Uncertainty Visualization and Hole Filling for Geometric Models of

Ancient Water Systems

Jeffrey Forrester

, William McVicker

, Timmy Gambin

, Christopher Clark

and Zo¨e J. Wood

California Polytechnic State University, San Luis Obispo, CA, U.S.A.

University of Malta, Msida, Malta

Harvey Mudd College, Claremont, CA, U.S.A.

Keywords:

Surface Reconstruction, Geometric Modeling, Hole Filling, Uncertainty Visualization.

Abstract:

Geometric data acquired via a scanning process can suffer from holes due to errors in the acquisition process,

noise, or challenges in merging multiple inputs together into a uniﬁed map. We present a straight forward

algorithm to ﬁll holes in incomplete evidence grids representing acquired geometric data. We also present our

methods to apply learning in order to statistically evaluate the proposed hole ﬁlling algorithm. This analysis

validates our proposed method for hole ﬁlling and additionally enables the construction of a probability dis-

tribution function to represent the accuracy of the ﬁlled data per model. During surface reconstruction, this

function can be used to visualize the certainty of the ﬁlled geometry via transparency and coloring giving

the user an understanding of the data’s accuracy. This work is motivated by a multi-year project to construct

educational visualizations of ancient water storage systems, i.e. cisterns and wells within churches, fortresses

and homes on the islands of Malta, Gozo and Sicily.

1 INTRODUCTION

Geometric data acquired via sensors and scanners of-

ten suffers from missing data due to noise, acquisi-

tion error, and geometric complexities. Mapped ob-

jects are generally closed surfaces and the representa-

tive data is considered incomplete until the surface is

closed. Surface reconstruction and hole ﬁlling are a

well studied problem, however challenges remain for

noisy data with signiﬁcant gaps. We present an algo-

rithm for hole ﬁlling acquired data with large gaps,

along with our methods to statistically evaluate this

algorithm and calculate an estimate of the ﬁlled holes

accuracy. During the statistical analysis stage of the

algorithm, a functional representation of certainty is

built. This functional representation can be seen as a

probability distribution function (pdf). Once surface

reconstruction and hole ﬁlling are completed, the pdfs

are incorporated in the ﬁnal visualizations of con-

structed models to illustrate the uncertainty of ﬁlled

regions.

This project is a part of a larger one, speciﬁcally

aimed at mapping and modeling ancient water stor-

age systems, i.e. cisterns, wells and water galleries

located in most houses, churches, and fortresses of

the islands of Malta, Gozo, and Sicily. Archaeologists

looking to study and document ancient water systems

have found the task can be difﬁcult, dangerous, and

expensive. The data used in this paper was gathered

through a series of underwater robot deployments in

which multiple sonar scans were gathered, then fused

into a map of the scene via SLAM algorithms (Si-

multaneous Localization and Mapping). Such maps,

which are typically evidence grids of probability val-

ues, can be treated as implicit volumes. Surfaces can

be extracted from these volumes via marching cubes

and then visualized. The input data for this project

includes both holes and noise (see Figure 7). In this

paper, we present our hole ﬁlling algorithm, including

statistical analysis of the certainty of ﬁlled regionsand

demonstrate resulting ﬁnal visualizations of the water

systems.

2 RELATED WORK

Surface Reconstruction and Hole Filling. In this

research we aim to create a tool for assisting archae-

ologists in their examination of underwater structures.

Surface reconstruction in the underwater setting is a

relatively new area of research with initial work com-

593

Forrester J., McVicker W., Gambin T., Clark C. and J. Wood Z..

Uncertainty Visualization and Hole Filling for Geometric Models of Ancient Water Systems.

DOI: 10.5220/0004229605930600

In Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information

Visualization Theory and Applications (IVAPP-2013), pages 593-600

ISBN: 978-989-8565-46-4

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

Figure 1: Geometric model extracted from an evidence grid created from sonar data. The model on the left is extracted

without application of our algorithm, while the model on the right has hole ﬁlling, pdf creation and uncertainty visualization

applied (red regions are less certain). This cistern is in a private home in Mdina, Malta.

pleted in the maritime setting (Pizarro et al., 2009)

and recent work in cenotes in Mexico (Fairﬁeld et al.,

2010). Similar to the work to map cenotes (Fairﬁeld

et al., 2010), this project involves the application of

marching cubes (Lorensen and Cline, 1987) to recon-

struct a geometric model of underwater data, however,

the cenote project relies on a very sophisticated and

signiﬁcantly larger more expensive ROV and sensors

that would be impractical in water systems explored

in this project. Our research requires mapping in tun-

nels with passages and access points that are relatively

small (e.g. 0.30m diameter at some points), requiring

a micro ROV with low payload capacity and minimal

sensors (i.e. scanning sonar, depth sensor, and com-

pass).

Managing holes in acquired data is a mature

ﬁeld with numerous research projects addressing hole

ﬁlling in both the surface and volumetric setting.

For an overview of many of the relevant research

projects related to polygonal repair, see the course

notes on ‘Geometric Modeling Based on Polyon-

gal Meshes’ (Botsch et al., 2007). Recent work

in surfaces has been conducted (Bac et al., 2008),

while seminal work in the volume setting includes,

VRIP (Curless and Levoy, 1996) and subsequently

volume diffusion (Davis et al., 2002), along with re-

cent work in the volume setting by (Janaszewski et al.,

2010). While there has been excellent work in the

ﬁeld of hole ﬁlling, the work presented in this paper

addresses the unique challenge of acquired data from

noisy sonar along with the challenge of showing in-

formation regarding conﬁdence in our ﬁlled geomet-

ric models.

Visualization and Uncertainty. For this project,

we use well established visualization techniques to

create visually appealing models and to convey uncer-

tainty (Pang et al., 1997; Schmidt et al., 2004). Uncer-

tainty visualization are relevant to many ﬁelds (Pang

et al., 1997; Schmidt et al., 2004; Pfaffelmoser et al.,

2011; Grigoryan, 2002). The work presented in this

paper contributes a method to statistically analyze and

quantify the certainty of ﬁlled regions that could then

be visualized with various uncertainty techniques.

Mapping via Underwater Robot Systems. This

project relies on data acquired from algorithms for

mapping with underwater robots, in particular, un-

der water Simultaneous Localization and Mapping

(SLAM) (e.g. (Williams et al., 2000), (Hern`andez

et al., 2009), and (Fairﬁeld et al., 2006)). The work of

Thurn (Thrun et al., 2005) includes a good survey of

the core techniques capable of fusing data from mul-

tiple sensors to create maps.

3 ALGORITHMS AND PRACTICE

This work is focused on reconstructing geometric

models from evidence maps representing Mediter-

ranean water storage systems (cisterns, wells and wa-

ter galleries). Speciﬁcally, the space being mapped

is discretized into a two dimensional grid of cells

with given a likelihood p

i, j

∈ [0, 1] of being occupied

(Thrun et al., 2005) and (White et al., 2010). The ev-

idence grid is extrapolated into three dimensions to

the appropriate height of the water system, and can

then be treated as volume data, and geometric models

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of the scanned data can be constructed via marching

cubes, similar to the work presented in (Forney et al.,

2011). Any cell with p

i, j

> t is considered an occu-

pied cell and associated with a wall in the model. The

threshold value, t, is used to deﬁne occupancy and is

generally set to values in the range .65 − .85. We re-

fer to cells in the grid at location (i, j) as x

, and the

probability of that cell, p(x

), as p

Due to the acquisition process, gaps in the ev-

idence grid are common and can lead to unwanted

holes in the reconstructed models. See Figures 4

and 2, for examples of incomplete evidence grids that

lead to geometric models with many holes, like those

seen in Figures 1 and 6. As the acquired data is known

to represent water tight structures, hole ﬁlling is nec-

essary to construct more realistic models.

3.1 Hole Filling

In summary, for our setting, hole ﬁlling is achieved by

ﬁtting a function, F, to the {x, y} position of a subset

of occupied cells surrounding identiﬁed holes. Next,

unﬁlled cells of the grid in the ‘hole’ that are crossed

by F are set to occupied in order to ﬁll the hole. To vi-

sualize the effectiveness of our ﬁtting method, we an-

alyze the accuracy of ﬁlled regions. For each model,

we execute a learning process on real data to compute

a pdf to represent the likelihood of the ﬁlled cell’s cer-

tainty as a function of neighboring cells being ﬁlled

and the distance from those cells.

3.1.1 Simulating Holes for Learning

Using acquired evidence grids, we wish to analyze

our hole ﬁlling on real data. We start by identify-

ing fairly complete sections of input data, represent-

ing walls of the water system. These walls are repre-

sented by long runs of connected occupied cells, into

which we introduce ‘simulated holes’. Speciﬁcally,

varying length valid segments of the evidence grid are

selected. A segment is a group of cells containing all

neighboring occupied cells. A neighboring cell is a

cell x

i±1j±1

. Valid segments have a minimum length,

min

, (in practice, l

min

= 9).

We create ‘simulated holes’ by knocking out the

middle section of valid segments by setting some

of the occupied cells in the segment to unoccupied.

When creating holes, we randomly choose segment

lengths to create a variety of hole sizes. This method

restricts our holes to a minimum size of around 3

cells, but allows them to be as large as a third of the

longest segment length. For the removed sections,

the occupied cells closest to the edge of the hole, one

from each side, are deﬁned as endpoints, e

and e

Figure 2: Deﬁnitions are shown in this ﬁgure from data

from the same cistern shown in Figure 1. Occupied cells

are shown in dark blue. Sample Points, S

are shown high-

lighted in light blue, endpoints are shown in orange and pink

samples, S

, correspond to a cubic polynomial ﬁt to the hole

between the two endpoints. Only the sample points, S

that

lay between the two endpoints will be ﬁlled.

Endpoint detection and matching in practice is dis-

cussed in Section 3.2.

To ﬁll each hole and measure the accuracy of each

ﬁt, we deﬁne the following (see Figure 2):

1. The shortest distance between a cell, x

and a

point on the ﬁtted curve F, f

as, kx

− f

k (for

higher order polynomials, this distance represents

the arc length along the curve).

2. S

is the set of occupied cells that were not re-

moved from the segment

3. F is the polynomial ﬁt computed using Gauss-

Jordan elimination on set S

4. S

is the set of cells in the evidence grid that lay

on the ﬁtted curve F

= {x

| |x

− f

| < 0.5,

− f

| < 0.5} (1)

5. S

is the subset of cells belonging to S

that lie

between the endpoints, e

and e

= {x

| S

≤ max(e

, e

≥ min(e

, e

< t} (2)

Given these deﬁnitions, in order to ﬁll the ‘sim-

ulated hole’, the unoccupied cells from S

are set to

ﬁlled: ∀x

∈ S

, p

= 1.

All observed water storage systems contain rela-

tively smooth man-made walls, thus we found the use

of varying order polynomials (linear, quadratic, or cu-

bic) worked well (documented shortly), however, our

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algorithm does not depend on the structure or ﬁtting

method of this function and any standard approach to

function approximation can be used.

3.1.2 Probability Distribution Function

Ideally, we would like to model the uncertainty of all

the ﬁlled cells in S

that we have introduced during

hole ﬁlling. To accomplish this, we can repeatedly

create simulated holes, then ﬁll them and measure

their accuracy against the original data. This process

allows us to compute a probability distribution func-

tion, η(d

), that models the likelihood that any cell

∈ S

has p

> t, as a function of the distance d

from x

to the nearest endpoint. That is, the pdf rep-

resents the likelihood that F sets the appropriate cells

to being occupied, depending on the distance from the

cell to the start of the hole.

η(d

) = P(p

′

> t | d

, x

∈ S

, p

> t) (3)

To compute this pdf, we learn from the evidence

grids by tabulating a histogram of cells of the ﬁlled re-

gion that match the original data. We analyze each ﬁt

by comparing the newly ﬁlled data with the ‘known’

real data. Speciﬁcally,for each cell x

∈ S

, we check

to see if the cell from the original evidence grid also

had an occupied cell at the same x

. This data is tabu-

lated into discrete bins, based on their distance along

F to the nearest endpoint to build up a histogram.

Speciﬁcally, for each unit along F, if the cell x

cor-

responds with an occupied cell in the original data, we

consider this point a hit, if not it is a miss. Distances

along F, denoted as d

, are measured along the arc of

the ﬁt from the nearest end point. For each type of ﬁt:

linear, quadratic, and cubic, hits and misses are tallied

into discrete bins for each order polynomial.

We build up the histogram data by iteratively re-

peating hole ﬁlling of simulated holes for each dataset

per ﬁt, and computing hits for each hole. Once the

samples have been collected, the percentage of hits

at given distances along the curves are computed for

each ﬁt. See Figure 3 for an example of the discrete

binned distance values of a completed histogram for

one evidence grid for a quadratic ﬁt. To create a con-

tinuous pdf for each model, the discrete binned hit

percentages at distances along the curve are ﬁt with a

cubic polynomial.

The goal of this project is to reconstruct models

that will be used to illustrate the shape of water sys-

tems, and it is important to distinguish between the

portions of the model reconstructed from acquired

data and the geometry introduced during the hole ﬁll-

ing, as well as the accuracy of the ﬁlled data. We use

each pdf in order to convey this information.

Figure 3: A histogram of tallied hits for a quadratic ﬁt

for the Tas Silg evidence grid. Bins are discretized at arc

lengths zero to 5 along F (measured from either end point,

so the fewest hits are in the middle of the hole at a largest

distance of 5 units from either endpoint)).

3.1.3 Application of Pdf Data for Visualization

Once the pdf information has been computed for a

given model, the data can be applied to cells in the

hole ﬁlling process. When ﬁlling a hole, each ﬁlled

cell is assigned a probability based on it’s distance

from the closest endpoint, d

. The pdf for the speciﬁc

order polynomial chosen as the best ﬁt is referenced

for certainty information and that data is assigned to

the newly ﬁlled cell. We describe the error metric

used for selecting the best ﬁt in Section 3.2. Thus,

every newly ﬁlled cell is given a value, γ, which rep-

resents the predicted likely-evidence of that ﬁt match-

ing the input model’s learned shape (or actual data).

We allow the user to visualize this uncertainty

by changing the coloring of the surface polygons for

those extracted from evidence cells with a value of γ.

For each vertex, red is added to the texture coloring

based on 1− γ, leading to less certain regions having

higher red colorings. Figures 1, 4, 7, and 6 all show

examples of the uncertainty visualizations generated

with our visualization system.

3.1.4 Validation

During this learning process, we also validate our hole

ﬁlling method overall. Speciﬁcally, we can tally over-

all the percentage of true positives, that is occupied

cells that we would ﬁll using our method that were

originally occupied in the input data. We can also

compute false positives, cells that were unoccupied in

the original data but are ﬁlled via our hole ﬁlling. See

Table 1 for results of the statistical analysis for seven

models evaluating the results of hole ﬁlling. Note that

all models perform very well with an average of 83%

true positives and only 17% false positives. Over-

all for all data sets the computed true positives are

> 68%, with the most challenging data being a very

long water gallery with over 87 holes.

We have presented our method to statistically ana-

lyze our hole ﬁlling method and compute a pdf which

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Table 1: Statistics for models.

Model name size positives false

of grid positives

Case Cutietta 120*120*26 94% 06%

College Garden 100*320*30 89% 11%

Gatto Pardo 132*109*25 75% 25%

Keyhole 150*100*30 87% 13%

Site 8 160*120*25 85% 15%

Tas Silg 187*175*26 83% 17%

Qanat 562*162*26 68 % 32%

Figure 4: The evidence grid and geometric model of the

Gatto Pardo cistern (private home in Mdina, Malta).

models the certainty of our hole ﬁlling, computed via

a learning process. We now describe how to apply our

hole ﬁlling to real holes in acquired data.

3.2 Hole Filling in Practice

For this expedition, investigators deployed a Video-

Ray ROV equipped with an underwater micron scan-

ning sonar, depth sensor, and two video cameras. The

ROV was lowered into cistern access points until it

was submerged. The investigators then tele-operated

the robot to navigate the underwater environment.

Due to the importance of acquiring accurate data, sta-

tionary sonar scans were takenusing a SeaSprite scan-

ning sonar mounted on top of the ROV. These sonar

measurements were used to generate evidence grids

that in turn, were used to generate geometric mod-

els of the cisterns (McVicker et al., 2012) and (White

et al., 2010), (Forney et al., 2011).

For real world acquired data, a number of subtle

challenges arise when applying hole ﬁlling to con-

struct a complete water tight model. In practice, we

ﬁnd two types of holes that must be ﬁlled: those due

to under-sampling, characterized by very small neigh-

boring segments and those due to missing data, char-

acterized by larger holes between deﬁned segments.

See Figures 2 and 7 for examples of both. We choose

to ﬁll small holes ﬁrst to build up information about

the shape of a model, thus we ﬁll holes in an iterative

fashion with the smallest holes being ﬁlled ﬁrst via a

distance threshold, E

, that is expanded only after all

holes within that range are ﬁlled. Our algorithm for

ﬁlling holes involves the following steps:

for E

= min;E

< H;E

+ +

Detect endpoints of valid segments

Match best pairs of endpoints

for any pairs with a size < E

ﬁll identiﬁed holes

Endpoint Detection. The ﬁrst stage in hole ﬁlling

is identifying the segments in the evidence grid. The

evidence grid created from the sonar data may contain

segments of varying thickness due to the reﬂection of

the sonar’s wide cone shaped beam off of organically

shaped walls. When identifying the endpoints of seg-

ments, we cannot make assumptions about the seg-

ment’s cell’s connectivity or endpoint locations (See

Figure 2). In order to compute approximate segment

endpoints, we identify the start and end cells of long

sequences or paths of occupied cells using two passes

of Dijkstra’s algorithm. We have found two passes

of Dijkstra’s to sufﬁce in practice to identify two rea-

sonable extremas of the segment to use as potential

endpoints. The algorithm proceeds as follows:

1. Using any occupied cell x

as the starting point,

a Dijkstra’s search traverses the neighborhood of

grid cells, x

i±1, j±1

, connecting to any unvisited

occupied cell. The cell in the resulting graph

which is reached via the longest path is identiﬁed

as the ﬁrst endpoint of the segment, e

2. A second pass is run using this endpoint, e

, as

the root node. The node at the end of the longest

path in this graph is identiﬁed as the second end-

point of the segment, e

Endpoint detection proceeds until all occupied cells

in the evidence grid have been traversed. This method

may miss extrema for segments that branch, however,

it has been found to work well in practice.

To account for noise in the evidence grid, a seg-

ment is considered a valid segment only if the distance

from endpoint to endpoint along its Dijkstra’s graph is

≥ 4, and a valid endpoint is any endpoint belonging

to a valid segment. We refer to the length along a Di-

jkstra’s graph from cell a to cell b as D

(a, b) where

the cost of traversing to a neighboring occupied cell

i±1, j±1

) is always 1. Once all valid endpoints have

been identiﬁed, we are ready to identify holes as the

cells with p

< t that lie between pairs of valid end-

point. In order to ﬁll these ‘holes’ (regions that are

unoccupied) properly, corresponding endpoints must

be identiﬁed and matched.

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Figure 5: Left: The complete geometric model of the Col-

lege Garden water system, including the correct reconstruc-

tion of three columns found in the cistern. Right: Evidence

grid and geometric model of Conca D’Oro Qanat.

Endpoint Matching. In practice, we do not know

which pairs of endpoints correspond to a hole and

matching endpoint pairs that are closest together is

not always accurate. Thus, given the collection of

endpoints, holes are identiﬁed via the use of pair-

wise testing, using two criteria: distance of end-

points to one another, and how well a given pair will

match the shape of neighboring segments. First, for

each iteration through the hole ﬁlling algorithm, end-

point matching is constrained to only potentially cre-

ate matches from any pair of endpoints, e

and e

whose Euclidean distance (ke

− e

k) < E

Once the set of potential matches has been iden-

tiﬁed, the algorithm measures how well the ﬁt of a

potential pair’s samples conforms to the local shape

of their segments. For each endpoint, all cells (from

the Dijkstra’s path) within a threshold distance, δ, are

identiﬁed as the endpoint’s sample points. The two

endpoint’s sample points are then ﬁt with a polyno-

mial, F, (as in Section 3.1.1). The best matched ﬁt

is the one with the smallest average distance from the

sample points to closest points on the ﬁt, F.

In order to measure the best matched ﬁt, S

is de-

ﬁned in practice as the set of all cells of the evidence

grid that fall within a given distance, δ, of each end-

point, e

and e

. That is:

= {x

| p

> t,

, e

) < δ ∨ D

, e

) < δ} (4)

Given these deﬁnitions, the shape matching error

is computed as:

Min Err =

∑

∀x

∈S

(kx

− f

(5)

Once all possible sets of endpoints have been

added to the list, we choose the matched endpoints

with the lowest error allowing each individual end-

point to be used only once. This matching process

allows us to identify the most likely endpoints that

correspond to a hole in the data.

Filling. Once best pairs of endpoints surrounding

holes have been identiﬁed, unoccupied cells within

holes are set to occupied by ﬁtting the function F to

the sample data, S

, (deﬁned above), for the best pair.

The order of F is chosen to minimize Min Err, (de-

ﬁned above) and hole ﬁlling is applied as deﬁned in

Section 3.1.1. That is, the unoccupied cells from S

are set to ﬁlled: ∀x

∈ S

, p

= 1.

Because F is determined by generating a func-

tion that matches local geometry, there must be well-

deﬁned local geometry available in order to ﬁll large

holes. Thus, we ﬁll smaller holes ﬁrst in order to build

up long segments in the evidence grid iteratively.

Final Visualizations. Once hole ﬁlling is complete,

the 2D evidence grid is extruded into 3D. The appro-

priate extrusion height is deﬁned via measured data

from each speciﬁc water system site. The walls of the

water systems are well modeled by the evidence grid,

but in the surface creation stage a ﬂoor is added af-

ter hole ﬁlling by ﬂood ﬁlling the interior of the now

closed model and adding an empty layer to the evi-

dence grid below the ﬂoor level. To remove small ex-

cess surface components due to noise (such as those

seen in Figure 1), volume smoothing is also applied

before surface extraction.

4 RESULTS

We have created geometric models for a dozen in-

dividual ancient water systems from the islands of

Malta, Gozo and Sicily. Each of these sites suffered

from incomplete data due to inaccuracies in the sonar

and mapping. These inaccuracies resulted in evidence

grids with numerous holes leading to extracted sur-

faces with holes. In this paper, we present a robust

hole ﬁlling algorithm, which ﬁlls in missing data in

the evidence grid, while honoring the shape of exist-

ing data. This hole ﬁlling algorithm results in water

tight meshes with boundary. We present the statistical

analysis used to validate this algorithm and to con-

struct a probability distribution function to model the

accuracy of the ﬁlled data. We also present details of

applying our hole ﬁlling method in practice. Using

the data gathered in our statistical analysis we also

present uncertainty visualizations of the ﬁlled surface

data. We believe this is the ﬁrst such complete system

for mapping and reconstructing underwater geometric

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Figure 6: On the left is the geometric model of the House Dar T’ana (private home on Gozo) cistern before hole ﬁlling and

on the right is the complete geometric model including uncertainty visualization of surfaces created via hole ﬁlling.

structures with hole ﬁlling and uncertainty visualiza-

tions from evidence grids.

We show surface reconstruction results and un-

certainty visualization for six different ancient water

systems, (the physical scale of the grids in this paper

range from 0.03m to 0.1m per cell):

• Figure 1, shows the cistern located in a private

home on Gozo, (with 18 ﬁlled holes).

• Figure 4, shows the cistern located in a private

home in Mdina, Malta, (with 17 ﬁlled holes).

• Figure 7 shows results from sonar scans of the wa-

ter system at Tas Silg (Malta). This ancient water

system, located at the 2,000 year old temple site,

was over 15 meters long and very complex with a

loop structure connecting two entry points via two

divergent channels, (with 42 ﬁlled holes).

• Figure 5 is the cistern at Wignacourt Museum

College Garden (Malta). The cistern is fairly com-

plex and includes several chambers and three pil-

lars visible in the model, (with 20 ﬁlled holes).

• Figure 6 shows the cistern located at House Dar

Ta’Anna - Upper courtyard, Gozo, which is a key-

hole shaped cistern, (with 9 ﬁlled holes).

• Figure 5 shows the qanat at the hotel Conca d’Oro

(Sicily). This very long water gallery (over 40

meters long) was a part of the historic paper fac-

tory and data collection included over 20 station-

ary sonar scans (with 87 ﬁlled holes).

Table 1 includes information on the size of

datasets used for this work and information about the

accuracy of our hole ﬁlling data for each model. Note

that all datasets include numerous holes, up to 87 for

the qanat in the hotel Conca d’Oro.

Figure 7: The top row shows both the sonar data and the

evidence grid for Tas Silg. This very large water system

located on an ancient temple site was over 15 meters long.

The bottom image shows the complete geometric model,

including uncertainty visualization of ﬁlled holes.

4.1 Conclusions

We have presented a robust straight-forward hole ﬁll-

ing algorithm, including statistical analysis of the cer-

tainty of ﬁlled regions and demonstrate resulting ﬁnal

visualizations of the water systems. The hole ﬁlling

performs very well even with very noisy input data

and missing segments, while still preserving the orig-

inal shape of the input data (such as the small con-

necting tunnel in Tas Silg, (Figure 7).

UncertaintyVisualizationandHoleFillingforGeometricModelsofAncientWaterSystems

599

Future improvements to this work, include prop-

agating probability errors from the mapping process

into the evidence grid to visualize the certainty of the

acquired data itself, including, incorporating a model

of the sonar uncertainty more completely throughout

the reconstruction process as done in (Pandey et al.,

2007). Finally, some of the water systems explored

contain interesting shape variation in their vertical ex-

tent and 3D mapping is an active area of our work.

REFERENCES

Bac, A., Tran, N.-V., and Daniel, M. (2008). A multistep ap-

proach to restoration of locally undersampled meshes.

In GMP’08, pages 272–289.

Botsch, M., Pauly, M., Kobbelt, L., Alliez, P., L´evy, B.,

Bischoff, S., and R¨ossl, C. (2007). Geometric model-

ing based on polygonal meshes. In ACM SIGGRAPH

2007 courses. ACM.

Curless, B. and Levoy, M. (1996). A volumetric method

for building complex models from range images. In

SIGGRAPH ’96, ACM Press.

Davis, J., Marschner, S., Garr, M., and Levoy, M. (2002).

Filling holes in complex surfaces using volumetric

diffusion. In Symposium on 3D Data Processing, Vi-

sualization, and Transmission.

Fairﬁeld, N., Kantor, G., Jonak, D., and Wettergreen, D.

(2010). Autonomous exploration and mapping of

ﬂooded sinkholes. Int’l. J. Rob. Res., 29(6):748–774.

Fairﬁeld, N., Kantor, G., and Wettergreen, D. (2006)). Real-

time SLAM with octree evidence grids for exploration

in underwater tunnels. In Journal of Field Robotics,

Vol 24, Issue 1-2, pp. 03-21.

Forney, C., Forrester, J., Bagley, B., McVicker, W., White,

J., Smith, T., Batryn, J., Gonzalez, A., Lehr, J., Gam-

bin, T., Clark, C., and Wood, Z. (2011). Surface recon-

struction of Maltese cisterns using ROV sonar data for

archeological study. In Proceedings of the 7th Int’t.

conference on Advances in visual computing.

Grigoryan, G. (2002). Probabilistic surfaces: Point based

primitives to show surface uncertainty. In Proceedings

Visualization 2002.

Hern`andez, E., Ridao, P., Ribas, D., and Batlle, J. (2009).

MSISPIC: A probabilistic scan matching algorithm

using a mechanical scanned imaging sonar. In Journal

of Physical Agents 3:311.

Janaszewski, M., Couprie, M., and Babout, L. (2010). Hole

ﬁlling in 3D volumetric objects. Pattern Recogn.,

43(10):3548–3559.

Lorensen, W. E. and Cline, H. E. (1987). Marching cubes:

A high resolution 3D surface construction algorithm.

In SIGGRAPH ’87, pages 163–169.

McVicker, W., Forrester, J., Gambin, T., Lehr, J., Wood, Z.,

and Clark, C. (2012). Mapping of ancient water stor-

age systems with an ROV for visualization: An ap-

proach based on fusing stationary scans within a par-

ticle ﬁlter. In IEEE ROBIO.

Pandey, A. K., Krishna, K. M., and Nath, M. (2007). Fea-

ture based occupancy grid maps for sonar based safe-

mapping. In Proceedings of the 20th Int’l. joint con-

ference on Artiﬁcal intelligence.

Pang, A. T., Wittenbrink, C. M., and Lodha, S. K. (1997).

Approaches to uncertainty visualization. The Visual

Computer, 13:370–390.

Pfaffelmoser, T., Reitinger, M., and Westermann, R. (2011).

Visualizing the positional and geometrical variability

of isosurfaces in uncertain scalar ﬁelds. In Computer

Graphics Forum (Proceedings of EuroVis 2011).

Pizarro, O., Eustice, R. M., and Singh, H. (2009). Large

area 3D reconstructions from underwater optical sur-

veys. IEEE Journal of Oceanic Engineering.

Schmidt, G. S., ling Chen, S., Bryden, A. N., Livingston,

M. A., Osborn, B. R., and Rosenblum, L. J. (2004).

Multidimensional visual representations for under-

water environmental uncertainty. IEEE Computer

Graphics and Applications, 24:2004.

Thrun, S., Burgard, W., and Fox, D. (2005)). Probabilistic

robotics. In MIT Press.

White, C., Hiranandani, D., Olstad, C., Buhagiar, K., Gab-

min, T., and Clark, C. (2010). The Malta cistern map-

ping project: Underwater robot mapping and localiza-

tion within ancient tunnel systems. In Journal of Field

Robotics.

Williams, S., Newman, P., Dissanayake, G., and Durrant-

Whyte, H. (2000)). Autonomous underwater simulta-

neous localization and map building. In Proceedings

of the 2000 IEEE Int’l. Conference.

APPENDIX

We wish to especially thank Christina Forney, Erik

Nelson, Jane Lehr and the ICEX teams (2011 &

2012). This material is based upon work supported by

the National Science Foundation, Grant No. 0966608.

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