3-D Projective Moment Invariants

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1 Journal of Information & Computational Science 4: * (2007) 1 Available at 3-D Projective Moment Invariants Dong Xu a,b,c,d,, Hua Li a,b,c a Key Laboratory of Intelligent Information Processing, Institute of Computing Technology, Chinese Academy of Sciences, Beijing , China b Key Laboratory of Computer System and Architecture, Institute of Computing Technology, Chinese Academy of Sciences, Beijing , China c National Research Center for Intelligent Computing Systems, Institute of Computing Technology, Chinese Academy of Sciences, Beijing , China d Graduate University of Chinese Academy of Sciences, Beijing , China Received 23 March 2006; revised 6 March 2007 Abstract 2-D projective moment invariants were firstly proposed by Suk and Flusser in [12]. We point out here that there is a useless projective moment invariant which is equivalent to zero in their paper. 3-D projective moment invariants are generated theoretically by investigating the property of signed volume of a tetrahedron. The main part is the selection of permutation invariant cores for multiple integrals to generate independent and nonzero 3-D projective moment invariants. We give the conclusion that projective moment invariants don t exist strictly speaking because of their convergence problem. Keywords: Geometric primitive; Moment invariant; Projective transformation; Permutation invariant 1 Introduction Projective geometry is a fundamental and important part in computer vision, for what we see is mainly perspective projection of 3-D scenes. Projective invariants have been studied for many years. Suk and Flusser in [12] proved that there are no projective moment invariants with finite terms, but they presented two infinite projective moments invariants. Except literature [12], cross-ratio is the most commonly used projective invariant. Projective invariant theories of differential and algebraic invariants are suggested to be used in computer vision in [14]. Lenz and Meer ([4]) used representation theory to derive projective and permutation (p 2 ) invariants of the four collinear and the five coplanar points configurations. The p 2 invariants Project supported by the National Natural Science Foundation of China (No and No ) and the National Key Basic Research Program (No. 2004CB318006). Corresponding author. address: xudong@ict.ac.cn (Dong Xu) / Copyright c 2007 Binary Information Press 2007

2 2 D. Xu et al. /Journal of Information & Computational Science 4: * (2007) 1 are insensitive to both projective transformations and changes in the labelling of the points. Suk and Flusser ([13]) presented two types of 2-D projective triangular invariants from crossratio and area of a triangle for recognition of deformed point sets. Projective depth, which is a new description of structure, was proposed by Shashua in [11]. He addressed the problem of reconstructing 3-D space in a projective framework from two or more views. Three projective invariants of six points in space were derived in [7] by using at least three uncalibrated images. Wu and Hu ([15]) applied projective geometric invariants to catadioptric camera models. Corrochano and Lasenby ([1]) used geometric algebra as a complete framework for the theory and computation of 3-D projective invariants formed from points and lines. A technique for extracting filled-in information in form documents is presented in [10] by using cross-ratios of five points. The computation of projective invariants in pairs of images from uncalibrated cameras was studied in [2]. Algebraic or geometric invariants are computed in projective space or directly from image measurements. 2-D moment invariants were firstly proposed by Hu ([3]) in 1962 for character recognition. In 1980, Sadjadi and Hall ([9]) first extended moment invariants from 2-D to 3-D. Lo and Don ([5]) constructed 3-D moment invariants with complex moments and group-theoretic technique. Rothe et al. presented the normalization method to determine invariants ([8]). They also transformed a set of points to a standard position with aspect to projective transformation. In this paper, we generalize projective moment invariants from 2-D to 3-D space, and select permutation invariant cores for generation of 3-D projective moment invariants. First, 3-D projective transformations are defined in Section 2, and 3-D projective moment invariants are constructed by multiple integrals of a special kind of invariant geometric primitive. In Section 3, we point out one obvious mistake in [12] and discuss the convergence problem for computation of projective moment invariants. We conclude the paper in Section D Projective Moment Invariants D projective transformations First, every point (x, y, z) in 3-D Euclidean space is shifted to 4-D space and gets its homogeneous coordinate (x, y, z, 1). Then 3-D projective transformation can be expressed as follows: x y z w = H x y z 1 = a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 x y z 1 (1) The point after projective transformation is ( x, y, z ). Here, we omit the situation that w w w transforms the point to infinity, that is w = 0. Homogeneous matrix H should be nonsingular. There are 15 degrees of freedom (DOF) for this transformation except the scale factor. It can be decomposed into the following 15 single-parameter transformations:

3 D. Xu et al. /Journal of Information & Computational Science 4: * (2007) 1 3 1) z = z x = σx 5) y = y z = z 9) z = z 13) x = x + α y = y x = x + ɛz y = y x = x/(1 + c 1 x) y = y/(1 + c 1 x) z = z/(1 + c 1 x) x = x x = x x = ωx 2) y = y + β 3) y = y 4) y = ωy z = z z = z + γ z = ωz x = x x = x + ηy x = x 6) y = τy 7) y = y 8) y = y + ρx z = z z = z z = z x = x x = x x = x 10) y = y 11) y = y + ξz 12) y = y z = z + δx z = z z = z + ζy x = x/(1 + c 2 y) x = x/(1 + c 3 z) 14) y = y/(1 + c 2 y) 15) y = y/(1 + c 3 z) z = z/(1 + c 2 y) z = z/(1 + c 3 z) Formulas 1) to 3) are translations, 4) is uniform scaling, 5) and 6) are stretching (nonuniform scaling), 7) to 12) are skewing (shearing), and 13) to 15) are pure projections. The scaling parameters ω,σ and τ are supposed to be positive. In Fig. 1, we show the visual effects of three kinds of single-parameter transformations for the sake of better understanding. (2) Fig. 1: The bunny model after transformations: nonuniform scaling (left), shearing (middle), pure projection (right). 2.2 Volume of tetrahedron under 3-D projective transformations Definition 1 3-D moments of order l + m + n of a 3-D density function ρ(x, y, z) are defined by the Riemann integrals: M lmn = x l y m z n ρ(x, y, z)dxdydz. (3) If the density function is piecewisely continuous and bounded in a finite region in 3-D Euclidean space, then moments of all nonnegative orders exist. Suppose an arbitrary point in the object has Cartesian coordinate p i = (x i, y i, z i ). The signed volume of a tetrahedron within four arbitrary points can be expressed as follows: V (i, j, k, l) = 1 6 x i y i z i 1 x j y j z j 1 x k y k z k 1 x l y l z l 1. (4)

4 4 D. Xu et al. /Journal of Information & Computational Science 4: * (2007) 1 This definition of signed volume of a tetrahedron also can be seen in [6]. Clearly it has the following two properties: a) If two labels in it are the same, then the result will be zero. b) If we exchange two labels in it, then the value will be the opposite of the original one. Suppose V (i, j, k, l) is the volume after the projective transformation. It is not difficult to prove that under translation 1) to 3), or skewing transformations 7) to 12), the value of signed volume keeps the same. Then, if scaling and stretching transformations 4) to 6) are carried out, we will have the following equation: V (i, j, k, l) = ω 3 στv (i, j, k, l). (5) If the points are transformed only under single-parameter transformation 13), then the volume becomes: V (i, j, k, l) = V (i, j, k, l)/[(1 + c 1 x 1 )(1 + c 1 x 1 )(1 + c 1 x 2 )(1 + c 1 x 3 )(1 + c 1 x 4 )]. (6) Based on the above analysis, the relationship between V (i, j, k, l) and V (i, j, k, l) under all the transformations 1) to 15) can be written as: V (i, j, k, l) = ω 3 στv (i, j, k, l)/[(1 + c 1 x 1 )(1 + c 1 x 2 )(1 + c 1 x 3 )(1 + c 1 x 4 )(1 + c 2 y 1 ) (1 + c 2 y 2 )(1 + c 2 y 3 )(1 + c 2 y 4 )(1 + c 3 z 1 )(1 + c 3 z 2 )(1 + c 3 z 3 )(1 + c 3 z 4 )]. (7) 2.3 Generation of 3-D projective moment invariants We set core(1, 2,, n) = [V (1, 2, 3, 4)V (1, 2, 3, 5),, V (i, j, k, l),, V (r, s, t, n)] 1, (i < j < k < l, r < s < t < n) to be reciprocal of the multiplications of n signed volumes, which involves n participating points. core(1, 2,, n) should satisfy the following conditions: A) Each term V (i, j, k, l) should satisfy i < j < k < l. All the volumes in core(1, 2,, n) should be arranged with ascending order. B) There are total n signed volumes in the denominator. C) Each label of point i(1 i n) should appear exactly four times in the denominator. D) Exchange the labels of any two points and regroup the orders of signed volumes in core(1, 2,, n), the value of core(1, 2,, n) should keep the same (not be the opposite of itself). Furthermore, make sure that there is no existing core which is the same as the exchanged one. E) core(1, 2,, n) cannot be divided into two or more parts like core(1, 2,, k)core(k + 1, k + 2,, n).

5 D. Xu et al. /Journal of Information & Computational Science 4: * (2007) 1 5 Explanation about these five restrictions is given below. They can help us choose the correct permutation invariant cores for integrals to construct projective moment invariants, and eliminate a huge number of duplicated and useless ones to save our energy. Condition A) is to avoid V (i, j, k, l) 0 and ensures that core(1, 2,, n) + ; Condition B) and condition C) ensure that the integral of core(1, 2,, n) is projective invariants; Condition D) is to avoid the integral result which is equivalent to zero, or avoid the dependent integral result which is the same as some existing one; Condition E) also guarantees that the projective moment invariants we get are independent of each other. Similar work about permutation invariants has been done in [2] and [4]. The multiple integral of the core(1, 2,, n) is as follows: MI(core(1, 2,, n)) = +,, + core(1, 2,, n)ρ(x 1, y 1, z 1 )ρ(x 2, y 2, z 2 ),, (8) ρ(x n, y n, z n )dx 1 dy 1 dz 1 dx 2 dy 2 dz 2,, dx n dy n dz n Theorem 1 MI(core(1, 2,, n)) is a projective invariant. Proof Suppose core (1, 2,, n) is the core after projective transformation. MI(core (1, 2,, n)) = = +,, + +,, + core (1, 2,, n)ρ(x 1, y 1, z 1)ρ(x 2, y 2, z 2),, ρ(x n, y n, z n)dx 1dy 1dz 1dx 2dy 2dz 2,, dx ndy ndz n core(1, 2,, n)ω 3n σ n τ n (1 + c 1 x 1 ) 4 (1 + c 2 y 1 ) 4 (1 + c 3 z 1 ) 4 (1 + c 1 x 2 ) 4 (1 + c 2 y 2 ) 4 (1 + c 3 z 2 ) 4,, (1 + c 1 x n ) 4 (1 + c 2 y n ) 4 (1 + c 3 z n ) 4 J ρ(x 1, y 1, z 1 )ρ(x 2, y 2, z 2 ),, ρ(x n, y n, z n )dx 1 dy 1 dz 1 dx 2 dy 2 dz 2,, dx n dy n dz n = MI(core(1, 2,, n)) (9) Here, J is the Jacobian of the multiple integral under projective transformation. J = ω 3n σ n τ n /[(1 + c 1 x 1 ) 4 (1 + c 2 y 1 ) 4 (1 + c 3 z 1 ) 4 (1 + c 1 x 2 ) 4 (1 + c 2 y 2 ) 4 (1 + c 3 z 2 ) 4,, (1 + c 1 x n ) 4 (1 + c 2 y n ) 4 (1 + c 3 z n ) 4 ] (10) The multiple integral of core(1, 2,, n) is just the 3-D projective moment invariants if we expand core(1, 2,, n) by polynomial form. That is to say, projective moment invariants of different orders can be constructed by the multiple integral frameworks. However, since the expansion of core(1, 2,, n) has infinite terms, 3-D projective moment invariants also have infinite terms of moments and include negative indices, like the deduction in 2-D space ([12]). 2.4 Examples of 3-D projective moment invariants If the number of participating points n equals to 4 or 6, we get four 3-D projective moment invariants. I 1 = MI([V 4 (1, 2, 3, 4)] 1 ) (11)

6 6 D. Xu et al. /Journal of Information & Computational Science 4: * (2007) 1 I 2 = MI([V 2 (1, 2, 3, 4)V 2 (1, 2, 5, 6)V 2 (3, 4, 5, 6)] 1 ) (12) I 3 = MI([V (1, 2, 3, 4)V (1, 2, 3, 5)V (1, 2, 4, 6)V (1, 2, 5, 6)V 2 (3, 4, 5, 6)] 1 ) (13) I 4 = MI([V (1, 2, 3, 4)V (1, 2, 3, 5)V (1, 2, 4, 6)V (1, 2, 5, 6)V (2, 4, 5, 6V (3, 4, 5, 6)] 1 ) (14) If n = 5, multiple integral of the core [V (1, 2, 3, 4)V (1, 2, 3, 5)V (1, 2, 4, 5)V (1, 3, 4, 6)V (2, 3, 4, 5)] 1 is equal to zero, because it doesn t satisfy the fourth condition. Proof The proof is simple if we exchange labels 1 and 2. MI([V (1, 2, 3, 4)V (1, 2, 3, 5)V (1, 2, 4, 5)V (1, 3, 4, 6)V (2, 3, 4, 5)] 1 ) = MI([V (2, 1, 3, 4)V (2, 1, 3, 5)V (2, 1, 4, 5)V (2, 3, 4, 6)V (1, 3, 4, 5)] 1 ) = MI([V (1, 2, 3, 4)V (1, 2, 3, 5)V (1, 2, 4, 5)V (2, 3, 4, 6)V (1, 3, 4, 5)] 1 ) = 0 (15) According to the five constrained conditions, there are only a finite number of projective moment invariants if we set the number of participating points n. If n < 4 or n = 5, there is no projective moment invariant. If n = 4, there is one projective moment invariant. The exact number of projective moment invariants hasn t been solved yet when n is a big number. 3 Problems of 2-D Projective Moment Invariants An invariant equivalent to zero In 2-D space, signed area A(i, j, k) of a triangle can be used to construct 2-D projective moment invariants in [12]. Since A 3 (1, 2, 3) = A 3 (2, 1, 3), this core doesn t satisfy the fourth condition and the first invariant they got is equivalent to zero. Feature descriptors are useless if they couldn t discriminate one object from another. Moments with negative indices 2-D projective moment invariants in [12] contain moments with both positive and negative indices. However, from Eq. (3), we can see that moments with negative indices may be infinite if the origin is inside the object. Series expansion The authors used a finite number of moments to approximate infinite series. Seriously speaking, the series expansion is not allowed because there is no guarantee that the series converges. There is also more than one kind of expansion. They themselves have also been aware of this in the conclusion part. Let s analyze the convergence problem directly from the definition formula Eq. (8). Each point (x i, y i, z i ), (1 i n) could be at any position of the whole 3-D object in the multiple integral. Any two points may have identical coordinates. core(1, 2,, n) will tend to infinity if some V (i, j, k, l) equal to zero. Hence, multiple integral formula Eq. (8) does not converge, and projective moment invariants don t exist strictly speaking.

7 D. Xu et al. /Journal of Information & Computational Science 4: * (2007) Conclusions and Future Work In this paper, we derive 3-D projective moment invariants by multiple integrals of a special kind of cores. If we demand projective invariants be composed of moments of all orders, then they should have infinite terms and contain negative indices. Projective invariants expressed by moments suffer from convergence problem and are hard to be used in practice. In the future, we will try to seek other kinds of projective invariants. Furthermore, we will attempt to apply existing 3-D projective invariants in projective space for object recognition to avoid camera calibration or reconstruction in 3-D Euclidean space. References [1] E. B. Corrochano and J. Lasenby, Analysis and computation of projective invariants from multiple views in the geometric algebra framework, in: M. A. Rodigues (Ed.), Invariants for Pattern Recognition and Classification, World Scientific, SG, 2000, pp [2] G. Csurka and O. Faugeras, Algebraic and geometric tools to compute projective and permutation invariants, IEEE Trans. Pattern Analysis and Machine Intelligence 21 (1999) [3] M. K. Hu, Visual pattern recognition by moment invariants, IRE Trans. Information Theory 8 (1962) [4] R. Lenz and P. Meer, Point configuration invariants under simultaneous projective and permutation transformations, Pattern Recognition 27 (1994) [5] C. H. Lo and H. S. Don, 3-D moment forms: their construction and application to object identification and positioning, IEEE Trans. Pattern Analysis and Machine Intelligence 11 (1989) [6] J. O Rourke, Computational Geometry in C, Cambridge University Press, UK, [7] L. Quan, Invariants of six points and projective reconstruction from three uncalibrated images, IEEE Trans. Pattern Analysis and Machine Intelligence 17 (1995) [8] I. Rothe, H. Susse and K. Voss, The method of normalization to determine invariants, IEEE Trans. Pattern Analysis and Machine Intelligence 18 (1996) [9] F. A. Sadjadi and E. L. Hall, Three-Dimensional moment invariants, IEEE Trans. Pattern Analysis and Machine Intelligence 2 (1980) [10] R. Safari, N. Narasimhamurthi, M. Shridhar and M. Ahmadi, Document registration using projective geometry, IEEE Trans. Image Processing 6 (1997) [11] A. Shashua, Projective structure from uncalibrated images: structure from motion and recognition, IEEE Trans. Pattern Analysis and Machine Intelligence 16 (1994) [12] T. Suk and J. Flusser, Projective moment invariants, IEEE Trans. Pattern Analysis and Machine Intelligence 26 (2004) [13] T. Suk and J. Flusser, The features for recognition of projectively deformed point sets, in: IEEE Int. Conf. Image Processing, 1995, pp [14] I. Weiss, Projective invariants of shapes, in: IEEE Conf. Computer Vision Pattern Recognition, 1988, pp [15] Y. Wu and Z. Hu, Geometric invariants and applications under catadioptric camera model, in: IEEE Int. Conf. Computer Vision, 2005, pp