COMPARISON IN THE HAUSDORFF METRIC OF

RECONSTRUCTION OF 3D URBAN TERRAIN BY FOUR

PROCEDURES

Dimitri Bulatov

Fraunhofer Institute of Optronics, System Technologies and Image Exploitation

Gutleuthausstrae 1, 76275 Ettlingen, Germany

John Lavery

Mathematics Division, Army Research Laboratory, P.O. Box 12211, Research Triangle Park, NC 27709-2211, U.S.A.

Keywords:

3D, Alpha shapes, Cubic spline, Hausdorff distance, Irregular data, Iso-surface, L

spline.

Abstract:

We computationally compare four procedures, namely, alpha-shapes, iso-surface extraction, gridﬁt and L

splines for geometric reconstruction of 3D urban structures represented by irregular point clouds with abrupt

changes in density. For signiﬁcant numbers of outliers, L

splines produce the most accurate reconstructions

both visually and when measured analytically in the Hausdorff metric but are more computationally expensive

than the other three procedures.

1 INTRODUCTION

Obtaining textured surfaces from point clouds re-

trieved by passive sensors is a topic of increasing in-

terest in computer vision and its applications, such as

automatic navigation and surveillance. Point clouds

obtained by photogrammetric methods are commonly

irregular and have abrupt changes of density because

the regions with high density of points resulting from

patches of good local coverage and textured areas are

interspaced with large patches of poor coverage. The

abruptly changing density of the point clouds that

we consider here is in contrast to the roughly uni-

form density of the point clouds obtained, for ex-

ample, from local LIDAR scanning of objects, for

which theoretically established conditions that guar-

antee reconstruction of a topologically correct sur-

face with error bounds are known (Amenta et al.,

2000). In addition to the irregular density, point

clouds produced from light, inexpensive cameras, es-

pecially from cameras without internal navigation ca-

pabilities, have a high level of noise and a signiﬁ-

cant number of outliers, i.e., points far from the sur-

face resulting, for example, from small moving ob-

jects, occlusions, or spurious matches in the areas

of roughly homogeneous intensity distribution which

are, in general, not easy to eliminate in the prepro-

cessing step. In some cases, it is possible to cope

with these negative properties using model assump-

tions, for example, the local 2.5D character of dig-

ital terrain maps (DTMs), which enables extraction

of roof planes (Haala N., 1998) and piecewise planar

structures in general (Mayer H., 2006), but, in gen-

eral, robust algorithms for reconstruction of 3D sur-

face for various topological types are not yet avail-

able and are therefore widely investigated. In this

context, numerous procedures have been considered

over the past two decades, including alpha shapes (α-

shapes) (Edelsbrunner and M

ucke, 1994), TMSs (tri-

angular mesh surfaces, called in the 2.5D case trian-

gulated irregular networks or, for short, TINs) (Bois-

sonnat et al., 2006), iso-surface extraction (Kazhdan,

2005; Hoppe et al., 1992), polynomial splines (Eck

and Hoppe, 1996), L

splines (Bulatov and Lavery,

2010; Lavery, 2001; Lin et al., 2006) and gridﬁt,

a widely used modeling tool available in MATLAB

(D’Errico, 2006).

In this paper, we will investigate the relative ac-

curacy of the surfaces produced by α-shapes, iso-

surface extraction, gridﬁt and L

splines.

These comparisons follow on and extend previ-

ous work on comparison of α-shapes with conven-

125

Bulatov D. and Lavery J. (2010).

COMPARISON IN THE HAUSDORFF METRIC OF RECONSTRUCTION OF 3D URBAN TERRAIN BY FOUR PROCEDURES.

In Proceedings of the International Conference on Computer Graphics Theory and Applications, pages 125-129

DOI: 10.5220/0002764501250129

 SciTePress

tional splines and L

splines (Bulatov and Lavery,

2009). While in (Bulatov and Lavery, 2009) the em-

phasis was on procedures that handled all steps of

the process from video as input to textured geomet-

ric models as output and merely visual comparison of

reconstruction results was the objective, the present

paper focuses on the geometric reconstruction por-

tion of this process and its quantitative evaluation.

This portion starts from a given cloud of points or

(oriented) patches and ends up with a 3D triangular

mesh. We choose an artiﬁcially constructed object

as the basis for comparison so that ground truth is

available. The standard deviation of Gaussian noise

and the outlier percentage is varied. Deviation of

the reconstructed surface from ground truth will be

measured not by orientation-dependent classical er-

ror measures such as root-mean-square, but rather by

the orientation-independent Hausdorff metric. We as-

sume here knowledge of a 2D (u, v)-parametrization

of a given point cloud S = {x

= (x

, y

, z

), m =

1, 2, . . . , M}, from which the object is to be recon-

structed.

In Section 2, we describe the Hausdorff metric in

which the comparisons will be carried out and the four

procedures that will be compared. Computational re-

sults are presented in Section 3. Concluding remarks

and discussion of future research are given in Section

2 HAUSDORFF METRIC AND

RECONSTRUCTION

PROCEDURES

In this section, we outline and motivate our choice of

the Hausdorff metric in which the comparisons will

be carried out and give brief descriptions of the four

procedures that will be compared, namely, α-shapes,

iso-surface extraction, gridﬁt and L

splines.

2.1 Hausdorff Metric

A fundamental issue when making comparisons is

the metric (measure of similarity) in which the com-

parisons are made. Conventional metrics such as

rms (root mean square—the square root of the av-

erage error) and generalizations thereof, such as the

norms (DiBenedetto, 2002), measure similarity

in ways inconsistent with human perception. For

many commonplace situations, including, for exam-

ple, thin walls in urban terrain, these metrics indi-

cate that two sets are nearly the same when observers

judge them to be dissimilar. In order to measure

completeness and correctness (Heipke et al., 1997)

of a geometric reconstruction (Seitz et al., 2006), it

is now common to use (generalizations of) the Haus-

dorff metric (Olson and Huttenlocher, 1997). The

Hausdorff metric is sensitive to outliers, a property

that makes it a suitable tool for evaluating surface re-

construction methods for practical applicability such

as automatic navigation. In our case, the outliers are

not input sample points but triangles of the result-

ing mesh that contain points far from the surface.

Some generalizations of the Hausdorff metric play

down the effect of outliers (Baddeley, 1992), but in

this paper we shall adopt the original Hausdorff met-

ric to perform the comparisons. We denote the dis-

tance from point x to mesh Y and the distance from

mesh X to mesh Y by d(x, Y ) = inf

y∈Y

d(x, y) and

d(X , Y ) = sup

x∈X

d(x, Y ), respectively. For our pur-

poses, d(x, y) is the Euclidean distance between x and

y. The Hausdorff metric for the “distance” from X to

Y is

(X , Y ) = max

{

d(X , Y ), d(Y , X )

}

. (1)

If Y is the ground-truth mesh, then the ﬁrst term

in (1) denotes correctness and the second term com-

pleteness of the reconstruction. Effective algorithms

are needed for estimating d(X , Y ) on the triangular

meshes considered here. We implemented the main

features of the method of (Guthe M., 2005), which

uses octrees for coarse identiﬁcation for a given ver-

tex x of the mesh X the part Y

′

⊆ Y to which com-

puting distance makes sense. Then we performed fast

computation of distance from x to every vertex, edge

and face of Y

′

. The minimum of these distances is an

estimate of d(x, Y ).

2.2 Alpha Shapes

Alpha shapes is the well known method introduced in

(Edelsbrunner and M

ucke, 1994). Given a point sam-

ple S, a triangle formed from a triple of points is added

to the list of triangles if no point of S lies in one of two

open balls of radius α around these points. Clearly,

for very small α, the α-shape will be S itself and, for

α → ∞, the convex hull of S will be obtained. Since

α-shapes are subsets of Delaunay triangulations, their

computation can be performed rather fast. Another

advantage of α-shapes is that they provide 3D models

without needing parametrization. On the other hand,

the procedure is not robust against outliers.

2.3 Iso-surface Extraction

The iso-surface extraction procedure that we use in

this paper is that given in (Kazhdan, 2005). Given

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126

a closed surface F , the procedure ﬁrst retrieves

the Fourier transform of the characteristic function

(χ(x) = 1 if x ∈ F and χ(x) = 0 if x 6∈ F ) from the

point set S and the set of oriented normal vectors us-

ing Stokes’s theorem (see (Kazhdan, 2005) for more

details). Then it obtains an implicit function that rep-

resents the surface by setting χ(x) = c where c is

suitable constant. Finally, it performs meshing us-

ing a marching cubes algorithm (Lorensen and Cline,

1987). This procedure has the advantages of not re-

quiring a parametrization and of being able to ob-

tain the normal vectors and their orientations from the

camera trajectory.

2.4 Gridﬁt

Gridﬁt in its standard implementation is an approxi-

mation procedure for ﬁtting 2.5D surfaces (D’Errico,

2006). We use gridﬁt to generate 3D parametric sur-

faces of the form (x(u, v), y(u, v), z(u, v)), where x, y

and z are 2.5D surfaces with respect to (u, v). Gridﬁt

solves a least-squares system of the type

(1− λ)||Ax− b||

+ λ||Bx||

, (2)

where the ﬁrst term is the data ﬁtting term and the sec-

ond term is the regularization term. The parameter λ

determines the balance between accurately ﬁtting the

data (small λ) and smoothing out the surface (large λ).

The version of gridﬁt used in this paper constructs a

surface that ﬁts the data approximately linearly on

triangles and regularizes the surface by approximate

equalization of the partial derivatives at the vertices

of the triangles.

2.5 L

-splines

Reconstruction with L

splines was described in (Bu-

latov and Lavery, 2010). Let (u

, v

) be the paramet-

ric postion of (x

, y

, z

). The key idea is to minimize

a functional that consists of

(1− λ)

∑

m=1

z(u

, v

) − z

+ (3)

(

+ 2

)dudv+ε

∑

nodes

(

)

and 12 analogous expressions involving x, y and z

over the manifold of cubic L

splines, that is, C

smooth piecewise cubic functions x, y and z on a given

(u, v)-grid. Similar to (2), the ﬁrst term in (3) ex-

presses how closely the data points are ﬁtted, the sec-

ond term expresses, by minimizing absolute values of

second partial derivatives, how close the surface is to

Figure 1: Model “house with overhanging roof” and point

cloud with no outliers. Left: view from side, right: view

from top, middle at top: parametrization in (u, v)-domain

(points on the ground, on the walls, on the horizontal, upper

and lower overhanging parts of the roof are marked in black,

red, green, cyan and yellow, respectively).

Figure 2: Surfaces extracted by α-shapes. Noise level 0.05

everywhere. Left and center: no outliers. Left: α = 5·10

σ,

center: α = 2 · 10

σ. Right: outlier percentage 0.01, α =

5· 10

σ.

a piecewise planar surface and the last term prevents it

from having a non-unique minimum. Just as was the

case for gridﬁt, the balance parameter λ determines

the trade-off between ﬁtting the data and smoothing

the surface. The third term prevents the functional

from having a non-unique minimum, and so any suf-

ﬁciently small, positive number can be taken for ε. As

in gridﬁt, the resulting mesh is formed by connecting

the neighboring spline knots.

3 COMPUTATIONAL RESULTS

The test object represented by the point cloud S must

be simple enough that it can be correctly evaluated

with the Hausdorff metric. On the other hand, S must

possess all properties of a point cloud obtained by

photogrammetric methods in urban terrain: gradient

discontinuities (characteristic for manmade objects),

high Gaussian noise, several outliers and varying den-

sity of points. In this paper, the point cloud S to be

used in the comparisons represents a house with an

overhanging roof (see Figure 1). The experiments

were carried out for levels 0.01 and 0.15 of Gaussian

Figure 3: Surfaces produced by iso-surface extraction. Left:

outlier percentage 0.0, right: outlier percentage 0.01.

COMPARISON IN THE HAUSDORFF METRIC OF RECONSTRUCTION OF 3D URBAN TERRAIN BY FOUR

PROCEDURES

127

Figure 4: Gridﬁt surfaces. Outlier percentage is 0.01 every-

where. Equally spaced grid. Left: λ = 0.2, right: λ = 0.33.

Figure 5: L

splines. Outlier percentage is 0.01 everywhere.

Equally spaced grid. Left: λ = 0.3, right: λ = 0.5.

noise and for percentages 0%, 1% and 10% of outliers

(i. e. points randomly chosen in the bounding box of

the object ) for x, y, z coordinates of the point (in the

case of iso-surface extraction also for normal vectors).

The density of points remained unchanged in all ex-

periments but was variable in different regions of the

data set. For each level of noise and outliers, we car-

ried out data-set generation, reconstruction and eval-

uation 10–15 times and computed the average of the

Hausdorff distances according to (1).

In the α-shapes procedure, we used α = 2· 10

and 5· 10

σ, values that produce the best results for

the point clouds of interest here. Here, σ is the stan-

dard deviation of the data-set coordinates. The results

are shown in Figure 2. Here and in what follows, the

Table 1: Hausdorff-metric errors for the surfaces of Figures

2–5. Explanation about choice of parameters is given in

Sec.2 and Sec. 3. The computing times are approximate,

since the algorithms were implemented in different pro-

gramming languages and average correction factors were

applied.

method noise outl. (%) parameters Hausdorff comp.

-distance time

0.01 0.26030

0.05

0.00 α = 2· 10

0.27441

0.01 0.22844

α-shapes

0.05

0.00 α = 5· 10

0.30783

≈ 2.0 sec.

0.00 0.17134

iso-surface 0.05

0.01 0.85048

≈ 4.5 sec.

0.00 0.20819

0.01 λ = 0.2 0.38543

gridﬁt 0.1 0.66349

eq. spaced 0.00 0.26813

0.01 λ = 0.33 0.35752

0.05

0.1 0.53354

≈ 1 sec.

0.00 0.19457

0.01 λ = 0.3 0.19387

0.1 0.45619

0.00 0.26196

0.01 λ = 0.5 0.24133

-splines 0.05

0.1 0.30898

≈ 240 sec.

point x on the approximating surface that yields the

value of the Hausdorff metric (the maximum distance

from the approximating surface to the ground-truth

mesh) is at the center of a red circle and a green line is

drawn from x to the closest point on the house model.

For iso-surface extraction, we consider the fact

that the result must be a closed surface and exclude

from consideration all triangles that lie outside of the

bounding box of the object. The bandwidth of the re-

construction (given by the voxel-grid for calculating

forward and inverse FFT) was chosen to be 64. The

orientation of the normals on points for the data set

described were computed according the position of

each point (on the ground, at the walls, on the roof,

on the underside of the overhang). The computational

results are presented in Fig 3.

For gridﬁt and L

splines, we generated

surfaces x(u, v), y(u, v), z(u, v) on the (u, v) do-

main [−1.15, 2.15] × [−2.15, 2.15]. The (u, v)-

parametrization is depicted on the right in Figure 1.

Although a robust iterative procedure of generating

parametrized data sets from similar point clouds

is given in (Bulatov and Lavery, 2010), for the

present paper, we chose manually suitable spatial

homographies for points on the ground, at the walls

and on the roof and under the overhang that transform

the points from different parts of the house into

the (u, v)-plane and preserve topological relations

between these points. Both surfaces were computed

for 50 × 50 equally spaced rectangular (u, v)-grids

and reasonable balance parameters λ = 0.2 and 0.33

for gridﬁt and λ = 0.3 and 0.5 for L

splines. Figure 4

shows the gridﬁt reconstruction with different balance

parameters. The reconstructions with L

splines are

given in Figure 5. In Table 1, we present the average

Hausdorff-metric errors between the reconstructions

in Figures 2–5 and the ground truth. As Figures 2–5

and Table 1 indicate, the α-shapes and iso-surface

extraction are not robust enough for high percentages

of outliers. In the current implementation of gridﬁt,

the lowest Hausdorff distances (the ones stated in

Table 1) for data sets with outliers were obtained

for equally spaced grids. For further reduction of

the Hausdorff distance, it is recommended to use L

splines, which perform signiﬁcantly better for data

sets with quite high percentages of outliers.

The time needed to calculate L

splines in their

current implementation is high because one must

solve a linear program, rather than a single linear

least-squares system. However, the computing time

of L

splines can be reduced by orders of magnitude

by applying a domain decomposition procedure intro-

duced in (Lin et al., 2006) as well as by using adaptive

triangular grids.

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128

4 CONCLUSIONS

There is an urgent need for accurate approximation of

irregular, 3D data in urban terrain as well as in geo-

metric modeling of irregular objects in general. The

Hausdorff metric is a widely recognized, orientation-

independent tool for measuring the accuracy of 3D re-

construction. The ﬁgures in Sec. 3 illustrate the cor-

relation between lower Hausdorff distance and bet-

ter reconstruction in the view of the user interested in

practical applications.

The four procedures described in this paper all

produced acceptable reconstructions for data sets

without outliers. However, once there is a signiﬁcant

number of outliers in the data, L

splines yield the best

results. Once implemented in a computationally efﬁ-

cient domain-decomposition framework and on more

ﬂexible triangular grids, L

splines will be computa-

tionally competitive with the other methods.

In the future, we will compare the procedures in-

vestigated in this paper with a wider class of recon-

struction procedures and will integrate these proce-

dures into complete reconstruction and texturing pro-

cedures that go all the way from the camera or other

sensors to the textured model.

ACKNOWLEDGEMENTS

We express our appreciation to Michael Kazhdan and

John D’Errico for them placing their well written and

well commented source codes for iso-surface extrac-

tion and gridﬁt, respectively, on the Internet.

REFERENCES

Amenta, N., Choi, S., Dey, T. K., and Leekha, N. (2000).

A simple algorithm for homeomorphic surface recon-

struction. In International Journal of Computational

Geometry and Applications, pages 213–222.

Baddeley, A. (1992). An error metric for binary images.

In Robust Computer Vision: Quality of Vision Algo-

rithms, Proceedings, International Workshop on Ro-

bust Computer Vision, pages 59–78.

Boissonnat, J., Cohen-Steiner, D., Mourrain, B., Rote, G.,

and Vegter, G. (2006). Meshing of surfaces, effec-

tive computational geometry for curves and surfaces.

Mathematics and Visualization. Springer.

Bulatov, D. and Lavery, J. (2009). Comparison of re-

construction and texturing of 3d urban terrain by l

splines, conventional splines and alpha-shapes. In

Proc. Internat. Conf. Computer Vision Th. Applic VIS-

APP, pages 403–409.

Bulatov, D. and Lavery, J. (to appear 2010). Reconstruction

and texturing of 3d urban terrain from uncalibrated

monocular images using l

splines. Photogr. Engr. and

Remote Sensing.

D’Errico, J. (2006). Surface ﬁtting using grid-

ﬁt.

http://www.mathworks.com/matlabcentral/

fileexchange/8998

DiBenedetto, E. (2002). Real Analysis. Birkh

auser Boston

Inc.

Eck, M. and Hoppe, H. (1996). Automatic reconstruction

of b-spline surfaces of arbitrary topological type. In

Proc. 23rd Annual Conf. Computer Graphics and In-

teractive Techniques, pages 325–334.

Edelsbrunner, H. and M

ucke, E. P. (1994). Three-

dimensional alpha shapes. ACM Trans. Graph., 13(1).

Guthe M., Borodin P., K. R. (2005). Fast and accurate

hausdorff distance calculation between meshes. In J.

WSCG, pages 41–48.

Haala N., B. C. (1998). Interpretation of urban surface mod-

els using 2d building information. In Computer Vision

and Image Understanding, pages 204–214.

Heipke, C., Mayer, H., Wiedemann, C., and Jamet, O.

(1997). Evaluation of automatic road extraction. In

In: International Archives of Photogrammetry and

Remote Sensing, pages 47–56.

Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., and

Stuetzle, W. (1992). Surface reconstruction from un-

organized points. In SIGGRAPH ’92: Proceedings of

the 19th annual conference on Computer graphics and

interactive techniques, pages 71–78. ACM.

Kazhdan, M. (2005). Reconstruction of solid models from

oriented point sets. In SGP ’05: Proceedings of the

third Eurographics symposium on Geometry process-

ing, pages 73–82.

Lavery, J. E. (2001). Shape-preserving, multiscale interpo-

lation by bi- and multivariate cubic l1 splines. Com-

puter Aided Geometric Design, 18(4):321–343.

Lin, Y.-M., Zhang, W., Wang, Y., Fang, S.-C., and Lavery,

J. (2006). Computationally efﬁcient models of urban

and natural terrain by non-iterative domain decompo-

sition with l

smoothing splines. In Proceedings of the

25th Army Science Conference.

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

A high resolution 3d surface construction algorithm.

SIGGRAPH Comput. Graph., 21(4):163–169.

Mayer H., B. J. (2006). Automated 3d reconstruction of

urban areas from networks of wide-baseline image se-

quences. In International Archives of the Photogram-

metry, Remote Sensing and Spatial Information Sci-

ences 37, pages 633–638.

Olson, C. F. and Huttenlocher, D. P. (1997). Automatic

target recognition by matching oriented edge pixels.

IEEE Transactions on Image Processing, 6:103–113.

Seitz, S. M., Curless, B., Diebel, J., Scharstein, D., and

Szeliski, R. (2006). A comparison and evaluation of

multi-view stereo reconstruction algorithms. In CVPR

’06: Proceedings of the 2006 IEEE Computer Society

Conference on Computer Vision and Pattern Recogni-

tion, pages 519–528. IEEE Computer Society.

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