A machine vision approach for detecting and inspecting circular parts

Transcription

1 A machne vson approach for detectng and nspectng crcular parts Du-Mng Tsa Machne Vson Lab. Department of Industral Engneerng and Management Yuan-Ze Unversty, Chung-L, Tawan, R.O.C. E-mal: (Fgures are not avalable n ths report) Abstract In ths paper, we present a machne vson approach for detectng and nspectng crcular parts and the parts wth crcular arcs on the contours. The method uses the Hough transform technque and utlzes the drectonal nformaton of a normal to the crcle at each boundary pont. A cubc polynomal curve fttng s used to estmate the normal and determne the concavty of the ftted curve at each gven boundary pont. The proposed Hough transform method s a two-stage procedure. The frst stage of the procedure uses a 2-D accumulator array to detect crcle centers. Then the second stage uses a 1-D accumulator array to detect the rad of crcles. The proposed method s robust to detect crcular parts wth partal occluson such as perpheral defects or burrs. For an mage of sze N N, the storage requrements are N 2 and the tme complety s bounded by (N+m)n, where m s the number of crcle centers detected n the frst stage and n s the number of boundary ponts n the mage. Keywords: Machne vson, Crcle detecton, Part nspecton, Hough transform

2 1. INTRODUCTION Detectng and locatng crcular parts from a dgtal mage s mportant n ndustral applcatons such as automatc nspecton and robotc assembly. The radus and the center of a crcle generally can be estmated by two dstnct approaches: the least-squares curve fttng and the Hough transform. Gven a set of coordnates whch represent the boundary of a part and presumably belong to a crcular arc, the parameters of the crcle can be estmated by mnmzng the least mean square errors between the gven boundary ponts and the curve [1, 2, 3]. For a comple part nvolvng crcular arcs and other non-crcular segments on the contour, the boundary ponts must be grouped nto meanngful crcular segments before the use of least-squares fttng methods. In addton, ths approach s very senstve to nose and occlusons of parts. The Hough transform (HT) technques [4] for analytc shapes use a constrant equaton to transform a set of feature ponts n mage space nto a set of accumulated votes n a parameter space. For each feature pont, votes are accumulated n an accumulator array for all parameter combnatons that satsfy the constrant equaton. At the end of votng, those array elements contanng large numbers of votes ndcate the presence of the shape wth the correspondng parameters. The HT converts a dffcult global detecton problem n mage space nto a more easly solved local peak detecton problem n a parameter space. The prmary benefts of usng the HT s ts robustness for nosy mages and occluded parts. Snce crcles are completely defned by three parameters, namely the radus r and the coordnates of the center ( ), the conventonal HT requres a three-dmensonal array to detect crcles n an mage scene. To search for a crcle whose center s wthn an mage of sze N N, and whose radus n not larger than N wth 1 pel

3 resoluton, would requre a three-dmensonal accumulator of sze N 3. Ths ncurs consderable storage and computaton. The use of the HT to detect crcles was frst ntroduced by Duda and Hart [5]. An edge detecton process s carred out to dentfy sgnfcant edge ponts. Then the postons of all possble center locatons are accumulated n a parameter space for all antcpated rad. Conker [6] used the gradent orentatons of two neghborng edge ponts to estmate the center of a crcle. Snce the center of a crcle must le along the gradent drecton of each edge pont on the crcle, the ntersecton pont of gradents s dentfed as the center. Ths approach can not dscrmnate concentrc crcles of dfferent rad and s dffcult to estmate the crcle center to the requred accuracy. Yp et al. [7] used parallel edge ponts to estmate the parameters of a crcle and an ellpse. Ths approach only requres a two-dmensonal accumulator, but wll fal to detect the occluded crcles on whch pars of parallel edge ponts are not presented smultaneously. In ths paper, we develop a fast two-stage Hough transform algorthm for detectng and nspectng partally occluded crcular parts. The occluson of parts may result from the perpheral breakdown, defects or burrs. The crcular portons of parts such as cams also can be detected and measured usng such algorthm. The proposed method conssts of a two-dmensonal accumulator to fnd crcle centers followed by a one-dmensonal accumulator to determne crcle rad. The frst stage begns wth a boundary followng process to fnd the sequence of boundary ponts of each part n the mage. A cubc polynomal curve fttng method s employed to estmate the normal and the concavty of the ftted curve at each boundary pont. Based on the normal drecton and concavty nformaton, the lne segment that the crcle center may le on s determned. The votes are collected n a parameter plane based on the coordnates of each pont on the resultng lne segment. At the end of the frst-stage votng process, those array elements contanng large numbers of votes ndcate the presence of crcle centers. The second stage of the

4 votng process calculates the radal dstances between each center detected n the frst stage and all boundary ponts n the mage. The local peaks n the 1-D accumulator then ndcate the presence of crcle rad for a gven crcle center. Ths paper s organzed as follows: In secton 2, the cubc polynomal curve fttng method for estmatng the normal and the arc concavty at a boundary pont s frst ntroduced. Then the detecton of crcle centers followed by the detecton of crcle rad s dscussed. In secton 3, epermental results are presented. The concluson s gven n secton THE HT WITH NORMAL DIRECTIONS Let the normal to a curve at pont p be the lne passng through p and perpendcular to the tangent there. Snce the center of a crcle must le along the normal drecton of each boundary pont on the crcle, the common ntersecton pont of these normals ndcate the center of the crcle. In ths paper, we estmate best, n the least-squares sense, the slope of a normal by fttng a small segment of a dgtal curve n an mage to a contnuous polynomal functon. Snce a small segment of crcular arcs shown n the dgtal mage may be represented by a straght lne, especally the one n the horzontal or vertcal drecton, or by a curve nvolvng an nflecton pont, a cubc polynomal functon s employed to appromate the small segment. The mage of scene parts s preprocessed by smple bnary thresholdng and boundary followng [8] n order to etract the boundary ponts. Let the sequence of n dgtal ponts descrbe a boundary P, P = { p = ( ), = 1, 2,..., n } where p +1 s adacent to p, and ( ) s the Cartesan coordnates of p n the mage.

5 Let p = ( ) be the boundary pont at whch the normal s to be estmated. The regon of support that defnes the neghborng ponts of p between ponts p -k and p +k for some nteger k s gven by N(p ) = { p - k + k } Shft all ponts p n N(p ) by (-, -y ) and translate p to the orgn of the new coordnate system. The sze of the regon of support k can be a predefned constant determned emprcally, or t can be selected adaptvely as proposed n [9]. The cubc polynomal functon s represented by y = f() = a + b + c 2 + d 3 where a, b, c and d are the four unknown coeffcents to be estmated. To mnmze the accumulated dstance errors between the observed ponts wthn the regon of support and the curve, the obectve functon s gven by mn F(a, b, c, d) = + k = k 2 3 [y - (a + b + c + d )] 2 Dfferentatng F(a, b, c, d) wth respect to a, b, c and d, and settng them to zero, we obtan where X = A -1 * B (1) X = [ a, b, c, d] T

6 A = B = [ 2 y, y, y, y 3 ] T For a resultng functon curve y = f(), the frst two dervatves of f at pont p wth the new translated coordnates (0, 0) are gven by f '(0) = b f "(0) = 2c f '(0) = b gves the slope of the tangent lne to the ftted curve at p. Thus, the slope of the normal lne at p s -1/b. The normal lne at p wth ts orgnal coordnates ( ) has the followng equaton: y = (-1/b) + (y + / b) (2) For a boundary pont p on the crcle, the center of the crcle must le on the normal gven by eq. (2). We can further determne the nterval of the normal lne that the center may le on by eamnng the concavty of the crcular arc. A curve f s concave upward f every tangent to the curve of f ntersects the curve only at the pont of tangency and otherwse les entrely below the curve of f. A curve f s concave downward f every tangent les above the curve of f. Therefore, a curve f s concave upward f f "() > 0, or f s concave downward f f "() < 0 for all on an open nterval.

7 Let ymn and yma denote the lower bound and the upper bound, respectvely, of the y coordnate of a crcle center. For an mage of sze NN, we may have y mn = 0 and = N 1. Consder a concave upward arc as shown n Fgure 1(a). The crcle center must le along the half-lne of the normal above the arc. Therefore, for a gven pont p = ( ) on the concave upward arc, the possble y-coordnate values of the crcle center must be larger than y,.e., the y coordnate of the crcle center s on the nterval [ y ma y ma ]. Smlarly, consder a concave downward arc as shown n Fgure 1(b). The crcle center must le along the half-lne of the normal below the arc. For a gven pont p = ( ) on the concave downward arc, the possble y-coordnate values of the center must be smaller than y,.e., the y coordnate of the crcle center s on the nterval [ y mn value on a normal lne at pont p = ( ]. For a gven y-coordnate value y, the correspondng -coordnate ) can be computed by rewrtng eq.(2) as follows: Or, = b y + b y + ' = f ( 0) y + f ( 0) y + The votng process for detectng the crcle centers s proceeded as follows: Let P = { p = ( ), = 1, 2,..., n} be the sequence of n boundary ponts of a part n the mage. ' For every boundary pont p = ( ), = 1, 2,..., n Estmate the cubc polynomal functon f() at pont p = ( ) usng eq. (1) For y, y, y = ymn ymn + 1, mn+ 1, mn+ 1 ma ma f f f f f f ( 0) ( 0) ( 0) > 0 < 0 = 0 ' Compute = f ( 0) y + f ( 0) y + '

8 Let A c ( ) = ( ) A c +1 End End Ths concludes the frst stage of the votng process for crcle center detecton. The accumulator elements ( ) A c contanng suffcently large numbers of votes aganst a predefned threshold T d are regstered as the canddates of crcle centers. These resultng centers are the nput to the second stage of the votng process for rad detecton. Let C = { c = (, y ), = 1, 2,..., m } denote a set of m centers of crcular arcs detected n the frst stage of the votng process. The votng process for detectng the rad of crcles s gven as follows: For every crcle center c = ( ) End Clean A R (r) for all r, = 1, 2,..., m For every boundary pont p = ( ), = 1, 2,..., n End Compute r = [( - Let A R (r) = A R (r) + 1 ) 2 + ( y - y ) 2 ] 1/2 If A R (r) s greater than the predefned threshold T d, r s the radus of the crcle centered at c. Ths concludes the second stage of the votng process for rad detecton. Note that a gven center c may have many rad f t s the common center of concentrc crcles.

9 The two-dmensonal accumulator array ( ) A c used n the frst stage can be elmnated as soon as the crcle centers are recorded n the set C. Ths released memory can be used n the second stage for the one-dmensonal accumulator array A R (r). Therefore, the searchng of a crcle whose center s wthn an mage of sze of N N, and whose radus s less than or equal to N (or s specfed wth the precson up to N 2 quantzaton ncrements), only requres a storage space of N 2. The tme complety of the proposed two-stage algorthm n the worst case s gven by N n + m n where N s the mage sze n the y dmenson, n s the number of boundary ponts n the mage, and m s the number of crcle centers detected n the frst stage. Snce the concavty nformaton of crcular arcs s utlzed, the number of possble y-coordnate values s less than N. N n and m n specfy the tme requrements for the frst stage and the second stage of the votng process, respectvely. 3. EXPERIMENTAL RESULTS Fgures 2(a) and 2(b) demonstrate a crcle wth a full boundary (2 ), and a half of the boundary ( ), respectvely. The crcles wth partal dstorton on the crcle boundares are also dsplayed n the fgures. The brght spots n the mages ndcate the common ntersecton ponts of the normal lnes, and are the locatons of crcle centers. The occluson of obects does not affect the sgnfcance of the center locaton. The more boundary ponts avalable on a crcle, the brghter (larger vote counts) the spots and more concentrated the ntersecton ponts to the deal center locatons.

10 In the eperments, the regon of support k = 8 s used for curve fttng. Twenty planar, dark crcular obects wth the rad rangng from 25 to 60 pels are tested usng the proposed algorthm. The resultng mean radus devaton s wthn 1. 2 pels. In order to evaluate the performance of the proposed method, ten synthetc crcles wth two rad and varous arc angles are also analyzed so that the estmated crcle centers and rad can be compared wth the deal parameter values. The ten crcles of study are generated n an mage of sze pels. The dgtal boundares of crcles are created such that ma { -1, y y -1 } = 1, Here ( ) are the coordnates of boundary ponts generated from the equaton of a crcle wth the radus r and the center at (254, 243) appromatng to the center of the mage,. e., ( - 254) 2 + (y - 243) 2 = r 2, Also, ( ) are truncated to ther nearest ntegers to represent the coordnates n mage space. Table 1 shows the performance of the estmated parameters of two synthetc crcles, one wth radus 100 pels and the other one wth radus 50 pels. Each crcle s tested under varous arc angles (arc lengths), rangng from 2 (100%), 3 /2 (75%), (50%), /2 (25%) to /4 (12.5%). Each dmenson of the parameter space s quantzed to resolve 1 pel changes. The errors of the estmated parameters, the and y coordnates of the centers and the rad, are wthn 1 pel for both crcles wth the arc angles larger than or equal to π(50%). All estmated errors of the crcle parameters for the small crcle of radus 50 are wthn 1 pel even though the arc angle s as small as /4 (12.5%). However, the estmated parameter errors are more sgnfcant (up to 5 pels) for the large crcle of radus 100 wth the arc angles smaller than (50%). Ths s due to the mpact of the angular devaton of the estmated normal for the large crcle. The accuracy of the estmated center locaton s determned by the precson of the normal slope derved from the cubc

11 polynomal fttng method. If an estmated normal has angular devaton φ wth respect to the deal normal, then the dsplacement from the deal center to the estmated normal lne s gven by r sn φ, where r s the deal radus (see Fgure 3). Therefore, the estmated parameter errors of the large crcle s larger than those of the small crcle. Epermental results have shown that the mean angular devaton of a normal lne usng the cubc polynomal fttng method s wthn 2 o. The applcatons of the proposed method to the detecton of crcular arcs of ndustral parts are demonstrated n Fgures 4 and 5. Fgures 4(a) and 4(c) show the real mages of two ol seals, one wth perpheral defect, and the other one wth burrs, respectvely. The estmated crcles of these two ol seals are presented n Fgures 4(b) and 4(d). Note that the etrudng portons (perpheral defect and burrs) of the ol seals can be easly detected n the mage snce they are located outsde the estmated crcles. The crosses "+" shown n the fgures mark the locatons of the estmated crcle centers. The estmated crcles concde wth the crcular portons of the boundares of the parts, as seen n Fgures 4(b) and 4(d). Fgure 5(a) llustrates the bnary mage of a plate cam, and Fgure 5(b) shows two detected crcular arcs on the contour of the cam. Fgures 5(c) and 5(e) show the profles of a brake shoe and a frcton plate, respectvely. The estmated crcles for the outermost crcular arcs of these two mechancal parts are llustrated n Fgures 5(d) and 5(f). 4. CONCLUSION In ths paper, we have presented a two-stage Hough transform method for detectng crcular parts wth partal occluson and parts wth crcular arcs on the contours. The resultng crcle of the proposed method can be further etended to detect the perpheral breakdown, defects or burrs of a crcular parts by evaluatng the varaton of radal dstances between the estmated center and boundary ponts.

12 The proposed method utlzes the drectonal nformaton of the normal at each boundary pont on the crcle to ncrease the computatonal effcency and reduce storage requrements. A cubc polynomal curve fttng s employed to estmate the slope angle of a normal and determne the concavty of the ftted curve. The frst stage of the votng process frst determnes the lne segment of the normal at each boundary pont and accumulates the occurrences of all coordnates on the lne segment n a 2-D accumulator. The second stage uses a 1-D accumulator to store the number of ndvdual dstances between each estmated crcle center and all boundary ponts. For an mage of sze N N, ths proposed method only requres an accumulator of sze N 2 and the tme complety s bounded by (N+m)n, where m s the number of crcle centers detected n the frst stage and n s the number of boundary ponts n the mage.

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