1
0.9
0.8
0.7

Step 2In summary, our improved evaluation method
was completed following these simple steps.

must be dealt with in a non-overlapping surface.
Where sP is the standard deviation of the P point
cloud distribution, and the multiplier “3” wasdetermined by our experience, according to thenormal distribution of the standard deviation.
Once, the overlapping surface has been
determined, the distance di between overlapping
surfaces can be visualized with a color mapping,which was formulated using Eq. (2).
(di −L1

Step 6

calculations

Step 5

distance

Figure 5: The color mapping used to visualize
distance.

Step 1qi
Figure 4: Overlapping
restrictions.

l1

Step 4dipk

Step 3s1

si

Precision, f (mm)

li

0.6
0.5
0

2

4

6

8

10

12

14

16

18

20

Outer-area expanding parameter j

Figure 6: Graph of precision, f for each expanding
parameter j.

4 Experimental results
The constructed Buddha models contain many– 140 –

The Journal of the Society for Art and Science, Vol. 21, No. 3, pp. 136 – 145 (2022)

decorations covering the whole body. According
to the archaeologists’ interest, we focus on the
body parts which might be molded from the same
template except for parts like the head, face andthrone in this experimental result. Because thoseparts are the special feature of each statue. Since
segmentation and hole-filling algorithms used inour study are applicable on cylindrically shaped
manifold surfaces, it is possible to verify thereliability of our methods on a few parts of thebuddha models such as the arm and chest. The
arm parts of each buddha seem more similar
compared to the chest, then, we consider the arm
parts in this study.
The graph of precision f is created for eachexpanding parameter j to confirm the efficiencyof the improved method, as shown in Figure
6. The vertical axis indicates the precision,f , derived by the Eq. (1) for the expanding
parameter j ∈ [1, 2, ..., 20] on the horizontal axis.
A sufficient expanding value was found by the
golden-section search algorithm at the 6th stepof the iteration. In the first step, j was 8, andthen 13, 5, 9, 10, and 11, according to the resultof the golden-section algorithm shown in Figure
6. The sufficient expanding value was 10 and theminimum precision was approximately 0.56 mm.
For the shoulder shown in Figure 7 (a), theouter area shape was rectangular.
For theupper-arm, the precision was approximately 0.24mm, and the outer area shape was cylindrical.
The generated points were uniformly distributed,
and the maximum number of control points was“7” in both parametric directions of the fitted
B-spline surface, according to the results from theknot insertion algorithm.
White Tara and Akshobhya statue arm
models were reconstructed with automaticallydetermined parameters using the improved
method.
The solid models of experimental
models are shown in Figure 7 for clear
visualization.
However, the surface fitting
approximation was better than the previous
result presented in the study [9]. The precisionof the approximated surfaces generated using theprevious method and improved method has beensummarized in Table 1.

(a) Reconstruction result (b) Reconstruction resulton
White
Tara’son
White
Tara’sshoulderupper-arm(c) Reconstruction result (d) Reconstruction resultonAkshobhya’son
Akshobhya’s
shoulderupper-arm

Figure 7: Solid models showing original surface withdecoration and its reconstructed bodysurfaces.
Table 1: precision of the approximated surfaces

Approximatedsurfaces
White Tara’s
shoulderWhite Tara’s
upper arm
Akshobhya’s
shoulderAkshobhya’s
upper arm

precision (mm)method [9]our method

0.73

0.56

0.48

0.24

0.70

0.54

0.28

0.28

4.1 Evaluation of the shape similarity
The reconstruction results made it possible toevaluate the similarity of the arm models. A
summary of the two reconstructed point clouds
has been presented in Table 2. The density of
the Akshobhya’s arm model was much higher
than White Tara’s arm model, according to thestandard deviation. As mentioned earlier, thescanned point cloud was registered from themultiple views. Therefore, some surfaces havea higher density, depending on the view points
used to reconstruct the surface. White Tara’s arm
model was nominated as the source cloud and
Akshobhya’s arm model was referred to as thetarget cloud. The results of the generalized-ICP– 141 –

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Table 2: Statistics of the input point clouds forevaluation

input cloudWhite Tara’s
arm
Akshobhya’sarm
5.02(mm)pointsstandard
deviation(mm)

59392

1.04

3179710.016
1.72(mm)15.84(mm)

5.02
1.36
0.28
0.68
0.00004

(a)

1.83
0.92
0.00610.14(b)2.38x10-7(c)Figure 8: (a) Color mapping distance results after
applying our implemented method, (b)distances calculated by point-to-point error
metric, and (c) differences between the
distances portrayed by (a) and (b).

algorithm were completed after 50 iterations, andthe convergence rate was approximately 1.89 mm.
In this experiment, a low convergence rate is
expected as two models belonged to different
statues.
The result of the implemented color mappingis shown in Figure 8 (a), where distances
between the two surfaces were derived using
the point-to-plane error metric.
Then, thecolor mapping is applied to the overlappedsurfaces where the blue points represent the
non-overlapping surfaces. The minimum distance
between the two surfaces was approximately
0.04 × 10−3 mm, and has been illustrated
by the cyan color. The mean distance wasapproximately 0.68 mm and is illustrated by thegreen color. The vertical color bar shows color
mapping representation of distance in which the
distance from 1.36 to 5.02 mm is represented byyellow to red color.

We applied the same color mapping to the
distance between two surfaces is derived bypoint-to-point error metric without considering
overlapping regions.
However, the distancegreater than 5.02 mm was represented by black.
The results are presented in Figure 8 (b). The
maximum distance was approximately 15.84 mm.
If the non-overlapped surface regions were larger,
the maximum distance would have been shown tobe greater using the point-to-point error metric;
as this was not the case, it is evident that our
improved evaluation method was able to visually
represent the distance precisely.
Next, the difference between the distances
calculated using our method and thepoint-to-point error metric was calculated
and illustrated using color mapping, as shownin Figure 8 (c).
The minimum difference
was approximately 2.38 × 10−7 mm, and the
maximum was 1.72 mm around the boundaryof the surfaces as illustrated by red color in
Figure 8 (c). The point cloud density differences
contributed to these differences.
The evaluation result in Figure 8 includes
the result of the approximation error, especiallyaround location where once decoration existed.
Therefore, we examine the similarity between thearm models with holes on the surface instead of
artificially generated points to fill the holes. The
evaluation result is presented in Figure 9.
The maximum distance between two surfaces,
calculated by our proposed method in Figure 9(a) was smaller than that shown in Figure 8(a). However, the maximum distance between
two surfaces calculated by the point-to-point
error metric in Figure 9 (b) was larger thanthe maximum distance shown in Figure 8 (b).
This is because the two models had large
holes and had larger non-overlapping surfaces.
Therefore, our proposed evaluation method
is robust for non-overlapping surfaces.
The
differences between the distances calculated using
our method and the point-to-point error metric
were illustrated using color mapping shown in
Figure 9 (c).
Figure 10 shows histograms of the distanceestimated by our work and by the point-to-point
error metric.
The horizontal axis indicates– 142 –

The Journal of the Society for Art and Science, Vol. 21, No. 3, pp. 136 – 145 (2022)

17.89(mm)4.12(mm)

1.77(mm)evaluation, as in Figure 9 (a). The average
distance between the two hole-filled surfaces wasless than 0.68 mm, and in the undecorated
arm model was 0.55 mm which is the mostimportant factor for archaeologists to verify theirassumption.

5 Conclusions

1.09
4.12
0.55

7.2x10

2.05

(a)

-6

1.03
0.01

(b)

0.19
0.09
4.7x10-7(c)Figure 9: (a) Color mapping distance results after
applying our implemented method, (b)distances calculated by point-to-point error
metric, and (c) differences between the
distances portrayed by (a) and (b).
105
histogram of distance estimated by point-to-point
histogram of distance estimated by our work
maximum distance between sufaces in point-to-point
maximum distance between surfaces in our work

104

103

102

In this study, we improved the hole-filling method
using the Golden-section-search and made shapesimilarity on two undecorated arm models forbetter evaluation. Undecorated complete arm
models were reconstructed using the improved
hole-filling method, which helped to achieve goodresults to evaluate shape similarity. Therefore,archaeologists can use it per their requirements:
the method allows them to easily analyze theTara statue’s body shape similarities, based on
the surface differences. We will use feedback fromarchaeologists to improve this evaluation method,
and will test the methodology on the other partsof the Buddha statues, such as the chest andelsewhere, in future.

101

100

2

4

6

8

10

12

14

16

Figure 10: Histograms showing distances estimatedusing our method and using point-to-point
error metric.

Acknowledgment
The author would like to thank the MJEED
JR14B16 project for financial support.

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The Journal of the Society for Art and Science, Vol. 21, No. 3, pp. 136 – 145 (2022)[19] R. P. Brent, Algorithms for minimization
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automation.

Kouichi Konno

Amartuvshin Renchin-Ochir

Amartuvshin Renchin-Ochir is an associate
professor at National University of Mongolia.
He received the B.Sc. and M.E. degrees fromthe National University of Mongolia in 2007
and 2010.
He earned his Dr.
Eng.
in
Design and Media technology from Graduate
School of Engineering from Iwate University
in 2019.
He worked on a lecturer of
School of Engineering and Applied Sciences
at National University of Mongolia from 2012
to 2016. His research interests include point
cloud processing, geometric modeling, computer
graphics, computer vision and robotics.

Kouichi Konno is a professor of Faculty ofEngineering at Iwate University. He received
a BS in Information Science in 1985 from the
University of Tsukuba. He earned his Dr.Eng.
in precision machinery engineering from the
University of Tokyo in 1996. He joined thesolid modeling project at RICOH from 1985 to
1999, and the XVL project at Lattice Technology
in 2000. He worked on an associate professorof Faculty of Engineering at Iwate University
from 2001 to 2009.
His research interests
include virtual reality, geometric modeling, 3D
measurement systems, and computer graphics.
He is a member of EuroGraphics.

Zorig Badarch

Badarch Zorig received the B.Sc. and M.E.
degrees from the Mongolian University of Scienceand Technology in 2004 and 2007. He worked
as a lecturer School of Engineering and Applied
Sciences at the National University of Mongolia
from 2012 till now.
His research interests
include geometric modeling, computer vision, and– 145 –