Roof Template Matching in Airborne LiDAR Data Cheng-Kai Wang Yi-Hsing Tseng Pai-Hui Hsu (Ph.D. Student, Professor), NCKU, Taiwan Assistant Professor, NTU, Taiwan
[email protected] [email protected] [email protected] KEYWORDS: Roof Template, Geometric Moments, Zernike Moments, Airborne LiDAR. ABSTRACT: Three-Dimension building modeling improves the visualization and applications of a city environment. Model-driven methods set up pre-designed models first, and then match models with data to estimate model parameters. One of the main challenges in this context is to choose the pre-designed models automatically. To achieve this goal, roof template matching is one of the ways. For matching purpose, the unique features of each object needs to be extracted first and then used for recognition. This paper presents two methods for obtaining the features which are Geometric moments and Zernike moments. In this research, we compare and discuss the results using both of them. The experiments show the feasibility of the generated features to recognize different type roofs. The discussion also covers the existence of noises that affect the results of generated features. 1. INTRODUCTION As the popularity and importance of digital earth, a lot of researches on 3D city reconstruction have been presented so far. The proposed approaches can be divided into two subjects which are model-driven methods and data-driven methods respectively. For model-driven methods, (Elberink 2008) points out that using model based approach will solve some of the problems with missing data. Another advantage is that staring with a predefined model, its topology implicitly describes the building hypothesis. However the biggest problem of model driven techniques remains the decision which model to choose. Searching a suitable model to fit point clouds automatically is quite a challenge. Several researches among the reconstruction methods implicitly cover this topic, such as (Jaynes, Riseman et al. 2003). In our research, the matching method is based on pattern recognition. And the most important key is to extract the features which sufficiently present each model. This paper introduces how the algorithms work on the point cloud data and show the potential for recognition purpose. 2. FEATURE EXTRACTION 2.1 Geometric Moments 2.2.1 Definition of Geometric Moments: The 3-D moments for solids are defined
M lmn
xl y m z n ( x, y, z )dxdydz
(2.1)
With the order l m n of a 3-D density function ( x, y , z ) . To connect the Geometric moments and the regular moments described by eq. (2.1), we introduce the invariant geometric primitives first. The primitives are measurements in 3-D Euclidean space which are invariant under rotation, such as distance between two points, area of two vectors. The geometric moments can then be considered as descriptions of the primitives using regular moments subject to the geometric invariants. If we set core ( p1 , p2 ,..., pn ) as the multiplication of invariant geometric primitives which involves n participating points. The invariant core can be expressed as the following polynomial form(Xu and Li 2008) n
core( p1 , p2 ... pn ) ai xijj 1 y ijj 2 z ijj 3 i
(2.2)
j 1
Where ai is the coefficient of the polynomial, and has been decided uniquely by core( p1 , p2 ... pn ) . 2.2.2 Invariants of Geometric Primitives: The invariants described in this paper are based on (Xu and Li 2008) which mainly use four invariant geometric primitives. They are the distance between two arbitrary points, the area between three arbitrary points, the dot product between two vectors, the signed volume of a tetrahedron between four points respectively and they are all invariant under rotation and translation in 3-D space. Following the geometric constraints, Xu derives several invariants. In this paper we chose six invariants for our features and eq. (2.3) shows two of them. 1 I1 7/3 ( 400 040 004 2 220 2 202 2 022 ),
000
I2
1 4 000
(2.3) (
2 300
2 030
2 003
3
2 120
3
2 102
3
2 012
3
2 210
3
2 021
3
2 201
6 ) 2 111
2.2 Zernike Moments 2.2.1 Definition of Zernike Moments: In 1934, Zernike introduced a set of complex polynomials which forms a complete orthogonal set over the interior of the unit circle, i.e., x 2 y 2 1 (Maaoui, Laurent et al. 2005). Let Z nm ( x, y ) Z nm ( sin , cos ) Rnm ( ) exp(im ) be the set of polynomials and the original function may then be approximated by(Teague 1980) f ( x, y ) Anm Z nm ( , ) n
(2.4)
m
Where n is a positive integer or zero; m is a positive and negative integer subject to constraints
n m even, m n ; is the length of vector from original to (x,y); is the angle between
vector and x axis in counter clockwise direction; Rnm ( x, y) is Radial polynomial defied as: ( m m )/2
R ( x, y ) m n
(1) s
s 0
(n s )! ( x 2 y 2 )( n 2 s )/2 n m n m s !( s )!( s )! 2 2
(2.5)
note : Rn m Rnm
The complex Zernike moment is by definition Anm [(n 1) / ] dxdyf ( x, y )[ Z nm ( , )]* ( An m ) * 1
(2.6)
2.2.2 Invariants of Zernike Moments: The scale invariant and translation invariant can be handled by the use of normalized over the interval of unit circle and translating the center of mass to the origin of coordinate system. We focus on the behavior of rotation and consider a rotation of a coordinate system as x ' x cos 0 y sin 0 (2.7) y ' x sin 0 y cos 0 The original function after rotation becomes f '( , ) f ( , 0 )
(2.8)
The new Zernike moments is therefore obtained by ( Anm ) ' [( n 1) / ] d d f ( , 0 ) Rnm ( ) exp( im 0 ) Anm exp( im 0 )
(2.9)
Obviously the Zernike moments have simple rotation transformation properties; each complex moment only acquires a mere phase factor on rotation. Following are several invariants derived from the Zernike moments(Teague 1980)
I1 ' A20 I 4 ' A31
I 2 ' A22 A2,2 A22 2
2
I 5 ' 2 A33 A31 cos(33 331 ) 3
I 3 ' A33
2
(2.10)
3. EXPERIMENT AND RESULTS To test the invariant of the Geometric primitives and Zernike moments, both the simulated point clouds and the sensed data were used which are shown in Figure 1. 3.1 Invariant Test on Geometric Moments 3.1.1 Test on Rotation: Table 1and Table 2 shows the results of the invariant test for rotation under the two kinds of test data. Note that the influence on the six invariants of the simulated
1
denotes the conjugate complex number
data is light but there are some invariants of the sensed LiDAR data have a little large variance. The large variance may be caused by the property of non-regular scattered point clouds. For calculating reasons, the points need to be segmented into a biggest voxel where only one point inside it. When rotating the coordinate system, the voxel corresponding to each point may change its size. This may result in a bit influence on the invariants.
(a) The simulated Point Clouds
(b) The sensed LiDAR data
Figure 1. Point clouds for invariant test
Rotation around axis z
Table 1. Invariant test on rotation factor for simulated data I1 I2 I3 I4 I5
I6
0
3.498463
2.149911
4.542290
0.220691
0.136614
2.329055
15
3.498443
2.149893
4.542230
0.220690
0.136614
2.329033
60
3.498436
2.149888
4.542207
0.220689
0.136613
2.329020
120
3.498456
2.149906
4.542266
0.220691
0.136614
2.329046
mean
3.498450
2.149899
4.542248
0.220690
0.136614
2.329038
(e-05)
1.223383
1.078579
3.693576
0.095742
0.050000
1.528616
Rotation around axis z
Table 2. Invariant test on rotation factor for sensed LiDAR data I1 I2 I3 I4 I5
I6
mean
0.9112
0.0065
0.2930
0.0050
0.0044
0.3308
0.0226
0.0019
0.0463
0.0028
0.0030
0.0282
3.1.2 Test on Broken Roofs: Figure 2 shows two cases of broken roofs. This broken situation may be caused by trees or any covers. The invariants are calculated as Table 3 which indicates that it has chance to recognize the broken roof as a complete one.
with hole
Table 3. Invariant test on Roofs with holes for simulated data I1 I2 I3 I4 I5
I6
Case 1
3.626593
2.312053
4.880842
0.233581
0.144418
2.457453
Case 2
3.284115
1.732086
4.246861
0.256151
0.143455
2.157720
(a) Case 1: the simulated Point Clouds with holes
(b) Case 2: the simulated Point Clouds with holes
Figure 2. Point clouds with hoes for invariant test 3.1 Invariant Test on Zernike Moments 3.1.1 Test on Rotation and noise: In this case, the sensed LiDAR data in Figure 1 (b) is used and re-locates inside a unit circle shown as Figure 3 (a). Figure 3 (b), (c), (d) show the reconstruction results of using 5, 10, 15 Zernike moments respectively. The invariants are then calculated under different rotations and are shown as Figure 4 (a). Figure 4 (b) shows the influences of noises which are added to the simulated data. Clearly there is less influence on invariants in both cases.
(a) The Sensed data in a unit circle
(d) Reconstruction using 15 Zernike Moments
(b) Reconstruction using 5 Zernike Moments
(c) Reconstruction using 10 Zernike Moments
Figure 3. The original model and reconstruction results of sensed data
0.8
0.8
0 15 60 120
Standard Deviation
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 0.1 0.5 2
I
5
0
0
1
1.5
2
2.5
3
3.5 number of I
4
4.5
5
5.5
(a) The invariants of sensed data under different rotations.
6
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
(b) The invariants of simulated data under different levels of noises.
Figrue 4. The invariants of the two kinds of test data. 4. DISCUSSIONS AND FUTURE WORKS This paper introduces two kinds of invariant features of Airborne LiDAR data. Take Figure 4 for an example, (a) and (b) can be considered as the signatures of the two kind roof types. Each signature is unique and can be used for roof template matching. Our future work will be focused on the set of roof database. Besides, the Zernike Moments we used in this paper is 2D. The 3D Zernike Moments (Novotni and Klein 2004) have also been developed and can be further applied to the Airborne LiDAR data. Finally, Comparing the two Methods, Geometric moments can be calculated directly while the Zernike moments need to translate the origin of the coordinate system to the mass of object. The more pre-processing it needs, the more influence the results would cause. Once the unique features are determined, they can be used in a classifier to recognize roofs. All of them are going to be our future works. REFERENCES: Elberink, S. O. (2008). PROBLEMS IN AUTOMATED BUILDING RECONSTRUCTION BASED ON DENSE AIRBORNE LASER SCANNING DATA International Society for Photogrammetry and Remote Sensing Beijing China. Jaynes, C., E. Riseman, et al. (2003). "Recognition and reconstruction of buildings from multiple aerial images." Computer Vision and Image Understanding 90(1): 68-98. Maaoui, C., H. Laurent, et al. (2005). 2D color shape recognition using Zernike moments. Image Processing, 2005. ICIP 2005. IEEE International Conference on. Novotni, M. and R. Klein (2004). "Shape retrieval using 3D Zernike descriptors." Computer-Aided Design 36(11): 1047-1062. Teague, M. R. (1980). "Image analysis via the general theory of moments*." J. Opt. Soc. Am. 70(8): 920-930. Xu, D. and H. Li (2008). "Geometric moment invariants." Pattern Recognition 41(1): 240-249.
