New Set of Rotationally Legendre Moment Invariants

Transcription

1 New Set of Rotationally Legendre Moent Invariants Khalid M. Hosny Abstract Orthogonal Legendre oents are used in several attern recognition and iage rocessing alications. Translation and scale Legendre oent invariants are achieved directly by using Legendre olynoials. Unfortunately this ethod can not convert into a rotational invariant for. This is a big disadvantage and liits the use of Legendre oent invariants in any vital alications. In this aer an indirect ethod is used in couting invariant Legendre oents as a linear cobination of translation scaling and rotation geoetric oent invariants. Geoetric oent invariants are couted exactly for syetric as well as non-syetric binary and gray-scale iages. According to the treendous reduction of the coutational colexity the roosed set of Legendre descritors is suitable for large size iages. The erforance of these descritors is evaluated by using a set of standard iages. Keywords Legendre oent invariants geoetric oents syetric iages. I. INTRODUCTION RTHOGONAL Legendre oents were introduced by OTeague []. These oents are used in a wide range of alications []-[7]. Teh and Chain [] reorted that orthogonal Legendre oents can be used to reresent an iage with iniu aount of inforation redundancy. Couter vision and attern recognition alications usually reuired the coutation of invariant features. These features can be obtained through Legendre oent invariants. The invariance of Legendre oents could be achieved ainly through two ethods. These ethods are direct and indirect. In the direct ethod Legendre oent invariants are couted directly by using Legendre olynoials [9]. The first drawback of this ethod is the high coutational deands esecially with big size iages and the higher order oents. A refined ethod is roosed [0] to overcoe this roble. The other drawback is the iossibility of deriving rotational invariance roerty. On the other side the indirect ethod easily achieves all kinds of invariance including the rotation invariance and is highly effective when the coutational colexity is treendously reduced. In this aer we roosed an indirect ethod that efficiently roduce Legendre oents descritors. The couted Legendre oents are invariants to translation Manuscrit received Noveber Khalid. M. Hosny is with the Couter Science Deartent Faculty of Couters and Inforatics Zagazig University Zagazig Egyt. (Corresonding author to rovide hone: ; e-ail: k_hosny@ yahoo.co). scaling and rotation. These oents works well for syetric as well non-syetric iages. Exact Legendre oents are couted as linear cobination of exact geoetric oents. A fast algorith is alied to accelerate the coutation rocess where the coutational seed is greatly reduced. A recurrence relation is derived to avoid the direct coutation of the factorial ters. The coefficient atrix is re-couted and stored for any future use. The rest of the aer is organized as follows: In section II an overview of Legendre oents is given. The roosed ethod is described in section III. Section IV is devoted to give detailed analysis of coutational colexity and soe exeriental results. Conclusion and concluding rearks are resented in section V. II. LEGENDRE MOMENTS The two-diensional Legendre oents of order for iage intensity function f x y are defined as: xp yf x ydxdy L P () where P x is the th-order Legendre olynoial defined as []: where P k k y k y () k 0 k k! k! k! k! (3) Legendre olynoial coefficients are reresented by. k The oerator n reresents an integer less than or eual to n. Legendre olynoial P x obeys the following recursive relation: P x x P x P x () with P 0 x P x x and. The set of Legendre olynoials P x fors a colete orthogonal basis set on the interval. The orthogonality roerty is defined as: 76

2 0 P xp xdx (5) A digital iage of size M N is an array of ixels. Centers of these ixels are the oints x i y where the iage intensity function is defined only for this discrete set of oints x. Since Legendre olynoials are defined only i y in the suare therefore the digital iage ust be aed into this suare. The aing transforations are defined as follows: x i ( i ) x (6-) y ( ) y (6-) with i 3... M and 3... N. xi xi xi y y y are saling intervals in the x -and y -directions resectively. In the literature of digital iage rocessing the intervals xi and y are fixed at constant values x i M and y N resectively. For this discrete-sace version of the iage "()" is usually aroxiated by using zeroth-order aroxiation (ZOA) as follows: N N P xi P y f xi y L ~ N i III. INDIRECT LEGENDRE MOMENT INVARIANTS Indirect ethod to get Legendre oent invariants reresent the D Legendre oents as a linear cobination of geoetric oent invariants or affine oent invariants. Geoetric oents are invariant to translation non-unifor scaling and rotation. A. Exact Geoetric Moent Invariants Translation invariance is achieved by shifting the iage so that the iage centroid x y is coincides with the origin of the coordinate syste. The centroid of D iage is: 0 0 x y () The central oents x x y y f x y dx dy (9) Where the (7) -order geoetric oents are defined as: M x y f x y dx dy (0) y using the binoial theore central geoetric oents sily reresented as: k x y k () k 0 0 k According to "()" exact coutation of geoetric oents [] results exact values for central oents. With S is the scaling factor in both x- and y-directions. Central oents after a unifor scaling are defined as: S () Scaling invariance could be achieved through the cancellation of the scaling factors. y setting eual to the unity the scale-noralized oents are: (3) Rotation through an angle about the coordinate origin is reresented by the following for: xcos y sin ycos x sin f x y dxdy () y using the binoial theore with "()" the oents of the noralized iage with resect to rotation could be written as follows: k k sin k 0 0 k (5) k cos k k Rotation noralization can be achieved by the aor rincial axis ethod [3] or colex oent's ethod []. The rincial axis oents are obtained by rotating the axis of the central oents until is zero. This ethod gives accurate results only in case of non-syetric iages and shaes while it fails with the N-fold syetrical obects with N >. Colex oents of order for iage intensity f x y are: function x i y x i y f x y c dx dy (6) where i. y using the binoial theore each colex oent can be exressed as a cobination of geoetric oents of the sae order or less as follows: k c i k k (7) k 0 0 k In the colex oent's aroach both original and rotating 77

3 iages have the sae agnitude values of these oents while the hase is shifted with the rotation angle as follows: i c e c () Rotation noralization via colex oents reuires being real and ositive. For Ic and Rc iaginary and real arts of the colex oents c c are the the rotation angle evaluated as follows: Ic tan (9) Rc Any non-zero colex oent could be used for the rotation noralization rocess. It is referable to kee the oent order as low as ossible. An algorith is roosed for autoatic selection of the noralization non-zero colex oent where Max is the axiu order of the oent [5].. Relation with Geoetric Moents The th-order Legendre oents are exressed as a cobination of geoetric oents of the sae order or less as follows: L i i (0) i0 0 The geoetric oent invariants are defined i by using "(5)." According to the "(3)" the exensive factorial coutations are avoided by ileenting the following recurrence relations: 0 0 (-) k k k k (-) k k k k (-3) k k with 0 0 and k. To reduce the coutational colexity "(0)" could be rewritten in a searable for as follows: L i0 Y i i 0 i Y i (-) (-) Legendre descritors of the third- and fourth-order are used to test the invariance. These selected descritors are: Third-order: Fourth-order: IV. NUMERICAL EXPERIMENTS In this section nuerical exerients with syetric and asyetric iages are erfored. Standard asyetric iage of baboon of size is used in the first exerient while the syetric iage of recycle logo of size in the second exerient. This iage is of 3-fold rotation syetry. (d) Fig. Original iage of baboon (b-d) rotated baboon iages with three different angles. 7

4 Legendre descritors of third- and fourth-order are used to test the invariance. Classical non-rotational Legendre oents [] and the rotational roosed ones are evaluated. The absolute errors for the three different rotations are lotted in figures (.a-c). It is clear that the absolute error of the non-rotational descritors show disturbed behavior and increased as the increasing of the order. On the other side the absolute error of the roosed descritors shows steady behavior. Their values decreased and aroach zero values as the descritor's order increased. Siilar to the first nuerical exerient the syetric iage of recycle logo is rotated by three different angles as shown in figures (3.b-d). The invariance absolute errors of the different rotation angles are lotted in figures (.a-c). It's clear that the errors of the roosed descritors are steady and aroach zero as the order increase. ased on these nuerical exerients both sets of lotted absolute errors ensure the validity of the roosed ethod. (d) Fig. 3 Original iage of recycle logo (b-d) rotated recycle logo iages with three different angles. Fig. Absolute invariance errors of the rotated baboon iages The original baboon iage as in figure (.a) is rotated by three different angles as shown in figures (.b-d). The set of 79

5 [] an-hoe Kwan Kok-Meng Ong and Raveendran Paraesran "Noise Reoval of ECG Signals using Legendre Moents" in Proc. of the 7th IEEE Annu. Conf. Engineering in Medicine and iology Shanghai [5] an-hoe Kwan and Raveendran Paraesran "Coarsion between Legendre Moents and DCT in ECG Coression" in Proc TENCON.IEE Region 0 Conf angkok 00. [6] E. Marengo M. obba M. Liarota E. Robotti and P. G. Righetti " Use of Legendre oents for the fast coarison of two-diensional olyacrylaide gel electrohoresis as iages" J. of Chroatograhy A Vol Nov [7] Y. Zhang S. J. Kiselewich and W. A. auson "Legendre and Gabor Moents for Vehicle Recognition in Forward Collision Warning" in Proc. IEEE Conf. Intelligent Transortation Systes Toronto [] C. H. Teh and R.T. Chin "On iage analysis by the ethod of oents" IEEE Trans. Pattern Analysis and Machine Intelligence vol. 0 no Aril 9. [9] C.-W. Chong R. Paraesran and R. Mukundan Translation and Scale Invariants of Legendre Moents Pattern Recognition vol. 37 no Jan. 00. [0] K. M. Hosny "Refined Translation and Scale Invariance of D and 3D Legendre Moents" Int. J. of Pattern Recognition and Artificial Intelligence subitted for ublication. [] K. M. Hosny "Efficient coutation of Legendre oents for gray level iages" Int. J. of Iage and Grahics vol. 7 no Oct [] K. M. Hosny Exact and Fast Coutation of Geoetric Moents for Gray Level Iages Alied Matheatics and Coutation vol. 9 no.. - June 007. [3] A. P. Reevrs R.J. Prko S. E. Andrews and F. P. Kuhl "Three- Diensional Shae Analysis using Moents and Fourier Descritors" IEEE Trans. Pattern Analysis and Machine Intelligence vol. 0 no July 9. [] Y.S. Abu-Mostafa and D. Psaltis "Iage Noralization by Colex Moents" IEEE Trans. Pattern Analysis and Machine Intelligence vol. 7 no Jan. 95. [5] K. M. Hosny "On the coutational asects of affine oent invariants for gray-scale iages" Alied Matheatics and Coutation vol. 95 no Febrauary Fig. Absolute invariance errors of the rotated iages of the recycle logo V. CONCLUSION This work rooses a new set of rotationally Legendre oent invariants. The roosed indirect ethod is fast where the treendous reduction in the coutational colexity enables the roosed set of descritors to deal with large iages and obects. The roosed Legendre descritors are couted for syetrical as well as nonsyetrical iages. Nuerical exerients clearly show the accuracy of these descritors. Khalid M. Hosny (M 00) received the.sc. M.Sc. and Ph.D. fro Zagazig University Zagazig Egyt in 9 99 and 000 resectively. Fro 997 to 999 he was a visiting scholar University of Michigan Ann Arbor and University of Cincinnati Cincinnati USA. He oined the faculty of Couters and Inforatics at Zagazig University where he held the osition of Assistant Professor. He is a eber of ACM and the IEEE couter society. Dr. Khalid works as a reviewer for nuber of international ournals. His research interests include atheatical odeling iage rocessing and attern recognition REFERENCES [] M. R. Teague Iage analysis via the general theory of oents" J. Ot. Soc. A. Vol August 90. [] J. Haddadnia K. Faez and P. Moalle Neural Network ased Face Recognition with Moent Invariants in Proc. Int. Conf. Iage Processing Thessaloniki [3] H. Qidaa and L. Radouane Robust Line Fitting in a Noisy Iage by the Method of Moents IEEE Trans. Pattern Analysis and Machine Intelligence vol. no Nov