3D Polar-Radius Invariant Moments and Structure Moment Invariants
3D Polar-Radius Invariant Moments and Structure Moment Invariants Zongmin Li1,2,3 , Yuanzhen Zhang1 , Kunpeng Hou1 , and Hua Li2 1
School of Computer Science and Communication Engineering, University of Petroleum, 257061, Shandong, P.R. China 2 Institute of Computing Technology, Chinese Academy of Sciences, 100080, Beijing, P.R. China 3 Graduate School of Chinese Academy of Sciences, 100039, Beijing, P.R. China {zmli, lihua}@ict.ac.cn [email protected], [email protected]
Abstract. A novel moment, called 3D polar-radius-invariant-moment, is proposed for the 3D object recognition and classification. Some properties of these new moments including the invariance on translation, scale and rotation transforms are studied and proved. Then structure moment invariants are given to distinguish complicated similar shapes. Examples are presented to illustrate the performance and invariance of these moments. With the help of these moment invariants, the 3D models are distinguished accurately.
1
Introduction
With the development of computer graphics and related software and hardware technologies, 3D models can be acquired easily and now play an important role in many mainstream applications such as mechanical manufacture, games, biochemistry, medicine, E-business, art, virtual reality, etc. Tools for acquiring and visualizing 3D models have become integral components of data processingAs a result, the need for the ability to retrieve models from large databases has gained prominence and a key concern of shape analysis has shifted to the design of efficient and robust matching algorithms. Shapes are described in a transformation invariant manner, so that any transformation of a shape will be described in the same way, and the best measure of similarity is obtained at any transformation. In this paper, we present a 3D content based retrieval method relying on 3D polar-radius invariant moments. We define the polar-radius invariant moment and its normalized moment, and the central polar-radius invariant moment and its normalized central moment. The translation, scale and rotation invariance of the normalized moment and normalized central moment are proved theoretically. Then we present structure moment invariants for complicated similar shapes. To support our new theory, an algorithm for object shape recognition is designed based on the new moments and experiments are conducted. Examples are presented to illustrate the performance of these moments. In the comparing L. Wang, K. Chen, and Y.S. Ong (Eds.): ICNC 2005, LNCS 3611, pp. 483–492, 2005. c Springer-Verlag Berlin Heidelberg 2005
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experiment of recognition of objects, 3D polar-radius invariant moments give an encouraging high recognition rates. And the complicated similar shapes can be distinguished. This paper is organized as follows. Firstly, some related works are summarized in Section 2. Then, The 3D polar-radius-invariant-moments representation is presented in Section 3,and the translation, scale and rotation invariance of these moments are proved. In section 4, structure moment invariants are proposed to distinguish complicated similar shapes. Finally, some retrieved examples are presented to analyze the validity of our method in Section 5. We conclude in Section 6 by summarizing our results and discussing topics for future work.
2
Related Works
Up to now, in the area of content-based 3D model retrieval, some original systems investigating theory and algorithm have been implemented, and some systems for general 3D objects retrieval have been introduced. For the latter, the first system was introduced in [1],which was followed by [2].A very present result is presented in[3].The feature of 3D objects extracted in them mainly include shape and color of 3D objects, as well as combination of the bottom level shape feature and semantic feature. According to the feature description of the 3D objects shape, the shape feature ex-traction consists of: (1) the shape feature extraction based on the analysis of geometric structure. It was dissertated in the papers [2,4-9].The feature extracted get a better description of information about 3D models structure. But it is applicable to some well required models and need a lot of computation to deal with the coordinates of the model. (2) the shape feature extraction based on the topological structure. The papers [10,11] introduced MRG (Multiresolution Reeb Graph) to obtain the feature of 3D objects. The MRG can well depicts the topological structure of 3D models and has a good stability to the shape transform of 3D models. But the drawback is much of computation and sensitive to the edge disturbance and noise. (3) the shape feature extraction based on the image of function. The method gets a detailed depiction in the papers [12-19]. The advantage of it is the simple feature convenient for the similarity matching. The disadvantage is some of important information lost in the course of the functional image. (4) the shape feature extraction based on the statistical characteristics. The merit of this method is that it is not required to standardize the model coordinate comparing with the above methods, and the feature extracted consists of global shape attribute, such as circularity, eccentricity, algebraic moments etc[20,21]. The feature is simple and can well satisfy the invariance of geometrical transform.The method is also applicable to the incompact and degenerate models and stable for the edge noise. The demerit is that the feature hasn’t a sufficient depict of 3D model and is sensitive to the topo-logical structure of 3D models.
3D Polar-Radius Invariant Moments and Structure Moment Invariants
3
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3D Polar-Radius Invariant Moments
We assume an object is a three-dimensional object represented by a set of vertices and a set of polygonal feces embedded in three dimensions. For object D, the three-dimensional polar-radius invariant moments of order p of a density f(x,y,z) are defined as Mp = rp dV, (1) D
where r is the distance from an arbitrary point of the object to the center of the object, (xc , yc , zc ) is center of the object D, and r = (x − xc )2 + (y − yc )2 + (z − zc )2 , (2)
xc =
1 V
D
xdV, yc =
1 V
D
ydV, zc =
1 V
zdV.
(3)
D
The central polar-radius invariant moments Mcp are defined as (r − r)p dV Mcp =
(4)
D
where r = V1 D rdV . The normalized moment of the polar-radius invariant moments and the normalized central moment of the central polar-radius invariant moments are defined as 1 r ( )p dV (5) Mnp = V r D Mncp =
1 V
( D
r−r p ) dV r
(6)
If an analog original object is digitized into its discrete version with its voxels, the integration of (5) must be approximated by summations. It has been a common prescription to replace Mnp in (5) with its digital version V 1 ri p ˆ Mnp = ( ) V i=1 r
(7)
V ri − r p ˆ ncp = 1 M ) ( V i=1 r
(8)
where V is the voxel summation of the object. The normalized moments Mnp of the polar-radius invariant moments and the normalized central moments Mncp of the central polar-radius invariant moments are invariants under translation, scale and rotation.
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A. Invariants Under Translation Under translation of coordinates, x = x + α, y = y + β, z = z + γ, where α, β and γ are constants. The distance from an arbitrary point of the object to the center of the object don’t change, and r don’t change too, therefore, Mnp and Mncp are invariants under translation. B. Invariants Under Scale D is the new object of D Under scale transformation, then r = αr, r = αr, V = α3 V , dV = α3 dV where α is a constant. Therefore αr p 3 r p 1 Mnp = V1 D ( ) dV = V (α)3 D ( αr ) (α) dV = Mnp .
r
Similarly (Mncp ) = Mncp . C. Invariants Under Rotation Under rotation transformation , the turning of an object by an angle φ about the center of the object is equivalent to that first rotate the object about one axis by the angle ϕ, then rotate the resulting object about another axis by the angle ψ. D denotes the new object rotate about one axis by the angle ϕ from D, and V denotes the volume of D . D denotes the new object rotate about one axis by the angle ψ from D , and V denotes the volume of D . Therefore, D = D = D, V = V = V , and 2 1 r = V1 D r · rdrd(θ + ϕ)dz = V D r drdθdz = r, r p 1 r p 1 Mnp = V D ( ) rdrd(θ + ϕ)dz = V D ( r ) rdrdθdz = Mnp . r
For D , the above form may be written as r p Mnp = V1 D ( r ) dV , where dV = r dr dθ dz . Then r = V1 r · r dr d(θ + ψ)dz = V1 (r )2 dr dθ dz = r , D D ( r )p r dr d(θ +ψ)dz = V1 ( r )p dr dθ dz = (Mnp ) . Mnp = D D r
r
We have the following expression: Mnp = Mnp . Similarly Mncp = Mncp .
4
Descriptions of Structure Moment Invariants
It is quite difficult to how to distinguish complicated similar shapes. Whether the “structure” of a picture is abundant or not, means that is it sharply or gently of the picture’s light intensity when varying with the position. The degree of abundant structure of the 2-D object is consistent with the integral as follows: I02 (x, y)dxdy (9) SF = F
That is to say, base on the premise that the total light energy is given definitely in area F, the bigger SF is, the more abundant the structure of the 2-D object is.
3D Polar-Radius Invariant Moments and Structure Moment Invariants
(a)
(b)
487
(c)
Examples of distribution of light intensity FigureFig. 1. 1.Examples ofthethe distribution of light intensity
F
I02 (x, y)dxdy = const
(10)
Just like the distribution of light intensity in Fig 1, the total energy of each one is 32. However, the integral value will be different after square of three integrable functions respectively: Fig 1 (a): I02 dx = 128 Fig 1 (b): I02 dx = 134 Fig 1 (c): I02 dx = 180 Namely the more abundant the structure is, the greater the value is. It is similar in 3D space. For this reason, in order to achieve the goal of recognition, we mapped the object function f(x) to another transformation space, then we got a new moment and we called it structure moment invariant: F (f (x)) · ψi (x)dx (11) µi =< F (f ), ψi >= Ω
Note F(f ) is function of f, and we can use the projection of object function f ∈ L2 on the area of Ω to define the moment µi which is used in analysis of object’s shape. The function on the area of Ω was defined as Ψ = {ψi } where i ∈ N. F can be linear transformation, and can be nonlinear transformation too. If g(x) = F (f (x)), then vi =< g(x), ψi >= Ω g(x)·ψi (x)dx. The complicated objects will be recognized through the existing pattern recognition methods.
5
Experimentation and Results
Firstly, we test the algorithm with three horse models, a bird model, and a pig model, given in VRML. Fig.a is represented with 3493 points and 6520 patches, Fig.b with 784 points and 1328 patches, Fig.c with 2129 points and 4034 patches, Fig.d with 4203 points and 1928 patches, and Fig.e with 11238 points and 6902 patches.
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2
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4
Fig.a Horse1 model: 1.the original, 2.after rotation, 3.after scale, 4.after translation
1
2
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Fig.b Horse2 model: 1.the original, 2.after rotation, 3.after scale, 4.after translation
1 2 3 4 Fig.c Horse3 model: 1.the original, 2.after rotation, 3.after scale, 4.after translation
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Fig.d Bird model: 1.the original, 2.after rotation, 3.after scale, 4.after translation
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Fig.e Pig model: 1.the original, 2.after rotation, 3.after scale, 4.after translation
In each model, we choose the original object as a pattern. We compute eighteen moments about each object and give some results about fig.(a,b,c) in table 1-3, and the distances between each pattern and models with Minkowski distance method in table 4. Experiments give a high recognition rates. From table 4, the recognition rate is 97% if the threshold is 5. Then, we find that horse2 is quite similar to horse3. In the horse2 column,fig.b2 is not recognized. we compute the Minkowski distance between horse2 and horse3 with three-dimensional polar-radius invariant moments and structure moment Invariants. From Table 5, the results about structure moment Invariants are better than the ones about three-dimensional polar-radius invariant moments.
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Table 1. Normalized moments and normalized central moments for fig.a 2 1 2 3 4
4
5
6
7
1.356 2.247 4.205 8.478 17.94 3.929e+001 1.351 2.229 4.158 8.363 17.66 3.860e+001 1.349 2.220 4.128 8.285 17.49 3.824e+001 1.357 2.248 4.207 8.476 17.92 3.919e+001
Central 2 1 2 3 4
3
3
4
5
6
7
0.355 0.178 0.353 0.363 0.548 7.202e-001 0.351 0.177 0.346 0.356 0.534 6.970e-001 0.349 0.172 0.343 0.352 0.535 7.059e-001 0.356 0.178 0.352 0.361 0.544 7.108e-001
8
9
10
8.825e+001 8.649e+001 8.588e+001 8.787e+001
2.023e+002 1.977e+002 1.971e+002 2.011e+002
4.721e+002 4.595e+002 4.608e+002 4.681e+002
8
9
10
1.069e+000 1.023e+000 1.056e+000 1.051e+000
1.563e+000 1.473e+000 1.554e+000 1.529e+000
2.401e+000 2.223e+000 2.407e+000 2.341e+000
Table 2. Normalized moments and normalized central moments for fig.b 2 1 2 3 4
4
5
6
7
1.261 1.880 3.149 5.714 10.95 2.182e+001 1.263 1.886 3.167 5.762 11.08 2.213e+001 1.264 1.890 3.174 5.775 11.09 2.213e+001 1.265 1.896 3.197 5.839 11.27 2.257e+001
Central 2 1 2 3 4
3
3
4
5
6
7
0.261 0.096 0.196 0.157 0.222 2.342e-001 0.262 0.097 0.199 0.160 0.228 2.430e-001 0.263 0.097 0.199 0.159 0.226 2.388e-001 0.265 0.100 0.203 0.165 0.234 2.500e-001
8
9
10
4.465e+001 4.545e+001 4.536e+001 4.646e+001
9.324e+001 9.524e+001 9.482e+001 9.751e+001
1.977e+002 2.027e+002 2.012e+002 2.077e+002
8
9
10
3.043e-001 3.177e-001 3.102e-001 3.256e-001
3.579e-001 3.776e-001 3.648e-001 3.862e-001
4.582e-001 4.875e-001 4.668e-001 4.961e-001
Table 3. Normalized moments and normalized central moments for fig.c 2 1 2 3 4
4
5
6
7
1.251 1.838 3.018 5.356 10.03 1.949e+001 1.247 1.824 2.984 5.279 9.863 1.915e+001 1.253 1.847 3.047 5.433 10.22 1.997e+001 1.252 1.839 3.023 5.368 10.06 1.957e+001
Central 2 1 2 3 4
3
3
4
5
6
7
0.251 0.084 0.175 0.128 0.181 1.792e-001 0.247 0.082 0.171 0.126 0.178 1.779e-001 0.253 0.086 0.179 0.134 0.189 1.899e-001 0.251 0.084 0.175 0.129 0.183 1.811e-001
8
9
10
3.894e+001 3.826e+001 4.010e+001 3.913e+001
7.939e+001 7.807e+001 8.217e+001 7.985e+001
1.644e+002 1.620e+002 1.710e+002 1.656e+002
8
9
10
2.291e-001 2.289e-001 2.437e-001 2.320e-001
2.565e-001 2.593e-001 2.765e-001 2.607e-001
3.209e-001 3.275e-001 3.483e-001 3.270e-001
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Table 5. The distances between each of horse2 and horse3
6
Conclusions and Future Work
We propose a new moment, called 3D polar-radius-invariant-moment, for determining the content-based similarity of three-dimensional objects. Two main issues are considered. The first is the invariance on translation, scale and rotation transform about these new moments. The second issue is the method for distinguishing complicated similar shapes. Experiments exhibit very good results. Further work is required in order to analyze the computation for these high order moments, and we will propose extensions and improvements. Moreover, we intend to elaborate a new feature vector with the combination of these new moment invariants.
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References 1. Paquet E., Rioux M. . A content-based search engine for VRML databases . In Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition , Santa Barbara, California, USA, 1998, 541-546. 2. Suzuki M., Kato T., Otsu N.. A similarity retrieval of 3d polygon models using rotation invariant shape descriptors. In: Proceedings of IEEE International Conference on Systems , Man, and Cybernetics, Nashville, Tennessee, USA ,2000, 2946-2952. 3. Yang YuBin, Lin Hui, Zhu Qing . Content-Based 3D Model Retrieval: A Survey. In:Chinese Journal Of Computers,China, Vol.27, No.10, Oct.2004. 4. Vranic D., Saupe D.. 3d model retrieval. In: Proceedings of Spring Conference on Computer Graphics ,Budmerice,Slovakia,2000,89-93. 5. Tangelder J., VeltKamp R.. Polyhedral model retrieval using weight point sets. International Journal of Image and Graphics,2003,3(1):1-21. 6. Zhang D., Hebert M.. Harmonic maps and their applications in surface matching. In: Pro-ceedings of IEEE Conference on Computer Vision and Pattern Recognition, Fort Collins, Colorado,USA,1999,2524-2530. 7. Zhang D., Hebert M.. Harmonic shape images: A 3d freeform surface representation and its application in surface matching. In: Proceedings of International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition, Sophia Antipo-lis ,France,1999,30-43. 8. Vranic D., Saupe D.. Description of 3d-shape using a complex function on the sphere. In: Proceedings of IEEE International Conference on Multimedia and Expo, Lausanne,Switzerland,2002,177-180. 9. Vranic D., Saupe D.. 3d shape Descriptor based on 3d Fourier transform. In: Proceedings of IEEE EURASIP Conference on Digital Signal Processing for Multimedia Communications and Services, Budapest, Hungary, 2001,271-274. 10. Hilaga M., Shinagawa Y., Kohmura T., Kunii T.. Topology matching for fully automatic similarity estimation of 3d shapes. In: Proceedings of ACM SIGGRAPH, Los Angels, USA, 2001, 203-212. 11. Bardinet E., Vidal S., Arroyo S., Malandain G., Capilla N.. Structural object matching. Department of Computer Science and AI, University of Granada ,Spain: Technical Report DEC-SAI-000303,2000. 12. Funkhouser T., Min P., Kazhdan M. et al.. A search engine for 3d models. ACM Transac-tions on Graphics,2003,22(1):83-105. 13. Cyr C., Kimia B.. 3d object recognition using shape similarity-based aspect graph. In: Pro-ceedings of IEEE International Conference on Computer Vision , Vancouver, Can-ada,2001,254-261. 14. Heczko M., Keim D., Saupe D., Vranic D.. A method for similarity search of 3d objects. In: Proceedings of German Database Conference (BTW),Oldenburg,Germany,2001,384-401. 15. Zhang D., Hebert M.. Harmonic maps and their applications in surface matching. In: Pro-ceedings of IEEE Conference on Computer Vision and Pattern Recognition, Fort Collins, Colorado,USA,1999,2524-2530. 16. Little J., Extended Gaussian images , mixed volumes, shape reconstruction. In: Proceedings of Annual Symposium on Computational Geometry, Baltimore, USA, 1985, 15-23. 17. Delingette H., Hebert H., Ikeuchi K.. A spherical representation for the recognition of curved objects. In: proceedings of IEEE International Conference on Computer Vision, Berlin, Germany, 1993, 103-112.
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18. Johnson A., Hebert M.. Using spin-images for efficient multiple model recognition in clut-tered 3d scenes. IEEE Transaction on Pattern Analysis and Machine Intelligence, 1999, 21(5), 433-449. 19. Saupe D., Vranic D.. 3d model retrieval with spherical harmonics and moments. In: Pro-ceedings of Germany DAGM Conference , Munich, Germany, 2001, 392-397. 20. M. Elad, P. Milanfar, and Gene Golub, Shape from Moments - An Estimation Theory Per-spective, the IEEE Trans. on Signal Processing on August 2002, 18141829 21. M. Novotni.3D Zernike Descriptors for Content Based Shape Retrieval, proceedings of The 8th ACM Symposium on Solid Modeling and Applications, June 2003,216225
School of Computer Science and Communication Engineering, University of Petroleum, 257061, Shandong, P.R. China 2 Institute of Computing Technology, Chinese Academy of Sciences, 100080, Beijing, P.R. China 3 Graduate School of Chinese Academy of Sciences, 100039, Beijing, P.R. China {zmli, lihua}@ict.ac.cn [email protected], [email protected]
Abstract. A novel moment, called 3D polar-radius-invariant-moment, is proposed for the 3D object recognition and classification. Some properties of these new moments including the invariance on translation, scale and rotation transforms are studied and proved. Then structure moment invariants are given to distinguish complicated similar shapes. Examples are presented to illustrate the performance and invariance of these moments. With the help of these moment invariants, the 3D models are distinguished accurately.
1
Introduction
With the development of computer graphics and related software and hardware technologies, 3D models can be acquired easily and now play an important role in many mainstream applications such as mechanical manufacture, games, biochemistry, medicine, E-business, art, virtual reality, etc. Tools for acquiring and visualizing 3D models have become integral components of data processingAs a result, the need for the ability to retrieve models from large databases has gained prominence and a key concern of shape analysis has shifted to the design of efficient and robust matching algorithms. Shapes are described in a transformation invariant manner, so that any transformation of a shape will be described in the same way, and the best measure of similarity is obtained at any transformation. In this paper, we present a 3D content based retrieval method relying on 3D polar-radius invariant moments. We define the polar-radius invariant moment and its normalized moment, and the central polar-radius invariant moment and its normalized central moment. The translation, scale and rotation invariance of the normalized moment and normalized central moment are proved theoretically. Then we present structure moment invariants for complicated similar shapes. To support our new theory, an algorithm for object shape recognition is designed based on the new moments and experiments are conducted. Examples are presented to illustrate the performance of these moments. In the comparing L. Wang, K. Chen, and Y.S. Ong (Eds.): ICNC 2005, LNCS 3611, pp. 483–492, 2005. c Springer-Verlag Berlin Heidelberg 2005
484
Z. Li et al.
experiment of recognition of objects, 3D polar-radius invariant moments give an encouraging high recognition rates. And the complicated similar shapes can be distinguished. This paper is organized as follows. Firstly, some related works are summarized in Section 2. Then, The 3D polar-radius-invariant-moments representation is presented in Section 3,and the translation, scale and rotation invariance of these moments are proved. In section 4, structure moment invariants are proposed to distinguish complicated similar shapes. Finally, some retrieved examples are presented to analyze the validity of our method in Section 5. We conclude in Section 6 by summarizing our results and discussing topics for future work.
2
Related Works
Up to now, in the area of content-based 3D model retrieval, some original systems investigating theory and algorithm have been implemented, and some systems for general 3D objects retrieval have been introduced. For the latter, the first system was introduced in [1],which was followed by [2].A very present result is presented in[3].The feature of 3D objects extracted in them mainly include shape and color of 3D objects, as well as combination of the bottom level shape feature and semantic feature. According to the feature description of the 3D objects shape, the shape feature ex-traction consists of: (1) the shape feature extraction based on the analysis of geometric structure. It was dissertated in the papers [2,4-9].The feature extracted get a better description of information about 3D models structure. But it is applicable to some well required models and need a lot of computation to deal with the coordinates of the model. (2) the shape feature extraction based on the topological structure. The papers [10,11] introduced MRG (Multiresolution Reeb Graph) to obtain the feature of 3D objects. The MRG can well depicts the topological structure of 3D models and has a good stability to the shape transform of 3D models. But the drawback is much of computation and sensitive to the edge disturbance and noise. (3) the shape feature extraction based on the image of function. The method gets a detailed depiction in the papers [12-19]. The advantage of it is the simple feature convenient for the similarity matching. The disadvantage is some of important information lost in the course of the functional image. (4) the shape feature extraction based on the statistical characteristics. The merit of this method is that it is not required to standardize the model coordinate comparing with the above methods, and the feature extracted consists of global shape attribute, such as circularity, eccentricity, algebraic moments etc[20,21]. The feature is simple and can well satisfy the invariance of geometrical transform.The method is also applicable to the incompact and degenerate models and stable for the edge noise. The demerit is that the feature hasn’t a sufficient depict of 3D model and is sensitive to the topo-logical structure of 3D models.
3D Polar-Radius Invariant Moments and Structure Moment Invariants
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485
3D Polar-Radius Invariant Moments
We assume an object is a three-dimensional object represented by a set of vertices and a set of polygonal feces embedded in three dimensions. For object D, the three-dimensional polar-radius invariant moments of order p of a density f(x,y,z) are defined as Mp = rp dV, (1) D
where r is the distance from an arbitrary point of the object to the center of the object, (xc , yc , zc ) is center of the object D, and r = (x − xc )2 + (y − yc )2 + (z − zc )2 , (2)
xc =
1 V
D
xdV, yc =
1 V
D
ydV, zc =
1 V
zdV.
(3)
D
The central polar-radius invariant moments Mcp are defined as (r − r)p dV Mcp =
(4)
D
where r = V1 D rdV . The normalized moment of the polar-radius invariant moments and the normalized central moment of the central polar-radius invariant moments are defined as 1 r ( )p dV (5) Mnp = V r D Mncp =
1 V
( D
r−r p ) dV r
(6)
If an analog original object is digitized into its discrete version with its voxels, the integration of (5) must be approximated by summations. It has been a common prescription to replace Mnp in (5) with its digital version V 1 ri p ˆ Mnp = ( ) V i=1 r
(7)
V ri − r p ˆ ncp = 1 M ) ( V i=1 r
(8)
where V is the voxel summation of the object. The normalized moments Mnp of the polar-radius invariant moments and the normalized central moments Mncp of the central polar-radius invariant moments are invariants under translation, scale and rotation.
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A. Invariants Under Translation Under translation of coordinates, x = x + α, y = y + β, z = z + γ, where α, β and γ are constants. The distance from an arbitrary point of the object to the center of the object don’t change, and r don’t change too, therefore, Mnp and Mncp are invariants under translation. B. Invariants Under Scale D is the new object of D Under scale transformation, then r = αr, r = αr, V = α3 V , dV = α3 dV where α is a constant. Therefore αr p 3 r p 1 Mnp = V1 D ( ) dV = V (α)3 D ( αr ) (α) dV = Mnp .
r
Similarly (Mncp ) = Mncp . C. Invariants Under Rotation Under rotation transformation , the turning of an object by an angle φ about the center of the object is equivalent to that first rotate the object about one axis by the angle ϕ, then rotate the resulting object about another axis by the angle ψ. D denotes the new object rotate about one axis by the angle ϕ from D, and V denotes the volume of D . D denotes the new object rotate about one axis by the angle ψ from D , and V denotes the volume of D . Therefore, D = D = D, V = V = V , and 2 1 r = V1 D r · rdrd(θ + ϕ)dz = V D r drdθdz = r, r p 1 r p 1 Mnp = V D ( ) rdrd(θ + ϕ)dz = V D ( r ) rdrdθdz = Mnp . r
For D , the above form may be written as r p Mnp = V1 D ( r ) dV , where dV = r dr dθ dz . Then r = V1 r · r dr d(θ + ψ)dz = V1 (r )2 dr dθ dz = r , D D ( r )p r dr d(θ +ψ)dz = V1 ( r )p dr dθ dz = (Mnp ) . Mnp = D D r
r
We have the following expression: Mnp = Mnp . Similarly Mncp = Mncp .
4
Descriptions of Structure Moment Invariants
It is quite difficult to how to distinguish complicated similar shapes. Whether the “structure” of a picture is abundant or not, means that is it sharply or gently of the picture’s light intensity when varying with the position. The degree of abundant structure of the 2-D object is consistent with the integral as follows: I02 (x, y)dxdy (9) SF = F
That is to say, base on the premise that the total light energy is given definitely in area F, the bigger SF is, the more abundant the structure of the 2-D object is.
3D Polar-Radius Invariant Moments and Structure Moment Invariants
(a)
(b)
487
(c)
Examples of distribution of light intensity FigureFig. 1. 1.Examples ofthethe distribution of light intensity
F
I02 (x, y)dxdy = const
(10)
Just like the distribution of light intensity in Fig 1, the total energy of each one is 32. However, the integral value will be different after square of three integrable functions respectively: Fig 1 (a): I02 dx = 128 Fig 1 (b): I02 dx = 134 Fig 1 (c): I02 dx = 180 Namely the more abundant the structure is, the greater the value is. It is similar in 3D space. For this reason, in order to achieve the goal of recognition, we mapped the object function f(x) to another transformation space, then we got a new moment and we called it structure moment invariant: F (f (x)) · ψi (x)dx (11) µi =< F (f ), ψi >= Ω
Note F(f ) is function of f, and we can use the projection of object function f ∈ L2 on the area of Ω to define the moment µi which is used in analysis of object’s shape. The function on the area of Ω was defined as Ψ = {ψi } where i ∈ N. F can be linear transformation, and can be nonlinear transformation too. If g(x) = F (f (x)), then vi =< g(x), ψi >= Ω g(x)·ψi (x)dx. The complicated objects will be recognized through the existing pattern recognition methods.
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Experimentation and Results
Firstly, we test the algorithm with three horse models, a bird model, and a pig model, given in VRML. Fig.a is represented with 3493 points and 6520 patches, Fig.b with 784 points and 1328 patches, Fig.c with 2129 points and 4034 patches, Fig.d with 4203 points and 1928 patches, and Fig.e with 11238 points and 6902 patches.
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Fig.a Horse1 model: 1.the original, 2.after rotation, 3.after scale, 4.after translation
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Fig.b Horse2 model: 1.the original, 2.after rotation, 3.after scale, 4.after translation
1 2 3 4 Fig.c Horse3 model: 1.the original, 2.after rotation, 3.after scale, 4.after translation
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Fig.d Bird model: 1.the original, 2.after rotation, 3.after scale, 4.after translation
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Fig.e Pig model: 1.the original, 2.after rotation, 3.after scale, 4.after translation
In each model, we choose the original object as a pattern. We compute eighteen moments about each object and give some results about fig.(a,b,c) in table 1-3, and the distances between each pattern and models with Minkowski distance method in table 4. Experiments give a high recognition rates. From table 4, the recognition rate is 97% if the threshold is 5. Then, we find that horse2 is quite similar to horse3. In the horse2 column,fig.b2 is not recognized. we compute the Minkowski distance between horse2 and horse3 with three-dimensional polar-radius invariant moments and structure moment Invariants. From Table 5, the results about structure moment Invariants are better than the ones about three-dimensional polar-radius invariant moments.
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Table 1. Normalized moments and normalized central moments for fig.a 2 1 2 3 4
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1.356 2.247 4.205 8.478 17.94 3.929e+001 1.351 2.229 4.158 8.363 17.66 3.860e+001 1.349 2.220 4.128 8.285 17.49 3.824e+001 1.357 2.248 4.207 8.476 17.92 3.919e+001
Central 2 1 2 3 4
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0.355 0.178 0.353 0.363 0.548 7.202e-001 0.351 0.177 0.346 0.356 0.534 6.970e-001 0.349 0.172 0.343 0.352 0.535 7.059e-001 0.356 0.178 0.352 0.361 0.544 7.108e-001
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8.825e+001 8.649e+001 8.588e+001 8.787e+001
2.023e+002 1.977e+002 1.971e+002 2.011e+002
4.721e+002 4.595e+002 4.608e+002 4.681e+002
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1.069e+000 1.023e+000 1.056e+000 1.051e+000
1.563e+000 1.473e+000 1.554e+000 1.529e+000
2.401e+000 2.223e+000 2.407e+000 2.341e+000
Table 2. Normalized moments and normalized central moments for fig.b 2 1 2 3 4
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1.261 1.880 3.149 5.714 10.95 2.182e+001 1.263 1.886 3.167 5.762 11.08 2.213e+001 1.264 1.890 3.174 5.775 11.09 2.213e+001 1.265 1.896 3.197 5.839 11.27 2.257e+001
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0.261 0.096 0.196 0.157 0.222 2.342e-001 0.262 0.097 0.199 0.160 0.228 2.430e-001 0.263 0.097 0.199 0.159 0.226 2.388e-001 0.265 0.100 0.203 0.165 0.234 2.500e-001
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4.465e+001 4.545e+001 4.536e+001 4.646e+001
9.324e+001 9.524e+001 9.482e+001 9.751e+001
1.977e+002 2.027e+002 2.012e+002 2.077e+002
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3.043e-001 3.177e-001 3.102e-001 3.256e-001
3.579e-001 3.776e-001 3.648e-001 3.862e-001
4.582e-001 4.875e-001 4.668e-001 4.961e-001
Table 3. Normalized moments and normalized central moments for fig.c 2 1 2 3 4
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1.251 1.838 3.018 5.356 10.03 1.949e+001 1.247 1.824 2.984 5.279 9.863 1.915e+001 1.253 1.847 3.047 5.433 10.22 1.997e+001 1.252 1.839 3.023 5.368 10.06 1.957e+001
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0.251 0.084 0.175 0.128 0.181 1.792e-001 0.247 0.082 0.171 0.126 0.178 1.779e-001 0.253 0.086 0.179 0.134 0.189 1.899e-001 0.251 0.084 0.175 0.129 0.183 1.811e-001
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3.894e+001 3.826e+001 4.010e+001 3.913e+001
7.939e+001 7.807e+001 8.217e+001 7.985e+001
1.644e+002 1.620e+002 1.710e+002 1.656e+002
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2.291e-001 2.289e-001 2.437e-001 2.320e-001
2.565e-001 2.593e-001 2.765e-001 2.607e-001
3.209e-001 3.275e-001 3.483e-001 3.270e-001
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Table 5. The distances between each of horse2 and horse3
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Conclusions and Future Work
We propose a new moment, called 3D polar-radius-invariant-moment, for determining the content-based similarity of three-dimensional objects. Two main issues are considered. The first is the invariance on translation, scale and rotation transform about these new moments. The second issue is the method for distinguishing complicated similar shapes. Experiments exhibit very good results. Further work is required in order to analyze the computation for these high order moments, and we will propose extensions and improvements. Moreover, we intend to elaborate a new feature vector with the combination of these new moment invariants.
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References 1. Paquet E., Rioux M. . A content-based search engine for VRML databases . In Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition , Santa Barbara, California, USA, 1998, 541-546. 2. Suzuki M., Kato T., Otsu N.. A similarity retrieval of 3d polygon models using rotation invariant shape descriptors. In: Proceedings of IEEE International Conference on Systems , Man, and Cybernetics, Nashville, Tennessee, USA ,2000, 2946-2952. 3. Yang YuBin, Lin Hui, Zhu Qing . Content-Based 3D Model Retrieval: A Survey. In:Chinese Journal Of Computers,China, Vol.27, No.10, Oct.2004. 4. Vranic D., Saupe D.. 3d model retrieval. In: Proceedings of Spring Conference on Computer Graphics ,Budmerice,Slovakia,2000,89-93. 5. Tangelder J., VeltKamp R.. Polyhedral model retrieval using weight point sets. International Journal of Image and Graphics,2003,3(1):1-21. 6. Zhang D., Hebert M.. Harmonic maps and their applications in surface matching. In: Pro-ceedings of IEEE Conference on Computer Vision and Pattern Recognition, Fort Collins, Colorado,USA,1999,2524-2530. 7. Zhang D., Hebert M.. Harmonic shape images: A 3d freeform surface representation and its application in surface matching. In: Proceedings of International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition, Sophia Antipo-lis ,France,1999,30-43. 8. Vranic D., Saupe D.. Description of 3d-shape using a complex function on the sphere. In: Proceedings of IEEE International Conference on Multimedia and Expo, Lausanne,Switzerland,2002,177-180. 9. Vranic D., Saupe D.. 3d shape Descriptor based on 3d Fourier transform. In: Proceedings of IEEE EURASIP Conference on Digital Signal Processing for Multimedia Communications and Services, Budapest, Hungary, 2001,271-274. 10. Hilaga M., Shinagawa Y., Kohmura T., Kunii T.. Topology matching for fully automatic similarity estimation of 3d shapes. In: Proceedings of ACM SIGGRAPH, Los Angels, USA, 2001, 203-212. 11. Bardinet E., Vidal S., Arroyo S., Malandain G., Capilla N.. Structural object matching. Department of Computer Science and AI, University of Granada ,Spain: Technical Report DEC-SAI-000303,2000. 12. Funkhouser T., Min P., Kazhdan M. et al.. A search engine for 3d models. ACM Transac-tions on Graphics,2003,22(1):83-105. 13. Cyr C., Kimia B.. 3d object recognition using shape similarity-based aspect graph. In: Pro-ceedings of IEEE International Conference on Computer Vision , Vancouver, Can-ada,2001,254-261. 14. Heczko M., Keim D., Saupe D., Vranic D.. A method for similarity search of 3d objects. In: Proceedings of German Database Conference (BTW),Oldenburg,Germany,2001,384-401. 15. Zhang D., Hebert M.. Harmonic maps and their applications in surface matching. In: Pro-ceedings of IEEE Conference on Computer Vision and Pattern Recognition, Fort Collins, Colorado,USA,1999,2524-2530. 16. Little J., Extended Gaussian images , mixed volumes, shape reconstruction. In: Proceedings of Annual Symposium on Computational Geometry, Baltimore, USA, 1985, 15-23. 17. Delingette H., Hebert H., Ikeuchi K.. A spherical representation for the recognition of curved objects. In: proceedings of IEEE International Conference on Computer Vision, Berlin, Germany, 1993, 103-112.
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18. Johnson A., Hebert M.. Using spin-images for efficient multiple model recognition in clut-tered 3d scenes. IEEE Transaction on Pattern Analysis and Machine Intelligence, 1999, 21(5), 433-449. 19. Saupe D., Vranic D.. 3d model retrieval with spherical harmonics and moments. In: Pro-ceedings of Germany DAGM Conference , Munich, Germany, 2001, 392-397. 20. M. Elad, P. Milanfar, and Gene Golub, Shape from Moments - An Estimation Theory Per-spective, the IEEE Trans. on Signal Processing on August 2002, 18141829 21. M. Novotni.3D Zernike Descriptors for Content Based Shape Retrieval, proceedings of The 8th ACM Symposium on Solid Modeling and Applications, June 2003,216225
