2D moment invariants have been successfully applied in pattern recognition tasks. The main difficulty of using moment invariants is the computational burden. To improve the algorithm of moments computation through an iterative method, an approach for fast computation of moments based on the 8-neighbor chain code is proposed in this paper. Then artificial neural networks are applied for 2D shape recognition with moment invariants. Compared with the method of polygonal approximation, this approach shows higher accuracy in shape representation and faster recognition speed in experiment

Han, Yan-ping

Houkes, Zweitze

Ji, Guang-rong

Wang, Guo-yu

Zheng, Bing

English

Ji, Guangrong

Wang, G.Y.

Houkes, Z.

University of Twente Research Information

1997 IEEE Internationill Conference on Intelligent Processing Systems Octobcr 28 - 3 I ,  Bcijing. Cliinii A New Method for Fast Computation of Moments Based on 8-neighbor Chain Code Applied to 2-D Object Recognition Guang-rong Ji, Guo-yu Wang, Zweitze Hokes*, Bing Zheng, Yan-ping Han (Dept. of Electrical Engineering, Ocean University of Qingdao, Qingdao, 266003, P. R. China) (* Dept. of Electrical Engineering, University of Twente, Enschede, The Netherlands) Abstract - 2-D moment invariants have been successfully applied in pattern recognition tasks. The main difficulty of using moment invariants is the computational burden. To improve the algorithm of moments computation through an iterative way, an approach for fast computa- tion of moments based on the 8-neighbor chain code is proposed in this paper. Then artificial neural networks are applied for 2D shape recognition with moment in- variants. Compared with the method of polygonal ap- proximation, this approach shows higher accuracy in shape representation and faster recognition speed in ex- periments. 11. FAST COMFUTATION OF MOMENTS A .  Moments computation in an iterative way A 2-D gray-level image can be represented With a continu ous functionffx,y). The moment offfx,y) in order of (ptq) is defined as mpq = j ~ x ' y q f ( x , y ) ~ d y ,  (1) where p,qEZ+, with Z+={x€Zlx>O} and Z belongs to set of all integers. According to unique theorem [6],  if f ( x , y )  is piece-wise continuous and non-zero over a finite region of the x-y plane, moments of all orders exist and {m,) are uniquely deter- mined by f (x ,  y) .  Conversely, the set of moments {mm ) determines f(x*Y) In many cases, shape of boundary is usually used for object recognition. After separa- tion from the background, the binary image of the object can be described by I. INTRODUCTION 2-D moment invariants is an efficient descriptor for 2D shape representation and recognition because of its invariant quality under the operation of translation, rotation and scaling [3] [6]. But the main problem of using moment invariants in practice is the computational burden. Some methods for fast computation of moments have been proposed, such as PO- lygonal-based method [7], 8-function [I I], IIR-filtering [SI and the Pascal-Green-based method [2], etc.. The polygonal- based method saves the calculation time by reducing the number of feature points through polygonal approximation. But it should be noticed that the process of polygonal ap- proximation is also time-consumed and such an approxima- tion may induce large deviation in representing some complicated 2D objects. In case of good quality in image acquisition, the boundary description with 8-neighbor chain an iterative way to compute moments with 8-neighbor chain code. For the purpose of fast recognition of 2D objects, we proposed a method to improve the moment computation Let denote the curve embedding the based on the approach of [7]. Then artificial neural networks ments as the feature descriptor. Compared with the method of polygonal approximation, the method proposed in this paper showed higher accuracy in shape representation, and faster recognition speed with preliminary experimental results. (2) f ( x , y )  = l,WX,Y) ED f ( x , y )  = 0, V(X ,Y> B: D ' where Hence, denotes the region of object in the image plane. can be simplified as code can accurately represent a 2D shape. Jiang [7] proposed mpq = JJxpyqdudy (3) D CO~~-~ected r gion D and assuming c to be Piece-wise smooth. curve-integration along C is counter-clocked, then the right side of (3) can be decomposed by applying Stokes theorem in the form are applied to finish the recognition task using invariant mo- If the function f ( x , y )  is integal Over the region D and the 0-7803-4253-4/97/$10.00 0 1997 IEEE - 974 - c, : 5 = x ,  , (4) ( 5 )  Y, and the components P(x,y) and Q(x,y) are chosen as [7] Case 3: In other directions, (6) 1 Q(x,r)=---xp+ly ' l ,P(x ,y)=O, 1 1 k ' k "  P +  1 Therefore, (3) can be formulated as c , : x = - - y + x  - - y  xp+lyqdy (7) where k is the slope of C, , m m  =tz Y J * '  1 1 The integration is along the closed-curve of object bound- ary C. This curve is digitized of chain code in computation. The curve consists of n line segmentsC.,V iE{O,1, .... n-I}. rPq = I[ (-Y + (x, -:Y, ))p+'Y4dY -, k linked by chain coding. Supposing C, starts at point < ( x , ,  y,)and ends at point <+l(x,+I, Each link is k determined by one of the eight directions. Now mpq in (7) can be computed with (13) Y ,  1 1 k - - ;- rp++l (9 + ( x ,  - - Y ,  )rp-i,q ((9 (8) 1 ')--I mpq =--C J.-cP+'y'dy P + l  r = O  p, Defining rpq as A* ,  r H ( i ) =  j x P + ' y q d y ,  then P. In case 3, the slope k can only be one of two possible val- ues because of 8-neighbor chain coding. Its value can be k-1 or k=- 1. Hence (9) r P'I ( i ) ~ ~ ~ r p - l , q + l ( i ) + ( x ,  T ~ , ) r ~ - , . ~ ( j )  (14) Equation (14) can be used for computing rpq(i)  itera- According to the method proposed in [7], moments can be computed out in an iterative way with (9). It can be noticed that in the 8-neighbor chain code representation for the closed curve C, each segment C, is inherently linked by the chain code. Concerning these characteristics, we proposed a way to improve the moment computation with the relation- ships among chain connections, so to reduce about half of the whole computational time. Let us think of possible cases along different chain directions for the computation of rpq tively. The finally required expression is directly obtained from the definition of rpq (i) by setting p=-1 : From the discussion above, we can see that only the item r-l.q (i) has to be computed separately, while the other orders of rpq (i) can be computed iteratively using (1 3) or (14). Case 1 : In the x-direction, B. Fast cornpitation of moments C , : y = y , , r , , ( i ) = O  (10) Case 2 : In the y-direction, From equation (15), the computation of rp4(i)  is deter- mined by *the computation of polynomial (y,"=,' -yp+') .  Considering the relationship of two adjacent points linked - 975 - with 8-neighbor chain code, there is y,+, = y ,  rtS with S E {OJ} . In case of6=1, we have Once the values of y,' , y," ,. . ., yp+', are known, computa- tion for y,il ,...,$++,I only need time for 1/2(q+l)(q+2) additions. After the boundary extraction of an object and applying S- neighbor chain coding to the closed curve C, the fast compu- tation of moments is implemented as below: For the starting point with i=O, compute the vector Using (16), compute the vector (y:+,,y:+, ,...,y,?'l) . Determine the direction of the ith segment C, . If C, is in the x-direction, set rw (i) =0, i=i+l and return to step 2, otherwise do step 4. Compute (i) and update ( y,' ,y,* ;.-,y:") with (Y,'+~, y :, ,. . ., y z '  ) , to prepare for cafculation of next segment. Determine the direction of C, , and use (11) or (14) to compute rN (i) . Let i=i+l, if i=n, the end point of the boundary has been reached, compute mpQ and stop. Otherwise return to step 2. (Y: 9 Y,' 3' ' ., yP+' 1 . L. Simulation of computational time and comparison As mentioned above, with the inherent iterative relation- ship of 8-neighbor chain code, the method for moment com- putation proposed in [7] is improved. Now the number of multiplication is reduced to a half time, for that the calcula- tion of the polynomial for r+(i) in (16) is finished with addition operations. According to (14) or (15), the computa- tion of rpq ( i )  just needs one multiplication. Such improve- ment was confirmed by the simulated experiment to compare both methods. Fig.2 shows result of computational time (in seconds). The curve b indicates the result with the method of [7] ,  while curve a indicates the result with our method. The order of moments is limited within p+q13. 111. APPLYING ARTIFICIAL NEURAL NETWORKS FOR MOMENT-BASED OBJECT RECOGNITION A .  Construction of the system A system of moments-based object recognition with neural networks is depicted in Fig.1. Because artificial neural net- works have advantages of parallel implementation, capability of learning and error tolerance, they have been widely used for recognition tasks. Since moment invariants can form the feature space for object description, it is suitable to construct the input vector of neural networks. The BP neural networks were used in our system. B. Extraction of moment invariants In order to eliminate the factors of rotation, translation and scaling in object measurement, the normalized central mo- ments are used as the invariant feature for 2D recognition. Under the constraint of the order of moments withii p+q53, there are seven functions for representation of the normalized central moments. These seven moment invariants were used to form the input vector of the neural networks. C. Experimental results Fig.4 are the five synthetic 2D objects that were used for recognition in our experiment. The BP networks consist of three layers. There are 7 input neurons corresponding to the 7 moment invariants and 5 output neurons corresponding to five patterns of the objects to be recognized. Each object was rotated in step of one degree over 360" and the seven moment invariants were computed out respec- tively. The results weFe used as the training vectors for the BP networks. The BP networks were trained 15,000 times. The error curve is depicted in Fig.3. I segmentation I 8-neighbor chain coding computation of moment invariants .c neural networks c5 output Fig. 1 A.block diagram of the recognition system - 976 - Fig 2 Computational time as a hnction of the number of sides of polygon Fig 3 The training error After training, the BP networks were used to recognize all the five objects in Fig.4. The result is shown in Tab.1, where the symbol (yl,y2,y3,y4,y5) denote the values of the five outputs and (a,b,c,d,e) denote the type of object as input. The certain output with value near to 1 corresponds to a certain input object while other outputs have the values near to 0. All the five objects were correctly classified in our experiments. Furthermore, Other objects which were not included in the training set, were also distinguished to those known objects. D. A comparison with the pofygon-bused method The polygon-based method applies polygonal ap'proxima- tion algorithm for shape analysis and recognition [4] [IO]. Usually, the feature points are extracted by detection of the local maximum of curvatures and the computational time is mainly consumed by the computation of the local maximum detection. According to the method proposed in [4], the shape of polygonal approximation is closely related to the a prior parameter Dth which is used as threshold for detection of local maximum of curvature (the readers are referred to [4] for more details). We noticed that in the computation of fea- ture extraction, number of multiplying operations increases exponentially as Dm increases. Therefore, to simplify the shape of an object by polygonal approximation with a larger Dth, which causes time reduction in moments computation, the computational burden in feature detection has to be im- posed in the stage of polygonal approximation. Tab.:! is the experimental results for the comparison of the whole time used for recognition by our method and the poly- gon-based method. The time T (in ms) indicate the results with our method and t (in ms) indicate the polygon-based method with different Dth. The five objects shown in Fig.4 were used in experiments. The results were obtained by car- rying out both methods on IBM-486 computer. IV. CONCLUSIONS AND DISCUSSION In this paper, we proposed an approach for fast computing moment invariants. 2-D objects are directly represented with 8-neighbor chain code. Based on the analysis of the inherent relationship among joints of boundary described by 8- TABLE I Experimental results of recognition with neural network b d e 0 10 010 010 0 IO 010 010 0 10 0 IO 0 90 0 1 0  010 0 10 090 0 10 0 10 0 IO 0 10 0 IO 0 IO 0 10 090  TABLE I1 The whole time used for recognition in experiments T D&=O 10 t 610 770 550 Fig. 4 The objects used in experieiits neighbor chain code, the method of moments computation in an iterative way has been improved. With the 7 fimctions describing the moment invariants, neural networks are di- rectly applied to implement the recognition tasks. The im- proved method for moments computation in iterative way proposed in this paper can save about half of the computa- tional time. The experimental results also showed that the whole recognition time taken in ow approach is less than that of the polyg,on-based method. Obviously, in case of noise absence, 8-neighbor chain code used in this paper can yield to more details in object representation and SO to high accu- racy of moments computation and recognition. But if the acquisition of images are noise-corrupted, the approach used in this paper is expected to be improved tlo keep robustness of object representation and validity of moments computation the same time. - 977 - V. ACKNOWLEDGMENTS IEEE Trans. ASSP, vol. 34, no. 3, 1986, pp.546-553. [6] M.K. Hu, “Visual pattern recognition by moment invari- ance”, IRE Ti-ans I n ?  Theory, no. 8 ,  1962, PP. 179-187. [71 X.Y. Jiang and H. Bunke, “Simple and fast computation This paper is supported by National “863” Foundation No: 03-05 and the National Science Foundation NO: 69675005. VI. REFERENCES Bing-Cheng Li, “A new computation of geometric mo- ments”, Pattern Recognition, no. 26. 1993, pp.109-113. Bing-Cheng Li and Jun Shen, “Fast computation of mo- ment invariants”, Pattern Recognition, no. 24, 1991, pp. Bing-Cheng Li, “The Moment Calculation of polyhedra”, Pattern Recognition, no. 26, 1993, pp. 1229-1233. C.C. Chang, S.M. Hwang and D.J.Buehrer, “A shape recognition scheme based on relative distances of feature points from the centroid” , Pattern Recognition , no. 24, M. A Hatamian, “Real-time two dimensional moment generating algorithm and its single chip implementation”, 807-813. 1991. pp. 1053-1063. of moments”, Pattern Recognition , no. 24, 1991, pp. [SI D.A. Mitzias and B.C. Mertzios, “Shape recognition with a neural classifier based on a fast polygon approximation technique”, Pattern Recognition, no. 27, 1994, pp. 627- 636. [9] A.P. Reeves, “The general theory of moment operations”, Purdue University Tech. Rep. TR-EE 8 1-37, October, 1981. [10]C.H. Teh. and R.T. Chin, “On the detection dominant points on digital curves’’, IEEE Trans. PAM1 vol. 11, [11]M.F. Zakaria, L.J: Vroomen, P.J.A. Zsombor-Murray and M. Kessel, “Fast algorithm for computation of moment invariants”, Pattern Recognition, no. 20, 1987, pp. 639- 643. 801-806. 1989, pp. 859-872. - 978 - 

A New Method for Fast Computation of Moments Based on 8-neighbor Chain CodeApplied to 2-D Objects Recognition

NARCIS 

A new computation of geometric moments”,

A shape recognition scheme based on relative distances of feature points from the centroid” ,

Fast computation of moment invariants”,

On the detection dominant points on digital curves’’,

Shape recognition with a neural classifier based on a fast polygon approximation technique”,

The general theory of moment operations”,

The Moment Calculation of polyhedra”,

A New Method for Fast Computation of Moments Based on 8-neighbor Chain CodeApplied to 2-D Objects Recognition

Abstract

Similar works

Full text

Available Versions

University of Twente Research Information

NARCIS