IMAGE RECONSTRUCTION FROM OMNI-DIRECTIONAL CAMERA Kai Guo and Zhuang Li

Transcription

1 IMAGE RECONSTRUCTION FROM OMNI-DIRECTIONAL CAMERA Kai Guo and Zhuang Li De 15, 2007 Boston University Department of Eletrial and Computer Engineering Tehnial report No. ECE BOSTON UNIVERSITY

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3 IMAGE RECONSTRUCTION FROM OMNI-DIRECTIONAL CAMERA Kai Guo and Zhuang Li Boston University Department of Eletrial and Computer Engineering 8 Saint Mary s Street Boston, MA De 15, 2007 Tehnial report No. ECE

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5 Summary This projet was ompleted as part of the assignment in EC 520. The objetive of this projet is to reonstrut the perspetive image from the omni-diretional image. We study the mapping relationship between the omni-diretional image and panorami image, and then we build the geometri projetion relation between the panorami image and the perspetive image. The simulation results indiate that the method used for image reonstrution an effiiently orret the geometri distortion. In order to improve the pereptual image quality, we use ubi interpolator and image enhanement. Additionally, Cirular Hough Transform is employed for automati detetion of the irle enter.

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7 Contents 1. Introdution Mirror design and onstant vertial resolution Panorami image unwrapping method Analysis of panorami image unwrapping error Interpolation methods Perspetive projetion from panorami image Color Enhanement Automati irle enter detetion Simulation results Conlusion...21

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9 List of figures Fig. 1 Illustration of the onstant vertial resolution 2 Fig. 2 The relationship between omni-image and the ylinder image 3 Fig. 3 Omni-image unwrapping 4 Fig. 4 Illustration of panorama distortion 6 Fig. 5 Bilinear interpolation 7 Fig. 6 Linear and Cubi interpolation 8 Fig. 7 2-D Cubi interpolation 8 Fig. 8 Projetion from the panorama to the virtual plane 10 Fig. 9 Planform of the projetion from the panorama to the virtual plane 10 Fig. 10 S-shaped mapping funtion 12 Fig. 11 Illustration of Cirle Hough Transform 13 Fig. 12a Omni-images: Lab 15 Fig. 12b Omni-image: Offie 15 Fig. 13a Panorami image: Lab (nearest-neighbor method) 16 Fig. 13b Panorami image: Offie (nearest-neighbor method) 16 Fig. 14a Panorami image: Lab (average filter) 16 Fig. 14b Panorami image: Offie (average) 16 Fig. 15a Panorami image: Lab (linear interpolator) 16 Fig. 15b Panorami image: Offie (linear interpolator) 16 Fig. 16a Panorami image: Lab (ubi interpolator) 17 Fig. 16b Panorami image: Offie (ubi interpolator) 17 Fig. 17a Panorami image: Lab (ubi interpolator + image enhanement) 17 Fig. 17b Panorami image: Offie (ubi interpolator + image enhanement) 17

10 Fig. 18a Perspetive image: Lab (view point position: y = 0. 10d ) 18 Fig. 18b Perspetive image: Lab (view point position: y = 0. 30d ) 18 Fig. 18 Perspetive image: Lab (view point position: y = 0. 50d ) 19 Fig. 18d Perspetive image: Lab (view point position: y = 0. 65d ) 19 Fig. 18e Perspetive image: Lab (view point position: y = 0. 65d ) 20 Fig. 19a Auto-deteted irle enter: Lab 20 Fig. 19b Auto-deteted irle enter: Offie 20 v v v v v

11 1 Kai Guo & Zhuang Li 1 Introdution With the widespread adoption of video sensor networks, it is desirable to maximum the overed area with minimum number of ameras, whih an redue the expense for ameras and the omplexity to install and alibrate ameras. However, onventional ameras only have very limited field of view that leads to the inreased network ost, need for mutual amera alibration and additional maintenane effort. Therefore, omni-diretional (atadioptri) system is devised to solve this problem. A typial system uses a speial mirror to reflet light rays from up to 360 degrees into a perspetive amera lens and forms an omni-diretional image (omni-image). This system is promising sine it an be pratially implemented in many areas, suh as video onferene, video surveillane, roboti vision and human ativity monitoring. As long as the atadioptri imaging system is setup and alibrated, it an view the surroundings all the time. But the omni-images aptured from atadioptri system have different properties ompared with the perspetive images in terms of lower effetive resolution and imaging deformation. Suh distortion leads to the diffiulty for the images to be diretly implemented; thus it is neessary to work out an effiient method to unwrap the omni-image. In this projet, we firstly analyze the potential reasons that result in the image distortion, inluding geometri imaging deformation and sampling system disrepany in the omni-image and panorami image. Then we use a pratial algorithm to unwrap the image by building the mapping relationship between two imaging systems. In order to further improve the unwrapping quality, several interpolation methods are ompared and we an find that the ubi interpolation ahieves the best performane. Although the panorami image is muh more omfortable for analysis than the omni-image, it is still not the best one for human observation; sine it is not a perspetive image. In order to solve this problem, we establish the relationship between the panorami image and the perspetive image whih an be used for geometri orretion. In pratial unwrapping, an important prerequisite is to find the oordinates of the omni-image s enter, whih is always manually ahieved. In this report, the irle

12 2 Kai Guo & Zhuang Li Hough transformation is used for auto enter detetion that has a satisfied auray. 2 Mirror design and onstant vertial resolution In this projet the mirror of the atadioptri system has onstant vertial resolution, whih means that objets at a (pre-speified) fixed distane from the amera s optial axis will always be the same size in the image, independent of its vertial oordinates. The geometry of the image formation of the atadioptri amera is shown in Fig. 1. If we onsider a ylinder of radius, d, aligned with the amera optial axis, we want to ensure that ratios of distanes measured in the vertial diretion along the surfae of the ylinder would remain unhanged when measured in the image. Suh invariane should be obtained by judiiously designing the mirror profile to yield a onstant vertial resolution mirror. By using this kind of mirror, the imaging distortion an be avoided; therefore, we are able to develop appropriate formulas for unwrapping without introduing transformation error. Fig. 1 Illustration of the onstant vertial resolution

13 3 Kai Guo & Zhuang Li 3. Panorami image unwrapping method The omni-image an be onsidered as the refleted ylinder (panorami) image around the mirror, as shown in Fig. 2. Fig. 2 The relationship between omni-image and the ylinder image Fig. 2 implies that in eah utting surfae, there exists a one-to-one mapping between the ylinder edge and the omni-image s radius, whih an be used to reover the unwrapped image. Fig. 3 depits the priniple of panorami unwrapping. Fig. 3 (a) denotes the omni-image with inner radius r and outer radius R. The region between r and R is the valid region that an be used for unwrapping, beause the region inside the inner radius is usually the imaging of the amera lens that is not ritial for unwrapping. Fig. 3 (b) shows the orresponding unrolled ylinder panorama. Aording to the fat disussed in ' ' ' setion II, for a pixel P ( X, Y ) in the ylinder image, the polar oordinates of the o o o orresponding pixel P X, Y ) an be determined with the following formula: o ( o o

14 4 Kai Guo & Zhuang Li Where ( x y ) m, m r m ' h o = Yo (1) ym 2π ' θ o = X o (2) x m are the desired dimensions of ylinder image in pixel units, r is m the radius of the mirror as seen in the irular panorama in pixel units, andθ o is in radians. The orret ratio of x / y m m is 2 π, beause the out perimeter of the mirror orresponds to the bottom row of the unwrapped image. Then we an find the mapping relationship between polar oordinates and retangular oordinates in the irular panorama: X o rm π ' = X + ho osθ o = X + Y X o (3) y m ' 2 o os xm Y o rm π ' = Y + ho sinθ o = Y + Y X o (4) y m ' 2 o sin xm Aording to the formula above, it is easy to alulate the oordinate in the omni-image when given a point in the ylinder panorami image. Note that ( X Y ) o, o is usually not an integer pixel; therefore, the pixel values annot be diretly obtained from the image aptured by the amera. Therefore, we need to use some interpolation approahes to estimate the pixel values. Fig. 3 Omni-image unwrapping

15 5 Kai Guo & Zhuang Li 4 Analysis of panorami image unwrapping error In order to orretly unroll the omni-image, it is worthwhile to analyze the possible reasons that result in the panorami image unwrapping error. Based on the disussion in Setion II, we have known that the oordinates in the omni-image mapped from ylinder panorama are frational values that an result in the unwrapping error. If a polar sensor with suffiient high resolution is used to apture the omni-image, this sort of error an be effiiently restrited. However, most of the ommerial ameras are based on the orthogonal sampling tehnique; therefore, the aptured omni-image has lower effetive resolution near the mirror vertex than near the mirror bottom, whih is one of the major soures of the unwrapping error. The other major soure of error an be analyzed aording to the geometri imaging priniple. This sort of error depends on the shape and position of the mirror and the influene of both fators will be disussed below. It is well known that the virtual image of a plane mirror has no deformation, i.e., no strething or shrinking in the size and no shape distortion. On the ontrary, a urved mirror will always lead to the virtual imaging distortion. The oni mirror an be onsidered as a ombination of two plane mirrors in the utting surfae; thus, the oni mirror has no imaging distortion, while other urved mirrors (suh as hyperboli mirror, paraboli mirror and hemisphere mirror) have suh sort of distortion. Despite the oni mirror an avoid the virtual imaging distortion, there is still another possibility an lead to the panorama distortion, as illustrated in Fig. 3. In Fig. 3 an objet AC is projeted onto the oni mirror and refleted to form an omni-image on the horizontal amera sensor and B is the enter of the objet. We an find that B is still the enter point in the virtual image whih indiates that oni mirror has no imaging distortion. Nevertheless, when the virtual image is projeted onto the sensor, B is no longer the enter point of AC, beause the virtual image is not parallel to the horizontal sensor. If the sensor is rotated from AC to A C, there is no projetion error any more and B is the enter point of A C.

16 6 Kai Guo & Zhuang Li We have analyzed the possible reasons to lead to the omni-image distortion and in the next setion we will disuss the effiient solution to redue the distortion. Fig. 4 Illustration of panorama distortion 5 Interpolation methods Although onstant vertial resolution mirror an avoid imaging distortion, the alulated pixels in the irular image does not orrespond exatly one to one to the pixels of panorami image so sub pixel anti-aliasing methods should be used. Interpolation is suh a method whih is ritial for approahing the better image quality from the unwrapped image. The ideal interpolation is the upsampling tehnique, but in spatial domain the SINC funtion is required for upsampling whih makes use of all the pixels to estimate an arbitrary pixel value, leading to the overload of omputational omplexity. For pratial implementation, several interpolation methods are used to approximate the urrent pixel value from its neighborhood pixels.

17 7 Kai Guo & Zhuang Li 1) Nearest-neighbor interpolation The nearest neighbor algorithm simply selets the value of the nearest point, without onsidering the values of other neighboring points at all. (e.g. if x=2.4 lies between 2 and 3, then x is equal to2). 2) Average (mean) interpolation The algorithm simply alulates the mean value of the neighboring points of the given point. The average (mean) filter an smooth image data and redue the noise. 3) Linear interpolation Linear interpolation is the simplest method of getting values at positions between the data points. The points are simply joined by straight line segments. In demanding situations, linear interpolation is often not aurate enough. 4) Bilinear interpolation Bilinear interpolation is an extension of linear interpolation for interpolating funtions of two variables on a regular grid. The key idea is to perform linear interpolation first in one diretion, and then in the other diretion. The result of bilinear interpolation is independent of the order of interpolation. Fig.6 shows that the four red dots show the data points and the green dot is the point at whih we want to interpolate. Bilinear interpolation first interpolates horizontally between the red points ( and ) to determine the blue points and, respetively. Then it interpolates vertially between the blue points to determine the green point. Fig.5 Bilinear interpolation

18 8 Kai Guo & Zhuang Li 5) Cubi interpolation Cubi interpolation is the method that offers true ontinuity between the segments. It requires more than just the two endpoints of the segment and the two points on either side of them. The points are joined by 3 rd order polynomial segments. Fig.6 shows that interpolation on a data set (red points) onsists of piees of polynomials (blue lines). Any point (purple point) an be alulated by the polynomials. The red line and blue line represent linear interpolation and ubi interpolation, respetively. The ubi interpolation is deemed as a good approximation of SINC funtion. Fig.6 Linear and Cubi interpolation The 2-D ubi interpolation interpolates between data points. It finds values of a two-dimensional funtion underlying f(x, y) the data at intermediate points. Fig.7 2-D Cubi interpolation

19 9 Kai Guo & Zhuang Li 6 Perspetive projetion from panorami image In the previous setions, the omni-image has been unwrapped to panorami image that is reonstruted onto the ylinder around the atadioptri system. However, the panorama may lead to some annoyed geometri distortion and is not appropriate for human observation; thus, it is desirable to form the perspetive image by projeting the ylinder image onto a virtual image plane. Limited by the nature of the perspetive image, the reonstruted image is usually onfined to a narrow range. The 3-D mapping relationship between panorama and perspetive image is shown in Fig. 8, and the 2-D planform is shown in Fig. 9. In Fig. 8, the plane OXY is the perspetive imaging plane; MNST is the urved plane of the ylinder whih is used to reonstrut the perspetive image. PQ is the axial line of the ylinder. V is the virtual view point. Sine the mirror used in our report (onstant vertial resolution) has no fous point, our atadioptri amera is not a single view system. Therefore, it is unlikely to form a perfet perspetive image and what we an do is to minimize the imaging error by adjusting the position of the virtual view point V, whih is always a manual job. VAA and VBB are the projetion lines, where points A, B are on the ylinder, and points A, B are on the plane. Aording to the geometry projetion priniples, we an formulate the relationship between the oordinate A' ( x p, y ) and the oordinate A x, y ). ( L α = (5) d d = d s (6) p + p d α width = 2d p tan (7) 2 d p = os( α / 2 x / d ) (8) x p = width d α x tan( 2 d p 2 ) (9) y p d = yv + ( y yv ) (10) d

20 10 Kai Guo & Zhuang Li Fig. 8 Projetion from the panorama to the virtual plane O x p A width x A d d p d α/2 Fig. 9 Planform of the projetion from the panorama to the virtual plane

21 11 Kai Guo & Zhuang Li where d is the radius of the ylinder, irle enter to the perspetive plane, d p is the perpendiular distane from the yv is the oordinate of the virtual view point whih shares the same x oordinates with the irle enter. Reformulate (9) and (10): x α 1 = d [ 2 d p tan 1 width ( x 2 p )] (11) y d = yv + ( y p yv ) (12) d p Given a point on the perspetive plane, we an alulate its orresponding mapping point on the panorama, whose oordinates are usually frational values. Thus, it is neessary to estimate the pixel value of the frational oordinates with interpolation tehnique. 7 Color Enhanement Image enhanement is a powerful tool for the improvement of digital image quality (for visual inspetion or for mahine analysis), without knowledge about the soure of degradation. In this report, one of the image enhanement algorithms, olor-mapping transformation, is implemented. As is known that the ontrast-strething tehnique an be used to inrease the dynami range of gray-levels, it is natural to extend it to the olor image proessing. The olor image transform is in the form:, where and are the olor omponents of the input and output images, n is the dimension of the olor spae of, and is referred to as full-olor transformation (or mapping) funtions. Here ubi-spline interpolation mapping funtion is used to adjust the ontrast and luminane of images, so that the histogram is smoothed. Fig. 10 shows the s-shaped mapping funtion.

22 12 Kai Guo & Zhuang Li Fig. 10 S-shaped mapping funtion This mapping funtion maps the input image with a full-olor referene bar. The two ending points an be used to adjust the luminane by moving up and down; while the two middle points an be used to adjust the image ontrast. The adjusting work is manually proessed until a satisfied image is obtained and the evaluation of image quality is a highly subjetive proess. 8 Automati irle enter detetion The irle enter position and the radius of the irular image are ritial for image unwrapping, and in some irumstanes the manual searh of irle enter may not be feasible, thus it s worthwhile to develop an algorithm that an automatially detet those parameters in an effiient and aurate way. One of the prevalent tools to ahieve this goal is the Cirle Hough Transform (CHT), whih an aurately find the irle enters and radii. Besides, the Hough tehnique has been found to be effetive and highly robust against olusions, noise and other artifats. The CHT is illustrated in Fig. 11. A set of edge points within the original image, are indiated by the blak irles. Eah edge point in the original image ontributes a irle of radius R to the output aumulator spae, and this ontribution is shown by the gray irles. The output aumulator spae

23 13 Kai Guo & Zhuang Li has a peak where these plotted irles overlap. In the other word, the peak is just the enter of the irle. Fig. 11 Illustration of Cirle Hough Transform For the Cirle Hough Transform alulation, a separate irle filter an be used for eah radius of irle to be deteted. This an be used to form the 3-dimensional parameter spae -, and R ( a a R domain), where two dimensions represent the oordinates of the irle enter (, ), and the third stands for its radius (R): (13) In (13), (, ) and R are the parameters and x, y are the variables orrespond to the points of the irle in the X-Y domain. For implementing Cirle Hough Transform, we need to re-analyze (13) in the x y a x a y R domain, where a, a R are regarded as variables and x, y are parameters. In this new domain, x y, we need to find the peak pixel value, whih is mostly likely to be the enter of the irle. The enter-detetion algorithm is based on the edge image (a binary image), whih an effiiently redue the omputational omplexity and improve the detetion auray.

24 14 Kai Guo & Zhuang Li 9 Simulation results In our experiment, we arry out the unwrapping from the omni-image to the panorami image, and then transform the panorami image to the perspetive image. Different interpolation methods (nearest-neighbor estimator, average filter, linear interpolator and ubi interpolator) are ompared. Fig. 12 shows the original omni-images and the unwrapped panorami images are shown in Fig. 13- Fig. 16, orresponding to different interpolation methods. Due to the limitation of the page size, the panorami image is only displayed at 20%; therefore, the differene is not visually distint. With areful omparison of the full size images, we an find that the ubi interpolation leads to the best visual quality. Thus, the following simulations are all based on the ubi interpolator. Fig. 17 shows the images after doing the image enhanement, whih an effiiently improve the visual quality. Fig. 18 shows the perspetive images with different positions of the view points. It is obvious that the view point is signifiant to the reonstruted image. In this example, when the view point is very high (as shown in Fig. 18 (a)), the perspetive image is onave distorted and vie versa. When the view point is properly seleted ( y = 0. 65d ), the distortion is minimized; beause the wall lines v and the fluoresent lamps are straight, whih is onsistent with the property of the perspetive image. Then we arry out the auto-enter detetion and the experimental results are shown in Fig. 19, in whih the enters are marked by green ross. The results indiate that the CHT algorithm has a good approximation of the irle enter.

25 15 Kai Guo & Zhuang Li Fig. 12 (a) Omni-image: Lab Fig. 12 (b) Omni-image: Offie

26 16 Kai Guo & Zhuang Li Fig. 13 (a) Panorami image: Lab (nearest-neighbor method) Fig. 13 (b) Panorami image: Offie (nearest-neighbor method) Fig. 14 (a) Panorami image: Lab (average filter) Fig. 14 (b) Panorami image: Offie (average) Fig. 15 (a) Panorami image: Lab (linear interpolator) Fig. 15 (b) Panorami image: Offie (linear interpolator)

27 17 Kai Guo & Zhuang Li Fig. 16 (a) Panorami image: Lab (ubi interpolator) Fig. 16 (b) Panorami image: Offie (ubi interpolator) Fig. 17 (a) Panorami image: Lab (ubi interpolator + image enhanement) Fig. 17 (b) Panorami image: Offie (ubi interpolator + image enhanement)

28 18 Kai Guo & Zhuang Li Fig. 18 (a) Perspetive image: Lab (view point position: y = 0. 10d ) v Fig. 18 (b) Perspetive image: Lab (view point position: y = 0. 30d ) v

29 19 Kai Guo & Zhuang Li Fig. 18 () Perspetive image: Lab (view point position: y = 0. 50d ) v Fig. 18 (d) Perspetive image: Lab (view point position: y = 0. 65d ) v

30 20 Kai Guo & Zhuang Li Fig. 18 (e) Perspetive image: Lab (view point position: y = 0. 65d ) v Fig 19(a) Auto-deteted irle enter: Lab Fig 19(b) Auto-deteted irle enter: Offie

31 21 Kai Guo & Zhuang Li 10 Conlusion The work in this report deals with the set of methods for omni-diretional image proessing. Firstly, we analyze the reasons that result in the imaging distortion of the atadioptri system. Then we establish the mapping relationship between the omni-image and the panorami image under the irumstane of the onstant vertial resolution mirror. We also build the geometri projetion relation between the panorami image and the perspetive image within a narrow range. The simulation results demonstrate that the image unwrapping an effiiently orret the geometri distortion. In order to improve the image quality, different interpolation algorithms are implemented and we find that the ubi interpolator outperforms others. Furthermore, the image enhanement is used to improve the subjetive visual effet by adjusting the ontrast and luminane of the image. Finally, we use Cirular Hough Transform to automatially detet the enter of the irle. The simulation results imply that the CHT has fairly good performane.

32 22 Kai Guo & Zhuang Li Appendix Below is listed Matlab soure ode developed for this projet. I. Omni-image -> panorami image %omni-diretional image proessing %This fution is used to unwrap the irular image to its panorami image. %The proportion of the output panorami image an be defined by user. l lear irle=double(imread('olor.jpg')); %inner radius: r1 %outer radius: r2 %oordinates in irle: X, Y %oordinates in square: Xs, Ys %angle in irle: theta %enter point in irle: enterx, entery %length of panorami image: Length %width of panorami image: Width %length and width adjuster: l_times, w_times r1=10; r2=272; enterx=272; entery=272; l_times=1; Length=round(l_times*2*pi*r2); w_times=1; Width=round(w_times*(r2-r1));

33 23 Kai Guo & Zhuang Li for w=1:width for l=1:length Ys=r2-w/w_times; theta=2*pi*l/length; X=enterX+Ys*os(theta); Y=enterY+Ys*sin(theta); xi(w,l)=x; yi(w,l)=y; end end square(:,:,1)=interp2(irle(:,:,1),yi,xi,'ubi'); square(:,:,2)=interp2(irle(:,:,2),yi,xi,'ubi'); square(:,:,3)=interp2(irle(:,:,3),yi,xi,'ubi'); square=uint8(square); figure,imshow(square,[]) II. Panorami image -> Perspetive image %omni-diretional image proessing %This fution is used to generate perspetive image from its panorami image. %User needs to define a paortion of panorami image whih will be used for %generating the perspetive image. %user defined portion of the panorami image: pano %The height and radius of panorami ylinder: d %view point position: yv %view point position adjuster: perent %width of the perspetive image: width

34 24 Kai Guo & Zhuang Li %height of the perpetive image: height %maximum length of the perpetive image: vertial_max %height of the perpetive image adjuster: vertial_offset l lear lose all square=imread('final1_4_1.jpg'); d=size(square,1); pano=double(square(1:300,1455:2023,:)); [pano_w,pano_l,h]=size(pano); alpha=pano_l/d; dp=d+30; perent=0.6; yv=d*perent; width=floor(2*dp*tan(alpha/2)); vertial_offset=0; vertial_max=round(yv+dp/d*(d-yv)); height=vertial_max-vertial_offset; for xp=1:width for yp=1:height x=d*(alpha/2-atan((width/2-xp)/dp)); d=sqrt((width/2-xp)^2+dp^2); y=yv+(yp-yv)*d/d; xi(xp,yp)=x; yi(xp,yp)=y;

35 25 Kai Guo & Zhuang Li end end pesp(:,:,1)=(interp2(pano(:,:,1),xi,yi,'ubi'))'; pesp(:,:,2)=(interp2(pano(:,:,2),xi,yi,'ubi'))'; pesp(:,:,3)=(interp2(pano(:,:,3),xi,yi,'ubi'))'; pesp=uint8(pesp); imshow(pesp(1:pano_w,:,:),[]) III. Cirle enter detetion % funtion result = hough_irle(i,step_x,step_y,step_r,step_angle,r_min,r_max,p) lear l aa=imread('olor2.jpg'); a=aa(:,:,1); b=edge(a,'sobel'); I=b; step_x=1; step_y=1; step_r=1; step_angle=0.05; [h,w] = size(i); r_min=max(h,w)/4; r_max=max(h,w)/2; p=1;

36 26 Kai Guo & Zhuang Li size_x = round(h/2/step_x)+1; size_y = round(w/2/step_y)+1; size_r = round((r_max-r_min)/step_r)+1; size_angle = round(2*pi/step_angle); hough_spae = zeros(size_x,size_y,size_r); [rows,ols] = find(i); eount = size(rows); hh= waitbar(0,'please wait...'); for i=1:eount for j=1:size_r for k=1:size_angle x = round(rows(i)-(r_min+(j-1)*step_r)*os(k*step_angle)); y = round(ols(i)-(r_min+(j-1)*step_r)*sin(k*step_angle)); if(x>h/2-h/4&x<h/2+h/4&y>w/2-w/4&y<w/2+w/4) hough_spae(x-round(h/4)+1,y-round(w/4)+1,j)= hough_spae(x-round(h/4)+1,y-round(w/4)+1,j)+1; end end end waitbar(i/100) end max_para = max(max(max(hough_spae))); %store the index for potential peaks in parameter spae index = find(hough_spae==max_para); length = size(index); for k=1:length par3 = floor(index(k)/(size_x*size_y))+1; par2 = floor((index(k)-(par3-1)*(size_x*size_y))/size_x)+1;

37 27 Kai Guo & Zhuang Li end par1 = index(k)-(par3-1)*(size_x*size_y)-(par2-1)*size_x; par11=eil(par1+h/4); par22=eil(par2+w/4); fprintf(1,'center %d %d radius %d\n',par11,par22,r_min+(par3-1)*step_r); lose (hh)