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Orthogonal moment operators for subpixel edge detection 297 here n20, and n-lml is an even positive integer. Now, the orthogonal complex polynomials Voo, V, Rm(e) is a radial polynomial defined as and V,os can be written as (n-|m2 Rnm(p)=∑ =1,V1=x+j,V20=2x2+2y n+ml The corresponding Zernike moments A the 2 original image data f(x, y) and A, s of the rotated Values of Vmm are orthogonal and satisfy the following Image are related by A,=A,,e, VRm(x, y)Vpe(x, y)dxdy nce n in equation is the imaginary +y2≤1 of A., we can write ifTn=p and m=q. (5) Im[A]=sin(D)Re[A,]-cos()ImA,]=0 Otherwise, the integral is zero. If f(x, y) is const (8 inside the unit circle, this relationship is valid for Amm where \ib/ and Re[a, are the imaginary and also. An important property of Zernike moments is that their values in an image and its rotated version real components of All, respectively. Therefore have a simplc relationship If an image is rotated by an angle the Zernike moments of the original image 中=an/Im[A13 RelAu] Aw and the Zernike moments of the rotated image Au Referring to Fig. 1 we can further calculate are related as A',=Aun e Joo 1√(1-x2) 2 h dy dx+2「 kdydx It is clear that Zernike moments merely acquire a phase shift on rotation and their magnitudes remain k constant. This property is useful for rotation invariant h+-ksin-(0)-kly(-72) pattern recognition and matching (2b) 2 f(x, y)(x-jydy dx 3. ZERNIKE MOMENT-BASED EDGE DETECTION 2k(1-12)3n2 Zernike moments are basically integral-based op- erators and therefore are good candidates for noise tolerant image processing. These moments are de- A20=∫∫f(x,y)(2x2+2y2-1)dydx pendent on the image data and weclaim that any features 2+y2≤1 resulting from the image intensity profile can 2kl(1-P2)32 nts. he 3 centrate on their use for subpixel-and pixel-level edge detection. Consider an ideal step edge, shown in Fig. 1. Solving these equations and substituting Aoo= Aoo k is the step height, h the background gray level, I the and A,=A,o, we get the other three step edge perpendicular distance from the center of the circular ernel and the edge makes an angle of with respect to the x-axis. If we rotate the edge by an angle -p, it will be aligned parallel to the y-axis. So, we have f(x, y)y dydx=0 where f(x, y) is the edge function after it is rotated Fig.1.Two-dimensional subpixel step edge model. Fig. 2. Circular kernel defined for a 5x 5 pixel area 298 S GHOSAL and R mehrotra parameters 219123115731 31160016001600123 10) 1573.60016001601573 1231.1600.16001600.123l 3A 029.1231.15731231.0219 2)3/2 +ksin-()+kl(1-12) ,0147 0000 0469 12) +.0147+0933.1253+0933+.0147 0933-,0640 0000 0933 .0469+.0640j+.0640+0640+0469 Thus three Zernike moments Aoo, Ao, and Aare 1253 0000 06401253 needed to calculate the four parameters of the step +j0 +j0 0 +j0 dg to estimate these moments the correspondin 0933.064000000640 0933 masks of any desired size can be obtained by evaluating 046 064001-064010640j-04691 the associated integral over each pixel assuming f(x, y) 0147 0469.0147 to be constant over that pixe l circular limits are taken .0147-0931253j-0930147 for integration as shown in Fig. 2. Masks of size 5 x 5 Au Mask are shown in Fig 3. The moments are estimated by correlating the image elements with the masks. The 0177.0595.0507.0595017 Zernike moment-based edge detector employs fewer 0595-.0492-.1004-04920595 asks compared to the geometric moment-based detector. The geometric moment-based edge detector 0507.1004÷.1516-,1004.0507 0595·.0492-.100-04920595 requires a total of six masks whereas the Zernike 0l7.0595050705950177 moment-based detector uses three masks (four real masks). Therefore, the proposed Zernike moment A Mask based edge detector is 33% more efficient compared to Fig 3. Zernike moment masks for subpixel edge detection the geometric moment-based detector 3. 1. Zernike moment-based pixel-level edge detector 3. 2. Stochastic analysis In this subsection, we develop a Zernike moment In this subsection, the influence of noise on the based method for pixel-level edge detection as a special performance of the proposed subpixel edge detector is case of subpixel-level edge detection. In pixel-level presented. We separately consider the effect of noise edges, the distance of the edge I from the center of the on edge location I and orientation Assume i.i.d. mask can be taken as zero and if the detected step size Gaussian noise n(x, y)with zero-mean and variance o' k is greater than a predetermined threshold then the is added to the pixel gray levcls of a sampled ideal edge given pixel is marked as an edge pixel. Now, if the Now, referring to equation(10), we can write the distance I of the edge from the center of the mask is detected random distance of the edge as ∑∑V2o(ij)f(, 20 0j=0 A L1 ∑∑(Re[v1(刃 f(i, jcos(刂)+Im[Vv1(门f(/)sn(中) i=0j=0 zero, equations(11 )and(12 )can be written as where f(i,j)=f(,j)+n(i, j)is the noisy image data and(w+1)xw+l) the dimension of the mask. this 3A k (13) response can be viewed as a deterministic part plus a random part M,m. Response of A,m to the noise can be written as A-0.5k nm=∑∑V(小 (16) i=0j=0 Using equations (9), (13)and( 14), pixel-level edge Hence nm is simply the weighted sum of(w+1)x(w+1) parameters,ic. the direction of the edge, step size k, independent, zero-mcan Gaussian random variables and background gray level h can be calculated. Thus In the presencc of noisc, I can be written as one complex mask for A, and one real mask for Aoo are required to specify the three parameters of a ixel-level edge Orthogonal moment operators for subpixel edge detection ilar in form to the pdf of the distance of a E{}=H=∑∑V20(j() ID step edge for the geometric moment-based edge operators proposed in reference(1). However, for the E(}==∑∑(Rev1朋)∫,cos(p) proposed set of operators, and ad are functions of the edge orientation. If A,=Rd=0, the above prol +Im[v1(,门sin(φ) bility density redi Figures 4 and 5 show the effect of noise on the the standard deviation uf g, can be written as performance of the proposed detector. The absolute mean location error of i can be defined as Gi,j) E (22) and the standard deviation of gd as The standard deviation of i can be defined as d2=a2∑∑(Rev1(,)2c0s2(p) t=0i=0 ∫1n(d-( +(Im[V1(])2sin2() Figure 4 shows the theoretical absolute mean location =a2∑∑〔Re[v1(G,门)2 error of an edge oriented vertically and passing through the center of a 5 x 5 window, with respect to snr The random variable I can now be viewed as the (20 log(h/o). The background gray level is 100 and quotient of two non-zero mean Gaussian random var- step is also 100. The location error is approximately iables. To completely parameterize this relationship, computed by numerical integration of equation(22) the covariance between the numerator and denom- The limits of integration are taken to be -15 and 15 inator random variables must be determined. Now in equations(22) and(23). pi(0) practically vanishes since Zernike moments are orthogonal eyond these limits even for very small SNRs. the ∑Σvm(,)Vm(,=0n≠porm≠4,(17) standard deviation of I for the proposed detector and for the log detector are plotted in Fig. 5 with respect After some algebraic manipulation, it is easy to show to the SNR. It can be shown that the standard that covariance E(9m,9al is zero. Now, if the two deviation of i for the LoG detector can be expressed Gaussian variables are uncorrelated, then they are as 27) necessarily independent. Hence, the distance equation is the quotient of two non-zero independent Gaussian +1 random variables So we can write SNR/10 plan,gd=Pgn(gnp(ga) (18) ke-k 12k e P()=∫ galpa(4)pa(g)d Simplifying the above equation, we get k p(=「 (y-n2(y-A) p -∞LTon0 2a2 aa is the standard deviation of the smoothing Gaussian (20) filter. Figure 5 shows the standard deviation of l for Pr()is the probability density function(PDF)of the three different values of g. note that for the Log detector, the theoretical mean location error is zero distance of the edge. For a 5 x 5 window size note, 04=0. 11220- and od=0.10050. After some math This theoretical formulation is based on the continuous ematical manipulation(shown in Appendix the PDF model of the step edge profile. The discrete subpixel for the distance of the edge can be written as edge locations are usually determined by resampling of the original image function at a higher resolution han the resolution of the input d then findi p{)= the zero-crossings of the loG detector as reported by Heurtas and Medioni(20)Thus, the standard deviation 2\Iu 2+Hn exp\2(102+o2) (ar-lpa) of I for the Heurtas-Medioni detector would be more (12l2+a2)15 than that obtained by equation (24). Table 1 compar the performance of the proposed detector to that of x erf udo+lu, od the Nalwa-Binford(N-B)detector(9)for comparable ond√(m2+12 (21)edge profiles. The ideal edge profile is blurred with normalized Gaussian function with o blur =0.6. These It is interesting to note that the pdf of the distance results show that the proposed detector is superior to of the 2D step edge for Zernike moment-based detec- the loG and Nalwa-Binford detectors and comparable 300 S GHOSAL and R mehrotra 03 Proposed-detect 0.25 0.2 do-to 0.15 0.05 10 30 40 SNR (in Fig 4. Estimated mean error of I VS SNR L。G(s⊥gma=0.3) "L。G(s⊥gma=0.8)”-- 0.7 do 3 0.2 10 15 20 25 30 35 40 SNr (1n dB) Fig. 5. Estimated standard deviation of I vs SNR Orthogonal moment operators for subpixel edge detection 301 Table 1 4 IMPLEMENTATIONS Std dev for Std dev for Numerous experiments have been conducted to test h/o n Detector proposed detector the efficacy of the proposed edge detection method with various gray level images ne of these ey 045 0.35 4 022 0.14 imental results are discussed in this section. 6 013 009 The main steps for pixel-level and subpixel edge 8 0.10 0.07 detection are outlined below Pixel-level edge detection o Calculate desired size masks using circular limits of integration as shown in Fig. 2. o Convolve these masks with the image points to with other existing subpixel edge detectors (refer to get Zernike moments Ao0, A1 and A20- plots shown in reference(23)) Evaluate from equation()using Re [All] and Theevaluation ofthe density function of the orienta Im[A1]masks. This gives the direction of the edge at tion of the 2D step edge is quite straightforward. a particular image point. Zernike moment masks Re[a and im [al are o Calculate A,=Re[a,= re[Au]cos(o)+ used to calculate the angle of the edge. The angle p is Im[au]sin(). Use this valuc of Aui in equation( 13) dctcrmindd by tan -(Im[Au]Re[AuJ). As before, to determine the step size k Im [A1] and re [au are random with gaussian a Calculate the background gray level h using distribution when Gaussian noise n(, y) is added to equation((4) the sampled edge image. Moreover, Re [aul and im [Au1 are independent of each other. Thus the angle is simply a quotient of two independent Gaussian random variables with the added transformation of tan"I. Then the pDf can be written as(z8) p(中)=e 1pcos(φ- u2cos2(小-中)/22 2 /(2r (φ-ψ) +erf (25) This density function Pa(o) is symmetrical about y=tan (uyH) such that the mean value of is tan(ay/ux).Ay is the mean of the numerator in equation(9)and 4 the mean of the denominator. o2 is the variance of both numerator and denominator d2=k2+12 Note that geometric moments can be expressed as of Zernike moments and re [al ar Im [Au]are basically equal to the geometric moments Mio and M So, the Pdf of the ed orientation for the Zernike moment-based detectors is the same as that for the geometric moment-based Fig hetic image with a subpixel edge Table 2. Subpixel edge parameters detected in Fig. 6 Measured Actual Column Angle,φ distance distance Step, k 1820059 04159 0.4124 5890 103 100 1820059 0.3810 0377558.50 100 820059 03461 034265814 1820059 0.3111 0.3077 5781 106 100 1820059 0.2762 0.2728 57.52 1820059 02413 02379 1820059 0206 02030570 09 1820059 0.1714 0.1682 5687 302 S GHOSAL and R. mehrotra THUFEIU Fig. 7. Original 256 X 256 Orthogonal moment operators for subpixel edge detection 303 除明 Fig 8. Detected edge maps by the proposed detector 304 S GHoSAL and r mehrotra The above steps are repeated for every point in the or other non-step edges are present. However, Zernike image plane to get step edge parameters at every point moment-based operators can be designed for ramp, in the image. Now postprocessing with proper threshold roof, or generalized step edges( combinations of step selection and edge thinning can be done to gct the final and ramp or step and roof edges)also (29) edge image. However, postprocessing is not the main topic of this paper. 5 CONCLUSIONS Subpixel edge detection In this paper an orthogonal complex moment-based technique for subpixel-and pixel-level step edge detec Calculate masks of desired size using circular tion is proposed. a total of three masks(one complex limits of integration as shown in Fig. 2 and two real)are required to compute all the par o Convolve these masks with the image points to ameters of a step edge with subpixel accuracy. Pixel- get Zernike moments Aoo, and All and A2o level edges can be detected using only two masks(one Evaluate from equation(9)using Re [All]and real and one complex ). Theoretical measures for the Im [Aul masks. This gives the direction of the edge at position and orientation delocalization of the detected a particular image point step edge in the presence of an additive noise are Calculate A20=A20 obtained Zernike moments are functions of the shape Calculate A1=Re [a1]=re [al]cos(o)+ and the size of objects in an image and hence any Im[A1]sin(). Use this value of All and Aio in features resulting from the image intensity profile can equation(10) to compute the distance of the edge, be detected by proper Zernike moments We have also 1. from the center of the mask developed a Zernike moment-based corner detection e Use equation(11)to determine the step size, k. method (30) We are currently investigating the use of Calculate the background gray level h using equa- Zernike moments for range image segmentation and (12) the detection of spatiotemporal features These steps basically specify all subpixel edge par- Acknowledgements-This work has been partially supported ameters. These subpixel edge parameters can result in by the NASA-Langley Research Center under grant NAG- simple postprocessing for pixel-level edge detection. 1-1276 by Biomedical Research Support Grant Program, Division of Resources, NiH, under grant BRSG SO RRO7114 Given an image point, an edge is detected if the 21, and the Center for robotics and manufacturing Systems detected step size of the University of Kentucky. The authors would like to k≥τ thank reviewers for useful suggestions that greatly improved the quality of this work where t is a chosen threshold value and edge distance <6 REFERENCES where 28 is less than the size of a pixel. In our im- 1.E. P Lyyers, O R. Mitchell, M. L Akey andAP.Reeves lementation 8 is taken to be 1/ 2 of the size of a pixel Subpixel measurements using a moment-based edge Thus, the subpixel edge distance I can be used to operator, IEEE Trans. Pattern analysis mach. 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Pattern Anal- GIRL and a houSe, edge maps obtained by using ysis Mach Intell. 7(6), 726-741 (1987) the subpixel-level detector are shown in Fig. 8. 11. A Kundu, Robust edge detection, Proc. IEEE Conf on The proposed edge detection method assumes an Computer Vision and Pattem Recognition, San Diego, California, pp 11-18( 1989) dcal step edgc model and hence its response deteriorates 12. D Mumford and J. Shah, Boundary detection by mini (thick and inaccurate)at locations where roof, ramp, mizing functionals, Proc. IEEE Conf on Computer vision

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