Edith Cowan University Research Online ECU Publications Pre. 11 1 Analysis of Hu's Moent Invariants on Iage Scaling and Rotation Zhihu Huang Edith Cowan University Jinsong Leng Edith Cowan University 1.119/ICCET.1.548554 This article was originally published as: Huang, Z., & Leng, J. (1). Analysis of Huâ s Moent Invariants on Iage Scaling and Rotation. Proceedings of 1 nd International Conference on Coputer Engineering and Technology (ICCET). (pp. 476-48).. Chengdu, China. IEEE. Original article available here 1 IEEE. Personal use of this aterial is peritted. Perission fro IEEE ust be obtained for all other uses, in any current or future edia, including reprinting/republishing this aterial for advertising or prootional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted coponent of this work in other works. This Conference Proceeding is posted at Research Online. http://ro.ecu.edu.au/ecuworks/635
Analysis of Hu s Moent Invariants on Iage Scaling and Rotation Zhihu Huang College of Coputer Science, ChongQing University, China Chongqing Radio & TV University, ChongQing, China Eail:Zh_huang@yeah.net Jinsong Leng School of Coputer and Security Science, Edith Cowan University, Australia Eail: j.leng@ecu.edu.au Abstract Moent invariants have been widely applied to iage pattern recognition in a variety of applications due to its invariant features on iage translation, scaling and rotation. The oents are strictly invariant for the continuous function. However, in practical applications iages are discrete. Consequently, the oent invariants ay change over iage geoetric transforation. To address this research proble, an analysis with respect to the variation of oent invariants on iage geoetric transforation is presented, so as to analyze the effect of iage's scaling and rotation. Finally, the guidance is also provided for iniizing the fluctuation of oent invariants. Keywords-Pattern recognition, Hu s oent invariant, Iage transforation, Sapatial resolution I. INTRODUCTION Moents and the related invariants have been extensively analyzed to characterize the patterns in iages in a variety of applications. The well-known oents include geoetric oents[1], zernike oents[], rotational oents[3], and coplex oents [4]. Moent invariants are firstly introduced by Hu. In[1], Hu derived six absolute orthogonal invariants and one skew orthogonal invariant based upon algebraic invariants, which are not only independent of position, size and orientation but also independent of parallel projection. The oent invariants have been proved to be the adequate easures for tracing iage patterns regarding the iages translation, scaling and rotation under the assuption of iages with continuous functions and noise-free. Moent invariants have been extensively applied to iage pattern recognition[4-7], iage registration[8] and iage reconstruction. However, the digital iages in practical application are not continuous and noise-free, because iages are quantized by finite-precision pixels in a discrete coordinate. In addition, the noise ay be introduced by various factors such as caera. In this respect, errors are inevitably introduced during the coputation of oent invariants. In other words, the oent invariants ay vary with iage geoetric transforation. But how uch are the variation? To our knowledge, this issue has never been quantitatively studied for iage geoetric transforation. Salaa[9] analyzed the effect of spatial quantization on oent invariants. He found that the error decreases when the iage size increases and the sapling intervals decrease, but does not decrease onotonically in general. Teh[1] analyzed three fundaental issues related to oent invariants, concluding that: 1) Sensitivity to iage noise; ) Aspects of inforation redundancy; and 3) Capability for iage representation. Teh declared that the higher order oents are the ore vulnerable to noise. Coputational errors on oent invariants can be caused by not only the quantization and pollution of noise, but also transforations such as scaling and rotation. When the size of iages are increased or decreased, the pixels of iages will be interpolated or deleted. Moreover, the rotation of iages also causes the change of iage function, because it involves rounding pixel values and coordinates[11]. Therefore, oent invariants ay change as iages scale or rotate. This paper quantitatively analyzes fluctuation of oent invariants on iage scaling and rotation. Epirical studies have been conducted with various iages. The ajor contributions of this paper include the findings the relationship aong the iage scaling, rotation and resolution, and the guidance of iniizing the errors of oent invariants for iage scaling and rotation. The rest of paper is organized as follows: Section II describes seven Hu s oent invariants. Section III discusses the variation of oent invariants when iages scale and rotate. Section IV analyzes the relationship between oent invariants and coputation. Finally, section V concludes the paper. II. MOMENT INVARIANTS Two-diensional (p+q)th order oent are defined as follows: pq = x p y q f ( x, y) dxdy p,q =,1,, L (1) If the iage function f(x,y) is a piecewise continuous bounded function, the oents of all orders exist and the oent sequence { pq } is uniquely deterined by f(x,y); 978-1-444-6349-7/1/$6. c 1 IEEE V7-476
and correspondingly, f(x,y) is also uniquely deterined by the oent sequence { pq }. One should note that the oents in (1) ay be not invariant when f(x,y) changes by translating, rotating or scaling. The invariant features can be achieved using central oents, which are defined as follows: μ pq where = p ( x x) ( y y) q f ( x, y) dxdy p, q =,1,, L 1 x = y = 1 () The pixel point ( x, y ) are the centroid of the iage f(x,y). The centroid oents μ pq coputed using the centroid of the iage f(x,y) is equivalent to the pq whose center has been shifted to centroid of the iage. Therefore, the central oents are invariant to iage translations. Scale invariance can be obtained by noralization. The noralized central oents are defined as follows: III. MOMENT INVARIANTS AND IMAGE TRANSFORMATION Iage geoetric transforation is a popular technique in iage processing, which usually involves iage translation, scaling and rotation. The translation operator aps the position of each pixel in an input iage into a new position in an output iage; the scale operator is used to shrink or zoo the size of an iage, which is achieved by subsapling or interpolating to the input iage; the rotation operator aps the position of each pixel onto a new position by rotating it through a specified angle, which ay produce non-integer coordinates. In order to generate the intensity of the pixels at each integer position, different interpolation ethods can be eployed such as nearest-neighbor interpolation, bilinear interpolation and bicubic interpolation. Fro the description in last paragraph, we can see that translation only shifts the position of pixels, but scaling and rotation not only shift the position of pixels but also change the iage function itself. Therefore, only the scaling and rotation cause the errors of oent invariants. We eploy two iages with 5 x 5 resolutions to study what extent the iage scaling and rotation can ipact the oent invariants, One is a coplex iage without principal orientation (iage A); the other is a siple iage with principal orientation (iage B). Figure 1 shows the two iages. μpq =, γ = ( p+ q+ )/, μ ηpq γ p+ q=,3, L(3) Based on noralized central oents, Hu[1] introduced seven oent invariants: φ + 1 = η η = ( η ) 4η 11 3 = ( η 3η 1 ) + (3η 1 μ ) 4 = ( η + η1 ) + ( η1 μ ) φ + φ φ + φ = ( η 3η )( η + η )[( η + η ) 3( η + η ) ] 5 + (3η )( η + η [3( η + η ) ( η + η ) ] φ = ( η 6 1 1 1 1 ) )[( η + η ) 1 + 4η ( η + η )( η + η ) 11 1 1 1 1 1 - ( η + η ) φ = (3η )( η + η )[( η + η ) -3( η + η ) ] 7 1 1 -( η -3η )( η + η )[(3( η + η ) 1 1 1 1 1 1 1 ] -( η + η ) 1 The seven oent invariants are useful properties of being unchanged under iage scaling, translation and rotation. ] Iage A Iage B Figure 1. Iages for Experients A. Moent Invariants and Iage Scaling As described above, iage scaling ay cause change of iage function, so the oent invariants ay also change correspondingly. Therefore, it is very necessary to study the relationship between oent invariants and iage scaling. We conduct the research by coputing seven oent invariants on iage A and iage B with resolution fro 1x to 5x5 in 1 increents. The Figure (a), (b) separately shows the results of oent invariant for iage A and B. Fro the diagras we see that the resolution of 1x1 is the threshold for iage A. The values have apparent fluctuations when the resolution less than the threshold but stability when the resolution ore than the threshold. Correspondingly, the threshold of iage B is 15x15. [Volue 7] 1 nd International Conference on Coputer Engineering and Technology V7-477
4 35 5 15 1 5 Resolution.648E- 1 5 1 15 5 35 4 45 5 (a) Moent Invariants for Iage A.656E- 1 45 9 135 18 5 7 315 36 Moent Value 1.E-6 1.E-6 φ φ 1.1E-6 1 45 9 135 18 5 7 315 36 φ4 5.E-11 φ5 4.8E-11 4.6E-11 1 45 9 135 18 5 7 315 36 Resolution 1 5 1 15 5 35 4 45 5 3.74E-1 (b) Moent Invariants for Iage B Figure. Moent invariants on iage scaling 3.71E-1 3.68E-1 φ4 1 45 9 135 18 5 7 315 36 Table. Fluctuation Ratio on Iage A Moent Resolution Resolution.84E- Invariants <Threshold Threshold.76E- φ5.58%.4%.68e- φ 47.% 1.6% 1 45 9 135 18 5 7 315 36 4 35 5 15 1 5 Moent Value 6.5% 1.41% φ4 88.6% 1.4% φ5 14.38% 5.68% 84.43% 1.84% 1.94%.59% The equation (4) is used to easure the fluctuation a series of data x. R ax(x) in(x) = 1% ean(x) (4) φ φ4 φ5 Where, ax(x), in(x) and ean(x) separately note the axiu, iniu and ean in data x. The data in Figure (a) are divided into two groups by the threshold 1x1 to separately copute the fluctuation by equation (4). The results are displayed in table. The axiu of fluctuation coes up to 14.38% in φ5 when resolution of iage less than threshold. But it decreases to 5.68% in φ5 when resolution of iage greater then threshold. B. Moent Invariants and Iage Rotation.64E- 3.7E-13-3.68E-13 3.66E-13 1 45 9 135 18 5 7 315 36 4.5E- 4.1E- 3.95E- 1 45 9 135 18 5 7 315 36 Figure 3. Moent Invariants on Iage B Rotated fro 1 o to 36 o in 1 o Increents Since the rotation of an iage can change the function of the iage ore or less, the oent invariants keep changing V7-478 1 nd International Conference on Coputer Engineering and Technology [Volue 7]
as the iage is rotated. In order to investigate the fluctuation of oent invariants result fro the iage rotation, we conduct the two kinds of experients, i.e., one kind is the different iages (A, B) with the sae resolution; another is the different resolutions of the sae iage. Each input iage is rotated fro 1 o to 36 o in 1 o increents. Figure 3 display the results of oent invariants for iage B rotated fro 1 o to 36 o, it shows the fluctuation of oent invariants becoes strong when the rotated angle near to 45,135, 5 and 315 degrees, while becoes weak as the rotated angle near to 9,18, 7 and 36 degree. In Section III A, we know the threshold of resolution on Iage B is 15x15. Does the sae iage have sae threshold between scaling and rotation? In order to answer this question, we produce a series of iages with resolutions fro 6x6 to 3x3 in increents using the sae original iage, and copute the oent invariants for each iage rotated fro 1 o to 36 o. Then, the fluctuation is coputed by equation (4) for each 36 iages. Figure 4 shows the trend of fluctuation of oent invariants. Fro the diagra, we can see that the threshold is different between iage scaling and iage rotation. The threshold of iage rotation on Iage B is not 15x15 but 4x4. 16 1 8 4 Fluctuation 6 9 1 15 18 1 4 7 3 Resolution Figure 4. Fluctuation of Moent Invariants on Iage B with Different Resolution φ φ4 φ5 Table shows the values of fluctuation for seven oent invariants on different resolution fro 6x6 to 3x3. We can see that the fluctuation decreases as the iage spatial resolution increases. The fluctuation alost coes up to 191.1% when the resolution is only 6x6, but rapidly decreases to 1.1% when the resolution is 7x7. The fluctuation obviously decreases as the resolution increases until to the threshold. However, the fluctuation does not onotonically decrease any ore when the resolution greater than 7x7. IV. MOMENT INVARIANTS AND COMPUTATION As discussed in Section III, we can reach the conclusion: The higher the resolution is, the lower the fluctuation is. However, the coputation of oent invariants will increase when the resolution increases. As a consequence, the research of relationship between the resolution of iages and coputation is necessary. We calculate the coputation of oent invariants for the resolutions fro 1x1 to 15x15 in increents on a PC (CPU P4.G, RAM 1G). Figure 5 shows the results. Fro the diagra, we can see that the relationship between the resolution and the coputation is non-linear. Therefore, we ust select an acceptable resolution to balance coputation and resolution on the real application. Table. Fluctuation of Moent Invariants on Different Resolution of Iage B Resolution φ φ4 φ5 6x6 18.7 39.9 184.7 193.8 1157.5 8.6 191.1 9x9 13.3 6.5 7.9 145.3 1118,1 194.7 84. 1x1 1.7 19.1 436. 19.9 947.6 14.7 517.4 15x15 7.4 13.6 38. 86.3 53. 98.9.1 18x18 4.5 8. 159. 51.5 37.3 57.7 14. 1x1 3. 5.6 88.1 36. 179.3 38.9 75.5 4x4 1.1 1.9 1.4 1.3 46. 1.8 19.7 7x7..3 1.8.4.9.5 1.1 x..5 1.4.3..5 1.3 3x3.1.3 1.9. 1.. 1.7 Coputation s 6 45 15 5 1 15 Resolution Figure 5. Coputation of Moent Invariants for Different Resolution V. CONCLUSION This paper has presented an analysis of fluctuation of Hu s oent invariants on iage scaling and rotation. Our findings ay be suarized as follows: (1) the oent invariants change as iages scale or rotate, because iages are not continuous function or polluted by noise; () the fluctuation decreases when the spatial resolution of iages increases. However, the change will not rearkably decrease as resolution increases if the resolution greater than the [Volue 7] 1 nd International Conference on Coputer Engineering and Technology V7-479
threshold; (3) the coputation increases quickly as resolution increases. Fro the experiental studies, we find that the choice of iage spatial resolution is very iportant to keep invariant features. To decrease the fluctuation of oent invariants, the iage spatial resolution ust be higher than the threshold of scaling and rotation. However, the resolution cannot be too high, because the coputation will rearkably increase as the resolution increases. Therefore, the choice of resolution ust balance coputation and resolution on the real application REFERENCES [1] H. Ming-Kuei, "Visual pattern recognition by oent invariants," Inforation Theory, IRE Transactions, vol. 8, pp. 179-187, 196. [] M. R. Teaque, "Iage Analysis via the General Theory of Moents," Journal of the Optical Society of Aerica, vol. 7, pp. 9-9, 198. [3] J. F. Boyce and W. J. Hossack, " Moent Invariants for Pattern Recognition," Pattern Recognition Letters, vol. 1, pp. 451-456, 1983. [4] Y. S. Abu-Mostafa and D. Psaltis, "Recognitive Aspects of Moent Invariants," Pattern Analysis and Machine Intelligence, IEEE Transactions on, vol. PAMI-6, pp. 698-76, 1984. [5] M. Schleer, M. Heringer, F. Morr, I. Hotz, M. H. Bertra, C. Garth, W. Kollann, B. Haann, and H. Hagen, "Moent Invariants for the Analysis of D Flow Fields," Visualization and Coputer Graphics, IEEE Transactions on, vol. 13, pp. 1743-175, 7. [6] C. Qing, P. Eil, and Y. Xiaoli, "A coparative study of Fourier descriptors and Hu's seven oent invariants for iage recognition," in Electrical and Coputer Engineering, 4. Canadian Conference on, 4, pp. 1-16 Vol.1. [7] J. Flusser, B. Zitova, and T. Suk, "Invariant-based registration of rotated and blurred iages," in Geoscience and Reote Sensing Syposiu, 1999. IGARSS '99 Proceedings. IEEE 1999 International, 1999, pp. 16-164. [8] Jan Flusser and Toas Suk, "A Moent-based Approach to Registration of Iages with Affine Geoetric Distortion," IEEE transaction on Geoscience and Reote Sensing, vol. 3, pp. 38-387, 1994. [9] G. I. Salarna and A. L. Abbott, "Moent invariants and quantization effects," in Coputer Vision and Pattern Recognition, 1998. Proceedings. 1998 IEEE Coputer Society Conference on, 1998, pp. 157-163. [1] Cho-huak Teh and Roland T. Chin, "On Iage Analysis by the Methods of Moents," IEEE Transactions on Pattern Analysis and achine Intelligence, vol. 1, pp. 496-513, 1988. [11] Rafael C. Gonzalez and Richard E. Woods, Digital Iage Processing(Third Edition): Prentice Hall, 8. V7-48 1 nd International Conference on Coputer Engineering and Technology [Volue 7]