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Computer-Assisted Reconstruction of Virtual
Fragmented Cuneiform Tablets
Sandra I. Woolley
Luis Hernandez Munoz
Electronic, Electrical and Systems Engineering,
University of Birmingham, Edgbaston,
Birmingham, B15 2TT, UK
Digital Humanities Hub,
University of Birmingham, Edgbaston,
Birmingham, B15 2TT, UK
School of Computer Science
University of Nottingham Ningbo China
199 Taikang East Road, Zhejiang Ningbo,
Centre for Creative Content and Digital Innovation,
University of Malaya,
50603 Kuala Lumpur, Malaysia
60629 Frankfurt/Main, Germany
Abstract—Cuneiform is one of the earliest known systems
of writing consisting of wedge-shaped strokes forming signs im-
pressed on clay tablets. Excavated cuneiform tablets are typically
fragmented and their reconstruction is, at best, tedious but more
often intractable given that fragments can be distributed within
and between different collections. Digital archives relevant to
cultural heritage, such as the Cuneiform Digital Palaeography
Project – and the Cuneiform Digital Library Initiative ,
, now make richly annotated artefact media available to wide
populations. Similarly, developments in computer-aided three-
dimensional reconstruction methods offer the potential to assist
in the restoration of broken artefacts. In this paper, a system
for providing computer assistance for identifying and orientating
matching fragment pairs is described. Metrics to grade the quality
and likely signiﬁcance of potential matches are also described.
Finally, the results from experiments with scans of laboratory
fabricated tablet fragments and genuine fragmented cuneiform
tablets are reported.
Keywords—Computational geometry, Iterative methods, Cost
function, Search problems, Computer graphics
Unlike jigsaw puzzles of thousands of pieces, which com-
puters can now easily solve , artefact fragments such as
broken pottery (potsherds) or cuneiform tablet fragments are
more complex three-dimensional objects and may belong to
an unknown number of complete or incomplete “puzzles”
whose surviving pieces may be eroded and may no longer ﬁt
well together. Computer-aided reconstruction of archaeological
fragments has been an active area of research in recent years
although most published work has been speciﬁc to the joining
of potsherds . A survey of computational reconstruction
The authors gratefully acknowledge the support of The Leverhulme Trust
research grant (F000 94 BP), and the multidisciplinary support of the Univer-
sity of Birmingham’s Digital Humanities Hub.
methods  observes that while approaches are sophisticated
and computationally efﬁcient, none claim to be ready for
deployment as archaeological tools. Impediments to automated
reassembly, aside from the practical difﬁculties associated with
obtaining three-dimensional scan sets, include the extremely
difﬁcult search problems, the lack of surface information inclu-
sion with object geometry and, signiﬁcantly, the resolution of
issues associated with large numbers of false-positive matches.
More recently, a hybrid human-computer approach has been
proposed to reﬁne computer and archaeological knowledge
bases with geometry, texture and feature knowledge to resource
match performance . Tested on sculptural fragments this
approach demonstrated good performance.
Adopting a hybrid human-computer approach, we have
designed a virtual collaborative environment for the recon-
struction of cuneiform tablets . A complete description
of the environment is detailed in a separate publication .
The design involves computer assistance on a number of
levels from simple quantitative join-quality feedback and semi-
automated alignment of nearby fragments (a form of “snap to
best ﬁt”) through to fully automated searching and joining of
fragments. Though, even in the latter case, the computer assists
human effort; human intervention is necessary to validate
candidate matches and eliminate false-positives.
Cuneiform fragment matching differs from problems such
as mosaic or potsherd matching in that tablet fragments have
signiﬁcant free-form variation in shape in all three dimensions.
However, to reduce the search problem, some assumptions
can be made about the expected size, shape, texture and
surface curvature properties of the completed tablet. In making
such assumptions, we can create additional criteria for the
assessment of match quality beyond those generally applied
in free-form 3D reconstruction algorithms –.
We propose several criteria for match quality assessment:
the distances between pairs of sampled points on the adjoining
surfaces, the interior area of the adjoining surfaces, the degree
of interlocking of the join, the continuity of the written surfaces
either side of the join, and the continuity of the slope of
the written surfaces either side of the join. These criteria are
used both for iteratively ﬁnding optimal matching fragment
orientations and also for ranking candidate matches; false
positives should be rejected and the most promising joins given
priority when presenting to users for validation. Typically, in
previously published reconstruction algorithms, match quality
is assessed by a more limited set of criteria such as measures of
inter-surface distances and size of overlapping area , 
but we have found, in this application, that the introduction
of additional criteria aids in the prioritisation process required
prior to human validation.
When a match is attempted between a pair of fragments,
an iterative process is used to ﬁnd the optimal translations
and rotations needed to minimise a cost function designed to
quantify the match quality. This process is repeated across
a search-grid of initial conditions in order to guarantee the
globally optimal match is identiﬁed. During the iterations,
the translations and rotations are applied to the model-view
matrix used by the graphical processing unit (GPU) of the
computer. This efﬁciently yields uniformly sampled depth
maps of the fragment from the irregularly sampled model data
of the archived scan . The cost function to be minimised is
calculated through analysis of the depth maps of the two frag-
ment surfaces. Although the use of GPU-generated projective
depth maps has been reported in previously published related
work , , the iterative algorithm and the associated cost
function presented here are novel and, while developed for
cuneiform fragments, could be applicable for other fragmented
II. MATCHING PROC ES SI NG
Upon attempting to match a pair of fragments, six degrees
of movement must be considered. If one of the fragments
is considered to be ﬁxed in space, the other may require
translation in three dimensions as well as rotation by the three
Euler angles in order to form a join. One of the degrees
of freedom can be eliminated if a restriction is imposed
that in any potential solution, the fragments must touch one
another but must not intersect. With ﬁve degrees of freedom
remaining, the search space is still prohibitively large for an
exhaustive search and an iterative approach is required. Due
to the intricate geometry of fragment surfaces, convergence
to a globally optimal solution cannot be easily guaranteed
so multiple iterations must be performed using a search grid
of different initial conditions. The ﬁrst part of our algorithm
involves some pre-processing applied to the fragment scans in
order to reduce the scale of this search.
When a tablet fragment is scanned, the ﬁrst part of our
process is to calculate the minimum-volume oriented bounding
box  of the vertices and to reorientate the vertex coordi-
nates so that the bounding box is aligned parallel with the x,
yand zaxes. For most cuneiform fragments of interest, we
expect at least one of the surfaces to contain script and, after
Fig. 1. Two virtual cuneiform tablet fragments with oriented bounding boxes.
The fragments, W 18349 (above) and Wy 777 (below) , are part of a letter
written in the Neo-Babylonian empire (626-539 BC, around the time of King
Nebuchadnezzar) and were found in the ancient city of Uruk (near modern-day
reorientation, this surface, being relatively ﬂat, will be roughly
parallel with one side of the bounding box. This side can be
identiﬁed using the statistical properties of the surface textures
or, more reliably, manually at the time that the fragment is
scanned. After identiﬁcation the fragment is rotated so that
the inscribed surface faces forwards as illustrated by the two
fragment scans shown in ﬁgure 1.
After these rotations, only the non-script sides of the
fragment are considered for potential matches. Furthermore,
when considering a pair of fragments, only the orientations of
the boxes with the inscribed surfaces facing forwards will be
considered. This reduces the number of possible permutations
to consider from 144 (each of the six faces of one box
presented to each of the six faces of the other with four possible
rotations of 0◦,90◦,180◦and 270◦) to just 16 (four faces of
one box presented to four faces of the other with no rotations
allowed). Also, because it is usually possible to determine the
direction of writing on the fragments, the number of possible
combinations can be further reduced to just four.
B. Pair-wise Matching
For each pair of faces, one of the fragments must be
translated and rotated to ﬁnd the best matching position. Three
rotations are allowed corresponding to the three Euler angles.
Translations are made in the left-right direction (xoffset) and
in the front-back direction (yoffset). An up-down translation (z
offset) is also made but this is calculated from the rotations and
the xand yoffsets such that the restriction that the fragments
must touch but not intersect is satisﬁed. There are, therefore,
only ﬁve independent positional parameters (three angles and
two offsets) that require optimisation.
Fig. 2. Example illustration of the depth map calculation performed by the GPU. (left) Depth map of the top face of the ﬁrst fragment, (centre) depth map
of the bottom face of the second fragment, (right) joined fragments and their summed depth map, z, after min(z) has been subtracted. The uniform, near zero,
depth values of the summed depth map indicate a constant distance between point-pairs and, therefore, a good quality match.
The approach we have adopted to optimise these ﬁve pa-
rameters for a given pair of fragments is based on the analysis
of two-dimensional projective depth maps of the surfaces to
be joined. The calculation of depth maps (also known as
z-buffers) is a part of the hidden-surface removal algorithm
performed whenever the GPU of a computer renders a three-
dimensional object. GPUs contain specialised processing units
within their architecture to efﬁciently perform the geometric
calculations needed to generate a z-buffer and the results can
be accessed via OpenGL . This allows us to generate
a uniformly sampled grid of depths of the surface of the
join from the irregularly sampled vertex data of the object
model. In operation, the vertex and face data of each fragment
are stored in array buffers in GPU memory. Depth maps are
then generated on each iteration by manipulating the model-
view matrix of one fragment before rendering the depth data
to an off-screen framebuffer object. Similar approaches have
been reported in previous work ,  but the subsequent
analysis of the depth map data presented here is novel.
Due to the intricate geometry of the surfaces of the join,
convergence to the globally optimal solution cannot be guar-
anteed by iteratively optimising all ﬁve of the rotation and
translation parameters. However, we have found that if the
xand yoffsets are ﬁxed, the three rotations can be reliably
estimated iteratively. This is due to the orientation applied
during the pre-processing which ensures that at least two of
the three angles are relatively close (within a few degrees) to
the optimal solution because of the approximate alignment of
the planes of the inscribed surfaces.
Although it is still necessary to systematically search
through a two-dimensional grid of possible xand yoffsets,
the reduction in computational complexity from the original
ﬁve-dimensional search space is signiﬁcant. For each potential
x, y offset in the search, the iterative estimation of the optimal
rotations uses a Nelder-Mead optimisation . The optimi-
sation aims to minimise a cost function which is calculated
at each iteration using the algorithm outlined in the following
function cost(x, y, φ, θ, ψ )
fragment0.rotate(φ, θ, ψ)
z←fragment0.depthmap() + fragment1.depthmap()
// Bring fragments into contact without intersecting
// Calculate average cost function of all sampled points
for n= 0 to N−1do
xand yare the offsets which are ﬁxed for each optimisation;
φ, θ and ψare the Euler rotation angles which are to be opti-
mised. The GPU calculates the depth maps for each fragment
by rendering an orthographic projection forming a uniform
grid of sampled distances from a ﬁxed observation plane to
the surface of the fragment. The rotations and translations are
applied to the model-view matrix of one of the fragments prior
to generating its depth map. The two depth maps are summed
and the minimum summed depth value subtracted from all
depths, thus satisfying the condition that the fragments must
touch but not intersect. For each element of the summed depth
map, z, the residual offset, zn, is converted to a cost-per-
sample by the function f(zn)to assess the goodness of ﬁt.
This function is averaged over the Ndepth samples to give
the overall cost function. Examples of the depth maps formed
from scanned laboratory fabricated tablet fragments are shown
in ﬁgure 2 and the geometry of this process is illustrated by the
side view in ﬁgure 3. Note that in both ﬁgures the fragments
form very good matches and the summed depth map shown in
ﬁgure 2 has a near-uniform value close to zero over the entire
surface of the join.
The reliability of the offset search and of the iterative
rotation optimisation are highly dependent on the function,
f(z). Examples of functions used in similar applications
include the Mean Square (MS) of the distances  and the
Largest Common Pointset (LCP) metric . Our initial testing
Fig. 3. Side view of the depths used to calculate the cost function. Note that
for a perfect match, all depth-pairs sum to the same value.
Fig. 4. The piecewise linear cost function used in our experiments. The
non-intersecting constraint means that zcan never be negative, hence only
positive values are shown.
with laboratory fabricated tablet fragments gave unsatisfactory
results with these methods. The MS cost function did not
perform well when a signiﬁcant proportion of the points in
the depth maps cannot meet due to chips and erosion of the
surfaces. Such features are over-emphasised by the squaring
process and lead to poor convergence and large numbers of
false-positive matches. The LCP approach avoids this problem
but requires a relatively ﬁne depth map resolution to converge
reliably. A linear cost function was also employed to attempt
to suppress the problems encountered with the mean square
approach and gave improved results but still had a tendency to
produce false-positive matches. The cost function we propose
as an alternative is a Piecewise Linear (PL) function designed
as a compromise between a linear function and the LCP
approach. The function, f(z), for the PL function is of the
f(z) = zfor z≤
γz +(1 −γ)for z > (1)
where is a threshold distance equivalent to the one used in
the LCP metric  and γ1is a constant. The form of
the function is illustrated in ﬁgure 4. In our experiments, the
values used were = 2 units (1 mm) and γ= 0.025.
The advantage of the PL cost function over the MS and
linear cost functions is that the impact of point-pairs that are
greater than apart is reduced. The reasoning is that these
point-pairs are so far apart that they are not, actually, part of the
matching surface and should not count towards the matching
metric. The small gradient, γ, helps to ensure convergence
when large numbers of points begin the optimisation with a
distance greater than . The advantage of the PL approach
over LCP is that the linear portion of the function in the
region z≤helps to ensure convergence and suppresses false-
positives, especially when the depth map is coarsely sampled.
Figure 5 compares the four cost functions under consideration
using the pair of fragments shown in ﬁgure 1. All except the
proposed PL function exhibit signiﬁcant false-positive peaks.
All of the plots in ﬁgure 5 exhibit multiple local minima
making convergence to the optimal solution using an iterative
process difﬁcult to guarantee. Our approach is to locate the
optimal offset using a multi-resolution search algorithm. The
ﬁrst pass searches the entire range of possible offsets over a
uniform grid of sample points with a relatively coarse 0.5 mm
separation. Subsequent passes reduce the extent of the search
to one quarter of the previous area, centred on the best result
from the previous pass, and reﬁne the resolution by a factor
of two so the total number of sample points remains the same.
This process continues down to a resolution of approximately
10 microns. Subsequent reﬁnement has not demonstrated ob-
servable improvement in the join with any of the fragment pairs
tested to date. This search process is outlined in the following
function search(xrange , yrange )
// Set initial centre of search and search resolution, d
while d > dmin do
for n=−Nxto Nxdo
for m=−Nyto Nydo
costn,m =solve(x, y)
(nmin, mmin )←arg min
// Set search centre for next pass
return (x0, y0)
where xrange and yrang e deﬁne the extent of the search area,
dinit is the search resolution for the ﬁrst pass (0.5 mm in our
case) and the function, solve(x, y), performs the Nelder-Mead
iterative rotational optimisation, using offsets xand y, and
returns the cost function for the optimal solution.
C. Match Ranking
The pair-wise matching technique described in the previous
section can be applied on a number of levels from a semi-
automated alignment of nearby fragments (a form of “snap to
best ﬁt”) through to fully automated searching and joining of
fragments. Regardless of the application, human intervention
will always be required following matching in order to validate
the computer generated match and eliminate false-positives.
Efforts to aid this process have already been described in
section II-B in the shape of the novel proposed piecewise-linear
cost-function. This has been shown to signiﬁcantly reduce the
incidence of false-positive matches.
Fig. 5. The four cost functions under consideration shown as normalised functions of xand yoffset. Piecewise Linear (PL), Linear (L), Largest Common
Pointset (LCP) and Mean Squares (MS) functions are shown.
Fig. 6. (i) Small overlapping join area (only four points are closer than ): a
low priority match. (ii) Larger join area: likely to be of greater signiﬁcance.
When the fully automated matching system is used, it is
desirable to prioritise the potential matches so that the human
validator can examine the most promising results ﬁrst. Ranking
different potential matches could be done simply on the basis
of the ﬁnal cost function of the join but this does not take
into account other factors that could indicate that one matched
pair is a more signiﬁcant candidate than another and is worth
investigating with a higher level of priority. In addition to the
measure of the distance between pairs of points on the depth
maps (section II-B), the quality or signiﬁcance of a match can
also be graded based on the following criteria:
•the overlapping area of the adjoining surfaces
•the degree of interlocking of the join
•the continuity of the written surfaces either side of the
•the continuity of the slope of the written surfaces
either side of the join
The overlapping area of the adjoining surfaces is estimated by
the number of points in the summed depth maps that have a
separation less than the threshold, . This metric is the same
as the common pointset size as used in the LCP cost function.
The greater the number of points, the larger the overlapping
area and, therefore, the more signiﬁcant the join is likely to
be. This estimation is illustrated in ﬁgure 6.
The degree of interlocking of a join is estimated by ﬁxing
the rotation angles, φ, θ, ψ, at their optimal values but moving
the offsets, xand y, by small amounts in each direction, i.e.
x←x±δx and y←y±δy (see ﬁgure 7). The average
increase in the cost function caused by these small translations
Fig. 7. (i) A join with tight interlocking before and after the translation by
δx. (ii) A loosely interlocking join; the average depth after translation is much
smaller in this case.
corresponds to the degree of interlocking; the tighter the join is
interlocked, the larger the average increase in the cost function
The last two criteria estimate the degree of any discon-
tinuity of the newly joined inscribed surface in terms of the
surface depth (ﬁgure 8i) and in the surface gradient (ﬁgure 8ii).
These discontinuities are jointly estimated by taking a second
depth map, this time of the joined tablets together viewed from
the front, and calculating the average value of the second-
derivative of depth with respect to distance. For a perfectly
smooth tablet and a perfect join, this will be zero across the
entire exterior surface.
The metrics described in this section have been tested using
scans of the real tablet fragments shown in ﬁgure 1 as well as
scans of laboratory fabricated fragments such as those shown
in ﬁgure 2. Unsurprisingly, the most signiﬁcant predictor of
match quality is the cost function used for the iterative rotation
optimisation. Even when a join is only partial due to damage to
the tablet fragments, the cost function still returns a low value
for correct joins by virtue of the piecewise-linear cost function.
The next most signiﬁcant metric is the overlapping area (the
common pointset size). This does not give extra information
about the quality of the join but does give an indication of
how signiﬁcant the match is likely to be.
The degree of interlocking does not provide information
about the quality or likely signiﬁcance of a match but does
provide information about the likelihood that a match is a false-
positive. Smoother surfaces with low degrees of interlocking
such as the edges of a tablet are more likely to ﬁnd relatively
well ﬁtting false-partners than more detailed surfaces that
interlock tightly. As such, when a tightly interlocking match
is discovered it is more likely to be a true-match and should
be given a higher level of priority.
Surface continuity has proved to be the least signiﬁcant of
the metrics under consideration. This is due, in part, to the
problems in reliably estimating this metric given the irregular
nature of the tablet surface. However, with further reﬁnement
to the algorithm used to calculate this metric, we believe it
Fig. 8. (i) A join exhibiting a discontinuity in surface depth. (ii) A join
exhibiting a discontinuity in surface gradient despite being continuous in
surface depth. (iii) A continuous join in terms of both depth and gradient.
will be possible to form a more robust estimator of surface
continuity, helping to identify false positive matches that have
escaped detection using the other criteria.
III. RES ULTS
The method and cost function described have been de-
veloped and tested using laboratory fabricated clay tablet
fragments. Figure 9 shows an example of a test tablet that had
been broken into four fragments before 3D scanning and pro-
cessing using the method described. In this case, the matches
were chosen based on the cost function alone and the tablet
completely reconstructed without human intervention. Note
that even though the join between the top two fragments is
quite badly chipped, the optimal orientation was still estimated
correctly. This is possible because of the use of the piecewise
linear cost function which places a reduced emphasis on the
join area suffering from the chipping when compared with the
linear or least squares functions.
Scans of the real tablet fragments shown in ﬁgure 1 have
also been tested. The resulting matched fragment pair is shown
in ﬁgure 10. This is the ﬁrst example of a pair of 3D scanned
cuneiform tablet fragments to be successfully joined by an
automated algorithm with no human intervention beyond the
initial scanning and pre-processing stages.
This paper presented a proposed 3D matching algorithm
designed speciﬁcally for fragmented cuneiform tablets. The
novel piecewise linear cost function employed by the algorithm
has proven successful in matching scans of both laboratory
fabricated test fragments as well as fragments of real cuneiform
Fig. 9. A laboratory fabricated clay tablet, broken into four fragments,
scanned and completely reconstructed using the proposed matching process.
tablets. This is the ﬁrst reported example of successful auto-
mated joining of scanned cuneiform tablet fragments.
Several metrics, in addition to the cost function, have
been proposed to aid in the ranking of potential matches.
Although the reconstruction shown in ﬁgure 9 was performed
using the cost function alone, this was only possible because
of the small number of fragments involved. With a larger
library of fragment scans to draw from, the likelihood of
false positives and the need for human validation becomes
increasingly important. This is when the additional match
quality criteria become valuable because they allow joins to
be ranked not only by the match quality, but also by the likely
signiﬁcance of the match.
Presently, although the fragment rotations are calculated
by an iterative optimisation process, the fragment offsets are
established by searching over a uniform grid. We plan to
improve the efﬁciency of the process by further optimisation
of the xy offset search using a second layer of iterative
optimisation (possibly employing some of the additional match
Fig. 10. The scanned tablet fragments, Wy 777 and W 18349 ,
successfully joined using the proposed piecewise linear cost function.
quality criteria). We also plan to acquire further scans of real
tablet fragments in order to further reﬁne the algorithm and
validate the importance of the match quality metrics.
Although further reﬁnements are planned, even in its cur-
rent form the proposed matching algorithm has the potential
to beneﬁt the Assyriology research community. There are
huge numbers of unmatched fragments in museums around the
world and, without an automated approach, the vast majority
of these would probably never be joined.
For the data acquisition with the 3D scanner of the
Heidelberg Graduate School of Mathematical and Computa-
tional Methods for the Sciences at the Interdisciplinary Center
for Scientiﬁc Computing, Heidelberg University, we would like
to thank Sonja Speck, Hubert Mara and Susanne Kr¨
The scanned tablets are published with the kind permission
of the German Archaeological Institute, Berlin.
The pre-processing computations described in this paper
were performed using the University of Birmingham’s
BlueBEAR HPC service, which provides a High Performance
Computing service to the University’s research community.
See http://www.birmingham.ac.uk/bear for more details.
For assistance and the ﬁring of the laboratory fabricated test
tablets, we would like to thank The Potters Barn, Sandbach,
Cheshire, UK (http://www.thepottersbarn.co.uk/).
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