Circular or elliptic fit of an object is  important  in target detection, shape analysis, and biomedical imageanalysis problems. Here, the problems of fitting circle or ellipse to an object in 2-D are considered. At first,the problems is converted to quadratic equation of single unknown by some constraints. Then its solutionsfor all the border points of the object are found and averaged. The major and minor axes of ellipse are presentedby least sum perpendicular distance of all points of the object. The other unknowns are found  using theequations of constraints. In the proposed method, the main constraint used for the circular and elliptic fit isthat the area of the fitting circle or ellipse is equal to the area of the object to be fitted. The approach appearsto be less sensitive to the object border noise and is computationally attractive. Some examples are presentedto show the effectiveness of the approach. A measure of degree of circularity, ellipticity, etc in fuzzy settheoretic framework is also proposed.Defence Science Journal, 2008, 58(6), pp.710-714, DOI:http://dx.doi.org/10.14429/dsj.58.169

Chaudhuri, Debasis

English

Defence Science Journal

Received 2 November 2007, revised 4 January 2008Defence Science Journal, Vol. 58, No. 6, November 2008, pp. 710-714Ó 2008, DESIDOC7101. INTRODUCTIONSeveral geometric shape properties are measuredfor recognition and description of two-tone objects indigital grid. Of these properties, circularity and ellipticityplay important roles in applications involving industrialparts inspection, biomedical image analysis1-4 andtarget detection5,6.Generally, the approach of locating a parametric curvecan be divided into two parts7. The first part detectsthe boundary of the curve. The second part estimatesthe parameters based on the boundary points found inthe previous step. A measure for circularity is proposedby Haralick8 where the centre of the fitting circle isassumed to coincide with the centroid of the objectborder. The radius r of the fitting circle is the averageof magnitude of radius vectors from the centre to theborder of the object. The standard deviation s of theseradii vectors wrt r is computed to find the degree ofcircularity represented in terms of r/s.Of all parametric curves, the elliptic shape has drawnthe maximum attention1,9-12 because circles are projectedinto ellipse in the perspective projection model. Varioustechniques are available in the literature about shapeanalysis problem  using object border points. Some ofthese use maximum chord of the object and the widthperpendicular to it as the major and minor axes of thefitting ellipse, respectively. A different approach is dueto Proffit13,14 who used two components of normalisedpolar Fourier expansion of border points to define theoptimised fitting ellipse. His results are invariant underObject Area-based Method for Elliptic and CircularFit of a Two-tone ImageD. ChaudhuriDefence Electronics Applications Laboratory, Dehradun-248 001ABSTRACTCircular or elliptic fit of an object is  important  in target detection, shape analysis, and biomedical imageanalysis problems. Here, the problems of fitting circle or ellipse to an object in 2-D are considered. At first,the problems is converted to quadratic equation of single unknown by some constraints. Then its solutionsfor all the border points of the object are found and averaged. The major and minor axes of ellipse are presentedby least sum perpendicular distance of all points of the object. The other unknowns are found  using theequations of constraints. In the proposed method, the main constraint used for the circular and elliptic fit isthat the area of the fitting circle or ellipse is equal to the area of the object to be fitted. The approach appearsto be less sensitive to the object border noise and is computationally attractive. Some examples are presentedto show the effectiveness of the approach. A measure of degree of circularity, ellipticity, etc in fuzzy settheoretic framework is also proposed.Keywords: Circularity, ellipticity, morphometry, shape analysis, pattern recognition, target detection.translation, rotation, and scaling. While this approach isconvenient for ellipse, it cannot be readily extended forellipsoidal fit. An optimal trial and error method to estimatethe parameters of an ellipse and circle to an object in 2-D as well as to estimate the parameters of sphere, spheroidor ellipsoid to an object in 3-D  using the border pointsof the object is proposed by Chaudhuri15. Almost all thetechniques work mainly on border points of the object,which is susceptible to border noise resulting from objectsegmentation and spatial resolution of digital grid.With the objective to find an alternative techniqueless dependent on the border, it is  assumed that thecentre of the ellipse coincides with the centroid of allpoints of the object and not with the centroid of theborder points only. The major axis of the ellipse is assumedto pass through the line of best fit to all points belongingto the object. The minor axis passes through the centroidand is perpendicular to the major axis direction. Also, thearea of fitting ellipse is assumed to be the same as thatof the object to be fitted. Since all these computationsare done on the area of the object, the results are lesssusceptible to noise. Next, an estimate of the major (orminor) axis length is found from the border points of theobject.2. ELLIPTIC AND CIRCULAR FIT2.1 Elliptic fitConsider an object A in R2. Let ( , ), 1, 2,...,i ix y i n=    ben pels belonging to A. Then the centroid ( , )x y  of A isdefined by711CHAUDHURI: OBJECT AREA-BASED METHOD FOR ELLIPTIC AND CIRCULAR FIT OF A TWO-TONE IMAGE1 11 1 ;   n ni ii ix x y yn n= == =å å                                             (1)It is required to find a line so that the sum of thesquared perpendicular distances of all ( , )i ix y A˛    fromthe line is minimum. Let q be the angle of the line withx-axis and the line passing through ( , )x y . Therefore, theequation of the line will betan  tan  0x y y xq - + - q =                            (2)From any point ( , ), 1, 2,...,i ix y i n=   the perpendiculardistance to Eqn (2) is( ) sin ( ) cosi i iq x x y y= - q - - qTherefore, the sum of the squared perpendicular distancewill be( ) ( ) 21sin cos  ni iiQ x x y y=Ø ø= - q - - qº ßåNow q is chosen in such a way so that Q will beminimum. Therefore, 0Q¶ ¶q =  gives2,15( ) ( )( ) ( )12 212tan 2ni iini iix x y yx x y y==- -q =Ø ø- - -º ßåå  and( ) ( )( )( )2 21 11cos 2  2sin 2  0n ni ii ini iix x y yx x y y= ==ì üq - - -í ýî þ+ q - - >å åå                         (3)Let  this line be considered  as best fitting line to A.It can be shown that if A is translated, then ( , )x y and theline are translated by the same amount. Also, the bestfitting line is invariant under scaling of A. In addition ifA is rotated by an angle j, the best fitting line is alsorotated by an angle j. It is assumed that(i) The centre of the fitting ellipse is ( , )x y . (It ensurestranslation invariance of fitting ellipse.)(ii) The major axis of the fitting ellipse to A coincides withthe best fitting line of A. Also the minor axis of thefitting ellipse is a line perpendicular to the best fittingline and passes through ( , )x y . (The assumptionsensure rotation invariance of the fitting ellipse. Also,it converts ellipse fitting problem to that of solvingtwo parameters a and b in Eqn. (4) below.)(iii) Let S be the area of A. Then the area of the fittingellipse is also S. (This assumption ensures scale invarianceof the fitting ellipse. Also, as shown below, it convertsellipse fitting problem that of solving a quadratic equation.)If the centre of the ellipse is the origin (0,0) while themajor and minor axes coincide with the x and y axes, respectively.Then it is well known that the equation of the ellipse is2 22 2 1x ya b+ =                                          (4)where 2a and 2b are the intercepts of the ellipse with thex and y axes, respectively. The area of the ellipse is abpwhile a b> .For fitting object A to the ellipse, A is translated androtated so that (x, y) coincides with the origin and the lineof best fit to A coincides with the x-axis of the coordinatesystem. Now according to assumption (iii), one has S ab= p                                            (5)Using  Eqn (5) in Eqn (4), one gets the quadraticequation in b2 given by2 2 4 2 2 2 2 0x b S b S yp - + =                               (6)For each pel on the border of A one gets a pair ofsolutions of b2 from the solutions obtained from all borderpels and use it as an estimate of b2 for the fitting ellipse.From this estimate, the major axis length a can be foundusing Eqn (5). However, the solutions should be examinedbefore these are accounted for averaging. Several situationsmay occur in solving Eqn (6) as below.(a) The solution may contain imaginary parts, which happenswhen ( )22 2  S xy< p . It signifies that an ellipse of areaS and the specified major and minor axis cannot passthrough the current border pel P. One would like tofind the ellipse that is nearest to the current borderpel and yet satisfies the constraints. In particular onelikes to find P¢  on the line OP so that the ellipsepasses through P¢  and P P¢  is smallest (Fig. 1). Let theslope of OP be m y x= .If one wants to find ( , )P x y¢ ¢ ¢  so that y x y x¢ ¢=   and( )2 22 x y S¢ ¢p = , i.e,1/ 2222 24Sxmæ ö¢ = ç ÷pŁ łThen one has 222 22Sbx=¢pwhile a is found using Eqn. (5).Figure 1. The case with complex solution to Eqn (6).MAJOR AXISP (x , y )P   (x  , y  )712DEF SCI J, VOL. 58, NO. 6, NOVEMBER 2008If one has a b< , then this pair of solution is notacceptable. One wants to find an ellipse so that a b‡and yet it is nearest to ( , )P x y .Actually under these constraints the ellipse is a circlewith 2 2a b S= = p . In brief, therefore, these circumstancesmay be taken care of using the following steps.• Check if the current border is such that ( )22 2  S xy< p .• If yes, compute1/ 22 22 22 2 2 2;4 2S Sx bm xæ ö¢ = =ç ÷ ¢p pŁ ł  and ( )22 a S b= pIf 2 2a b‡ , then accept the solutions. Otherwiseaccept 2 2a b S= = p .(b) The solutions may be real. For the two solutions of2b , say 21b  and 22b  one gets two 2a , say 21a  and 22ausing Eqn. (5). Since one should get 2 2b a£ , onedisregards the solution, if any, for which 2 2b a> . Ifhowever, both solutions satisfy 2 2b a£  one disregardsthe one that corresponds to 2 21 2max  { , }a a . (In thiscase, one of the solutions tries to make major axislength unnecessary lengthy to meet the area constraintof assumption (iii). Such situation may occur near toor on the major axis. Then, one shall get one of theib 's equal to zero and its corresponding ia  equal toinfinity, which must be disregarded).When a and b are found from the averaged resultswith due considerations of (a) and (b), the fitting ellipsecan be drawn. Choice of border pels should be done carefully.Equispaced points on the border should not be taken. Thisis because, if A has a thorn-like protrusion or intrusion,then points on those portions will be more heavily weighed.A better approach is to use equal area sweeping points.If the area swept out by a line from the origin as it movesfrom a border point j to j+1 is equal to that while movingfrom point j+1 to j+2, then one says that j,  j+1 and j+2are equal area sweeping points. If one want to considern border points, then the area swept between j and j+1should be np  ab.2.2 Circular fitThe circular fit to A can be obtained at once if oneassumes that the area of A is equal to the fitting circle areapr2, where r is the radius of the circle. Then the circle canbe drawn with ( , )x y  as the centre and radius r S= p .The estimate of r obtained by this method is less sensitiveto border noise and it is computationally attractive. Also,this method avoids the problem of choosing the appropriateborder points encountered in other methods8,16. A comparativestudy of this method with that of Haralick's method8 isgiven.3. RESULTS AND DISCUSSIONTo test the effectiveness of the proposed techniques,at first, a set of 2-D object borders was taken to test forthe circular fit. Three types of circular fit were considered.The first type is the proposed method where the centroidof all pels within the object is taken as the centre of circlewhile its area is equal to the area of the fitting circle.• In the second type, the centre was chosen as abovebut the radius of the circle is taken equal to the averageof radius vector lengths from this centre to the borderof the object.• In the third type, the centre was found as the centroidof the border of the object while the radius is foundby second method.• The error of fit is measured by two approaches: (a)the area of mismatch, and (b) the absolute sum ofdifference between radius vector and radius of allborder points.For both kinds of error of fit, it was found that thethird type of fit is always inferior to the first and secondtypes. Of the first and second types, the differences inerror is very small and for one figure the first type of fitis slightly better while for the other figure, it is slightlyinferior. A typical result with two circles is shown in Fig. 2and is presented in Table 1. From these results we can drawthe conclusions that the first and second type of fit yieldidentical results. Also, the centroid of all of the pels inthe object is a stable and better measure for the centreof the fitting circle.The last observation appears to be true in case ofelliptic fit as well. When the centroid of all pels in theobject is used for the centre of the fitting ellipse, the errorsof fit of both types are less than the errors if the centroidof border is the centre of the ellipse.  Figure 2. Two objects and their circular fits. Figure 3. Two objects and their elliptic fits.713CHAUDHURI: OBJECT AREA-BASED METHOD FOR ELLIPTIC AND CIRCULAR FIT OF A TWO-TONE IMAGETwo typical object borders with their elliptic fits accordingto the present approach are shown in Fig. 3. Also, in Fig.4, an ideal ellipse is computed by noise and then the presentmethod is applied to find the fitting ellipse. The computationalefforts necessary is of the order of N for the major andminor axes direction where N is the number of pels in theobject. To find major and minor axes lengths, an order onn real computation is necessary, where n is the numberof chosen border points.The degree of circularity and ellipticity may be modelledunder fuzzy set theoretic framework17. Considering errorof fit of type (a), let S and Se   be the area of the objectA and its area of mismatch with the fitting circle or ellipse.Then, the fuzzy degree of circularity or ellipticity of A maybe represented as( ) 12eXSASbØ øm = -Œ œº ß                                    (7)where X denotes circularity or ellipticity and b is anexponential fuzzifier to control the shape of ( )X Am . Notethat, ( ) 1X Am = when  0eS = and ( ) 0X Am = when 2eS S= .4. CONCLUSIONAn approach of fitting closed conic sections in 2-Dis proposed. The approach is based on converting theproblem into quadratic equation of one variable by someconstraints. The centroid of the object is constrained tobe the centre of the fitting conic section, while the axesof the fitting conic are constrained to be the lines of bestfit to the object. For elliptic fit, the area of the fitting ellipseis constrained to be equal to the area of the object. TheError for Fig. No. Type of error of fit First type fit Second type fit Third type fit Approach (a) error of fit 255 259 285 (i) Approach (b) error of fit 67.3 69.1 75.06 Approach (a) error of fit 124 120 144 (ii) Approach (b) error of fit 68.9 37.8 43.95 Table 1. Comparative study between the proposed method and the Haralick’s methodapproach appears to be less sensitive to the object bordernoise and is computationally attractive. Experimental resultspresent effectiveness of the approach. A measure of degreeof circularity, ellipticity, etc in fuzzy set theoretic frameworkis also proposed.REFERENCES1. Nakagawa, Y. & Rosenfeld, A. A note on polygonaland elliptical approximation of mechanical parts. PatternRecognition, 1979, 11,133-42.2. Sychra, J.J.; Bartels, P.H.; Taylor, J.; Bibbo, M.& Weld,G.L. Cytoplasmic and nuclear shape analysis forcomputerized cell recognition. Acta Cytologica, 1976,20, 68-78.3. Paton, K.A. Conic sections in chromosome analysis.Pattern Recognition, 1970, 2, 39-51.4. Belson, M.; Dudley, A.W. & Ledley, R.S. Automaticcomputer measurements of neurons. Pattern Recognition,1968, 1, 119-28.5. Zhu, C. & Wang, R. A fast automatic extraction algorithmof elliptic object groups from remote sensing images.Pattern Recognition Letters, 2004, 25(13),1471-478.6. Zhang, S.C. & Liu, Z.Q. A robust real-time ellipse detector.Pattern Recognition, 2005,38(2), 273-87.7. Chaudhuri, D. & Samal, A. A simple method for fittingof bounding rectangle to closed regions. PatternRecognition, 2007, 40, 1981-989.8. Haralick, R.M. A measure of circularity of digital figures.IEEE Trans. System, Man and Cybernetics, 1974, SMC-4, 1974, 394-96.9. Ellis, T.J.; Abbood, A.& Brillault, B. Ellipse detectionand matching with uncertainty. Image Vision Comput.,1992,10, 271-76.10. Rosin, P.L. A note on the least squares fitting of ellipse.Pattern Recognition Lett.,  1993, 14, 799-08.11. Safaee-Rad, R.; Tchoukanov, I.; Benhabib, B.; & Smith,K. C.; Accurate parameter estimation of quadratic curvesfrom gray-level images. CVGIP: Image Understanding,1991, 54, 259-74.12. Wu, W. & Wang, M. Elliptic object detection by usingits geometric properties. Pattern Recognition,  1993,26, 1499-509.13. Proffit, D.; The measurement of circularity and ellipticityon a digital grid. Pattern Recognition,  1982, 15, 383-87.14. Proffit, D. Normalisation of discrete planer objects. Figure 4. An ellipse with distorted border and its fitting ellipseby proposed method.714DEF SCI J, VOL. 58, NO. 6, NOVEMBER 2008Pattern Recognition, 1982, 15, 137-45.15. Chaudhuri, D. Elliptic and circular fit of a two-tineimage using optimal trail and error method and their3-D extensions. In International Conference on Opticsand Optoelectronics Theory, Devices, and Applications,(ICOL), Narosa Publishing House,  1998,  2,pp. 817-22.16. Albano, A. Representation of digitised contours interms of conic arcs and straight-line segments. Comp.Graphics Image Process., 1974, 3, 23-33.17. Zadeh, L.A. Fuzzy sets, Information and Control, 1965,8, 338-53.ContributorsDr D. Chaudhuri  obtained his MSc inApplied Mathematics from JadavpurUniversity, Kolkata. He received his PhDin image processing and pattern recognitionfrom Indian Statistical Institute, Kolkata.He has worked at Indian Statistical Institute;Institute of Armament Technology, Pune;and Integrated Test Range, Chandipur.He was a visiting Assistant Professor atUniversity of Nebraska, USA in the Deptof Computer Science and Engineering for 2003-2004. Presently,he is working as a Scientist at Defence Electronics ApplicationsLaboratory, Dehradun. He has more than 12 years experiencein the field of image processing, pattern recognition, computervision and remote sensing with extensive systems developmentandtheir  implementation. He has extensive experience in thefield of automatic target detection from satellite imagery.  Hehas published more than 28 papers in reputed internationaljournals.

Object Area-based Method for Elliptic and CircularFit of a Two-tone Image

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Object Area-based Method for Elliptic and CircularFit of a Two-tone Image

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