56 M. HAGARA, P. KULLA, EDGE DETECTION WITH SUB-PIXEL ACCURACY BASED ON APPROXIMATION OF EDGE Edge Detection with Sub-ixel Accuacy Based on Aoximation of Edge with Ef Function Mioslav HAGARA, Pete KULLA Det. of Radio and Electonics, Slovak Univesity of Technology, Ilkovičova, 8 9 Batislava, Slovakia mioslav.hagaa@stuba.sk, ete.kulla@stuba.sk Abstact. Edge detection is an often used ocedue in digital image ocessing. Fo some actical alications it is desiable to detect edges with sub-ixel accuacy. In this ae we esent edge detection method fo -D images based on aoximation of eal image function with Ef function. This method is veified by simulations and exeiments fo vaious numbes of samles of simulated and eal images. Results of simulations and exeiments ae also used to comae oosed edge detection scheme with two often used moment-based edge detectos with sub-ixel ecision. Keywods Edge detection, sub-ixel accuacy, image ocessing.. Intoduction Physical contou is one of the most imotant oeties of an object. In ode to extact the contou of an object, we must detect the edges foming that object. So a lot of methods wee develoed to detect edges. One can use simle edge detectos like Sobel, Kisch, Pewit [], moe sohisticated Canny edge detecto [] o obust mohological edge detectos [], [4]. All these methods efom edge detection with a ixel accuacy. Sometimes it is useful to detect edges with sub-ixel ecision. Most edge detectos at sub-ixel level fall in thee gous: fitting, moment-based and inteolationbased methods. The methods of the fist gou use continuous functions, such as hyebolic tangent [5] o B- sline [6], to fit samles of image function. Then sub-ixel edge location is found as inflection oint of continuous function. Anothe fitting methods use a local enegy function [7] o wavelets [] to detemine the edge aametes. Inteolation-based methods achieve the sub-ixel accuacy by inteolating the image data to obtain a fine gid of ixels. Then usual edge detectos, such as Canny [8] o LoG oeato [9], ae alied to esized image. Moment-based methods aly statistical moments to detemine unknown edge model aametes. One can use gay level moments [0, ], satial moments [, ], Fouie-Mellin moments [4] o Zenike moments [5, 6]. Some industial alications, e.g. measuement of the objects with high ecision, need to detect edges with subixel accuacy in -D images. Fo such a task we intoduce in this ae sub-ixel edge detection method based on aoximation of eal image function with Ef function. We comae the oosed algoithm with two often used sub-ixel edge detectos: gay level moment (GLM) edge oeato [0] and satial moment (SM) edge detecto []. This ae is oganized as follows. Section includes edge models. In section we descibe moment-based edge detection methods which we use fo comaison with ou oosed algoithm. This algoithm is intoduced in section 4 esults. Section 5 and section 6 include simulations and exeiments. In section 7 conclusions ae made.. Edge Models Fo analysis of oosed edge detectos and thei veification by simulations thee basic models ae mostly used. Ste edge (Fig. ) is a simlest model and is eesented by ste function []: h, x l f ( x) () h k, x l The model is chaacteized by thee aametes: backgound intensity h, edge contast k, and edge location l. Fig.. Ste edge. In eal images the bightness changes gadually and am edge (Fig. ) is moe suitable [7]:
RADIOENGINEERING, VOL. 0, NO., JUNE 0 57 h, x l k f ( x) h x l, l x l () l l h k, l x Ram edge has fou aametes: backgound intensity h, edge contast k, edge beginning l and edge end l. Location of the edge l is equal to the aithmetic aveage of l and l. Fig.. Ram edge. The thid model is closest to eal edge because it esects defocusing, o bluing due to the effects of the oint sead function of the otic system. This model (Fig. ) is called blued edge and is eesented by function [7]: k x l f ( x) ef h whee ef(x) is defined as [8]: ef ( x) x t e 0 dt (). (4) This model has fou aametes: backgound intensity h, edge contast k, edge location l and edge bluing σ. Fig.. Blued edge.. Moment-Based Edge Oeatos Tabatabai and Mitchel oosed gay level moment (GLM) edge oeato fo -D image [0] based on the fist thee moments m, m, m of the inut data sequence: mi n n j x i j... i,, (5) whee x, x,... x n ae image samles. Let suose that they ae the samles of ideal ste edge (Fig. ) and h is a numbe of samles with gay level h (they ae the ixels on the left of the edge). If we define the densities and as: h, (6) n n h, (7) n then solution of thee equations: m h h k, (8) h h h h m k, (9) m, (0) k with thee unknown vaiables h, k, esults in: s, () 4 s whee s h m, () k () m m mm, (4) m m. (5) In the case of eal image, h =n. is not intege and eesents sub-ixel edge location. Anothe sub-ixel edge detecto [] is based on satial moments (SM) of continuous function f(x) of ode, which ae defined: M x f ( x) dx. (6) Let function f(x) eesents ste edge (Fig. 4) and x is fom - to + (to simlify calculations). Then equation (6) fo =0, and can be witten as: M M M 0 h dx k l h x dx k x dx k l dx h k l, (7) l h x dx k x dx h k l, (8) l. (9)
58 M. HAGARA, P. KULLA, EDGE DETECTION WITH SUB-PIXEL ACCURACY BASED ON APPROXIMATION OF EDGE The solution of these equations esults in fomulas fo edge location l: l M M 0, (0) M edge contast k M k, () ( l ) and backgound intensity h h M k l. () 0 Fig. 4. Edge model fo satial moment edge detecto. 4. Edge Detecto Based on Aoximation with Ef Function (AEF) We oose the sub-ixel edge detecto based on aoximation of eal image function f (i) with function f a (i), which is equal to blued edge model () and has fou aametes - h, k, l and σ. The coe of this AEF edge detecto is aametic fitting by minimizing a diffeence between the eal image function f (i) and function f a (i). This diffeence is defined: N i f ( i) f ( i) E( h, k, l, ) () a whee N is a numbe of samles. Minimizing the diffeence E(h,k,l,σ) gives subixel edge location l. Edge detection algoithm based on aoximation consists of thee stes: Edge detection with ixel accuacy. Initial values estimation of aametes h, k, l and σ. Paametic fitting by minimizing diffeence function E(h,k,l,σ). The fist ste can be done by any edge detection method with ixel accuacy. We use the simlest way, we find maximum of discete deivative of function f (i): df ( i) f ( i ) f ( i) (4) and we denote i max fo which df (i) eaches its maximum as the edge osition. Initial value of edge location we set to l 0 = i max. To estimate initial value of σ we aly deivation of continuous function f a (x) [8]: k x l f a ( x) ex. (5) Fo x max =l, x d =(l-σ) and x u =(+σ) we can deive: fa ( xd ) fa ( xu ) ex 0.5. (6) f ( x ) f ( x ) ex(0) a max a max In the case of f (i) if we find i d < i max and i u > i max which satisfy: and df ( id ) 0.5 (7) df ( i ) max df ( iu ) 0.5 (8) df ( i ) max and ae as close as ossible to i max, we can estimate σ 0 = 0.5(i u - i d ) and also h 0 = f (i d ) and k 0 = f (i u )- f (i d ). Fo aametic fitting by minimizing diffeence function E(h,k,l,σ) we aly Matlab function fminseach which uses the simlex seach method [9]. 5. Simulations We did all simulations in ogam Matlab (vesion 7..0.67). Let thee is -D image senso which consists of elements with width w and ga g between two senso elements (Fig. 5b). Let the bightness aound the edge is constant in time and vaies only in the diection x accoding to () (Fig. 5a). Then simulated noiseless outut signal f s (i) of the i-th senso element (Fig. 5c) can be calculated: f s ( i) ct iw / a iw / k x l ef h dx (9) whee c is senso integal sensitivity and T a is accumulation time. Fo simulations we can set ct a =. Because the ga between two senso elements is vey small we can set w=. Then noiseless outut signal f s (i) of the i-th senso element is: f s ( i) i0.5 i0.5 k x l ef h dx. (0) Fo all simulations in this section the backgound intensity used in (0) is h=0. and edge contast is k=. The fist simulations of noiseless signal insect how the edge location eo deends on actual osition of the edge fo diffeent values of bluing aamete σ = 0.5,,
RADIOENGINEERING, VOL. 0, NO., JUNE 0 59 and 5. Numbe of samles of outut signal f s (i) is N = 4. The esults of these simulations ae esented in Tab., and gahically ae inteeted in Fig. 6. slightly unfocused images (σ = ) the theoetical accuacy of GLM and SM is etty good (bette then 0.0), fo stongly unfocused images (σ = 5) come close to 0.07. AEF method has (excet fo σ = 0.5) zeo location eo. It's undestandable, because the function unde which an outut signal is geneated fo simulation is equal to the function used fo aoximation. Simulations with noisy signal and exeiments with eal images hel to detemine eal oeties of AEF. Fig. 5. a) Bightness aound the edge, b) elements of image senso, c) simulated noiseless outut signal. σ = 0.5 σ = l AEF GLM SM AEF GLM SM -0,5 0 0,000 0,00 0 0,0007 0,008-0,4 0,0007-0,0096 0,0008 0 0,0005 0,00-0, 0,00-0,056 0,0006 0 0,0004 0,007-0, 0,00-0,057 0,0004 0 0,000 0,00-0, 0,0009-0,0097 0,000 0,000 0,000 0,0006 0 0 0 0 0 0 0 0, -0,0009 0,0097-0,000-0,000-0,000-0,0006 0, -0,00 0,057-0,0004 0-0,000-0,00 0, -0,00 0,056-0,0006 0-0,0004-0,007 0,4-0,0007 0,0096-0,0008 0-0,0005-0,00 0,5 0-0,000-0,00 0-0,0007-0,008 σ = σ = 5 l AEF GLM SM AEF GLM SM -0,5 0 0,008 0,0 0 0,04 0,067-0,4 0 0,00 0,008 0 0,094 0,0509-0, 0 0,007 0,006 0 0,046 0,08-0, 0 0,00 0,004 0 0,0097 0,055-0, 0 0,0006 0,00 0 0,0049 0,07 0 0 0 0 0 0 0 0, 0-0,0006-0,00 0-0,0049-0,07 0, 0-0,00-0,004 0-0,0097-0,055 0, 0-0,007-0,006 0-0,046-0,08 0,4 0-0,00-0,008 0-0,094-0,0509 0,5 0-0,008-0,0 0-0,04-0,067 Tab.. Edge location eo of simulated noiseless signal. The edge with bluing σ = 0.5 is vey close to ste function and such a case does not occu in eal images. Real values of bluing in well-focused images ae aoximately equal to σ =. One can see fom simulation esults that fo this value the ecision of all thee methods esented in this ae is theoetically vey high. Fo Fig. 6. Edge location eo of simulated noiseless signal fo bluing aamete: a) σ =0.5, b) σ =, c) σ = and d) σ =5. To add noise to signal defined in (0) we aly Matlab function andn, which etuns a seudoandom, scala
50 M. HAGARA, P. KULLA, EDGE DETECTION WITH SUB-PIXEL ACCURACY BASED ON APPROXIMATION OF EDGE value dawn fom a nomal distibution with mean 0 and standad deviation. Values of function andn we multily with 0.008, so we get signal-to-noise atio 7dB. We calculated noisy signal hunded times and fo each noisy signal we used successively 4, and samles aound the edge to find sub-ixel edge location. Then we calculated edge location eo fo all hunded ealizations. Fo these hunded eos we detemined the standad deviation and ue q u and lowe q d 5% quantiles. Ue 5% quantile q u means, that thee is 5% obability that the eo will be bigge than q u. Lowe 5% quantile q d means, that thee is 5% obability that the eo will be smalle than q d. We did calculations mentioned above fo diffeent values of bluing aamete σ = 0.,, and 5. The esults ae esented in Tab. and Tab., some of them ae gahically inteeted in Fig. 7 and Fig. 8. σ = 0.5 σ = N AEF GLM SM AEF GLM SM 4 0,0 0,07 0,044 0,06 0,067 0,0454 0,0 0,08 0,07 0,06 0,069 0,05 0,09 0,0 0,06 0,087 0,087 0,04 σ = σ = 5 N AEF GLM SM AEF GLM SM 4 0,06 0,044 0,046 0,048 0,046 0,048 0,05 0,06 0,05 0,074 0,055 0,04 0,07 0,08 0,078 0,77 0,07 0,0447 has bette ecision fo smalle numbe of samles. Howeve, fo sufficient numbe of samles AEF and GLM ae moe ecise than SM. Fig. 7. Standad deviation of edge location eo of simulated noisy signal. Tab.. Standad deviation of edge location eo of simulated noisy signal. σ = 0.5 AEF GLM SM N q u q d q u q d q u q d 4 0,0-0,007 0,04-0,09 0,09-0,09 0,05-0,07 0,05-0,04 0,065-0,069 0,05-0,04 0,067-0,049 0,047-0,047 σ = AEF GLM SM N q u q d q u q d q u q d 4 0,04-0,06 0,05-0,0 0,098-0,0867 0,04-0,07 0,04-0,09 0,0644-0,0695 0,06-0,054 0,07-0,065 0,0445-0,0478 σ = AEF GLM SM N q u q d q u q d q u q d 4 0,045-0,046 0,047-0,047 0,099-0,085 0,05-0,0545 0,0509-0,058 0,0649-0,0675 0,07-0,077 0,066-0,06 0,0544-0,058 σ = 5 AEF GLM SM N q u q d q u q d q u q d 4 0,088-0,0858 0,0876-0,086 0, -0,097 0,4-0,49 0,06-0,09 0,08-0,0876 0,507-0,509 0,69-0,484 0,0865-0,09 Tab.. Ue q u and lowe q d 5% quantiles of edge location eo of simulated noisy signal. Fom the esented esults it is clea that fo small bluing aametes (σ = 0.5 and ) the numbe of samles used fo calculations is not significant. This is because visual function in this case vaies only in the vicinity of edge. Fo sufficient numbe of samles the ecision of AEF and GLM methods deends only on bluing aamete (see Fig. 7 and Fig. 8). By contast, SM method Fig. 8. Ue q u and lowe q d 5% quantiles of edge location eo of simulated noisy signal. 6. Exeiments Fig. 9. Souce fo test images (ca engine valve). Fo exeimental veification we shot few images of ca engine valve (Fig. 9). As a backgound we used PC monito so we got good edge contast. Monito was not immediately behind the valve to get backgound without textue (due to un-focusing). We used camea Nikon D00 with mega-ixel esolution. We shot some images with automatic focusing and some with manual focusing so we got images with diffeent values of bluing aamete. Uncomessed images wee conveted fom oiginal NEF
RADIOENGINEERING, VOL. 0, NO., JUNE 0 5 fomat to 4 bits colo TIFF fomat. Fo comutations we used detail of 50x80 ixels aound the edge (Fig. 0). We woked only with geen channel of image. a) b) c) Fig. 0. Test images: a) valve.tiff, b) valve.tiff and c) valve.tiff. We alied AEF, GLM and SM edge detection methods to one hunded adjacent ows to comute the edge location with the sub-ixel accuacy. Since the valve must be efectly staight, comuted sub-ixel edge ositions should ceate a staight line, which can be eesented as olynomial (x) = a x + a 0. We used the Matlab function olyfit to find the coefficients a and a 0 of a olynomial (x) that fits the data. Diffeence between the comuted edge osition and the value of the olynomial can be consideed to be the edge location eo. In such a way we comuted the edge location eos fo all ocessed ows. Standad deviation and 5% quantiles wee also comuted. Fig.. Row n.6 of test image valve.tiff. To illustate examles, thee ae selected ows of test images valve.tiff, valve.tiff and valve.tiff in Fig., Fig. and Fig.. Comuted edge locations and aoximating staight lines ae in Fig. 4, Fig. 5 and Fig. 6. Fo each ow of the test image the edge osition was calculated using a diffeent numbe of ixels aound the edge N = 4, and. Calculated olynomial coefficients a and a 0, standad deviation, 5% ue q u and lowe q d quantiles of edge location eo of test images ae esented in Tab. 4. Fig.. Row n.6 of test image valve.tiff. Fig.. Row n.6 of test image valve.tiff. Fig. 4. Edge locations and aoximating staight line of test image valve.tiff : a) AEF, b) GLM, c) SM.
5 M. HAGARA, P. KULLA, EDGE DETECTION WITH SUB-PIXEL ACCURACY BASED ON APPROXIMATION OF EDGE Fig. 6. Edge locations and aoximating staight line of test image valve.tiff : a) AEF, b) GLM, c) SM. Fig. 5. Edge locations and aoximating staight line of test image valve.tiff : a) AEF, b) GLM, c) SM. valve.tiff (σ = 0,85) N a a 0 std q u q d AEF 0,0084 9,84 0,0 0,048-0,048 4 GLM 0,008 9,84 0,09 0,055-0,05 SM 0,0084 9,7 0,058 0,088-0,099 AEF 0,0084 9,85 0,0 0,048-0,04 GLM 0,008 9,85 0,09 0,057-0,048 SM 0,0085 9,76 0,05 0,086-0,095 AEF 0,0084 9,86 0,04 0,044-0,04 GLM 0,008 9,87 0,04 0,056-0,057 SM 0,0080 9,8 0,05 0,074-0,085 valve.tiff (σ =,6) N a a 0 std q u q d AEF 0,0058 4, 0,06 0,04-0,04 4 GLM 0,0058 4, 0,07 0,7-0,4 SM 0,0059 4, 0,07 0,07-0,6 AEF 0,0057 4,8 0,068 0, -0,06 GLM 0,0058 4, 0,078 0,5-0,40 SM 0,006 4, 0,088 0,8-0,60 AEF 0,0059 4,87 0,09 0,45-0,65 GLM 0,006 4,78 0, 0,8-0,7 SM 0,0064 4,7 0,79 0,8-0,85 valve.tiff (σ = 4,50) N a a 0 std q u q d AEF 0,0080 44,97 0,088 0,8-0,57 4 GLM 0,0074 44,90 0,04 0, -0,8 SM 0,006 45,4 0,65 0,4-0,6 AEF 0,007 44,7 0,48 0,4-0,79 GLM 0,005 44, 0,5 0,4-0,47 SM 0,007 44, 0,59 0,775-0,605 AEF 0,0050 4,54 0,45 0,4-0, GLM 0,004 4,77 0,497,4-0,746 SM 0,000 4,8 0,6,460-0,97 Tab. 4. Polynomial coefficients a and a 0, standad deviation, 5% ue q u and lowe q d quantiles of edge location eo of test images. Fig. 7. Deendence of accuacy (ue and lowe quantiles) on the bluing aamete σ (fo AEF with N = 4).
RADIOENGINEERING, VOL. 0, NO., JUNE 0 5 In this ae, sub-ixel edge detection method based on aoximation of eal image function with Ef function (AEF) is esented. This method is veified though simulations and exeiments and comaed with two othe methods: GLM and SM. Although these methods ae not ecent, we chose them fo comaison because they ae designed imaily fo -D images. Newe methods that ae mentioned in the intoduction ae geaed towads the -D images and fo -D images do not achieve such accuacy. Thee ae also few methods secifically designed fo -D images [0], [] but they ae hadwae oiented in ode to achieve geate seed and thei accuacy is not vey high. The esults obtained fom exeiments ae faily well consistent with the esults of simulation fo small values of the bluing aamete (σ ), which coesond to the well-focused images. Achieved accuacy of edge location fo eal images is about ±0.05 of ixel. Fo slightly (σ ) and stongly (σ 5) unfocused eal images the exeimental esults ae moe diffeent fom the esults of simulation. This may be because the bightness aound the edge does not fit exactly the used Ef function. It looks like the image on the ight of the edge has diffeent aamete σ as the image on the left, obably due to diffeent distances of backgound and foegound fom the camea. But it is tyical fo the eal situations. Howeve, the accuacy of AEF method is about 5 0 % bette than GLM and much bette than SM. We also found that detection accuacy can be affected by the insufficient numbe of ixels used fo comutation and we ecommend woking with the numbe of ixels that is 8 0 bigge then bluing aamete σ. How to estimate this aamete one can find in section 4. Fo some industial alications such as contactless measuement of objects to imove the esolution of system can be desiable. As an examle we can give magnesite bicks, which ae used in blast funace as lining. These bicks ae manufactued by comession and tansvese dimension has to be checked with geat accuacy. Imovement of installed hadwae is usually limited by the cost and ealization. In the case, that comutational cost is not a citical attibute, the esented method fo sub-ixel edge detection can constitute an aoiate solution to this oblem. Fig. 8. Deendence of accuacy (ue quantile) on the numbe of ixels used fo comutation (fo AEF). Fom the obtained esults one can see that the accuacy of all esented methods (AEF,GLM,SM) get wose with inceasing bluing aamete σ (Fig. 6). Also, it can be concluded that fo small values of σ the numbe of ixels used fo comutation is not imotant (Fig. 7). This is not tue fo lage values of σ and accoding to the exeiments we can say that equied numbe of ixels is aoximately equal to 8 0 σ. 7. Conclusion Refeences [] PRATT, W. K. Digital Image Pocessing. 4 th ed. Hoboken: John Wiley & Sons, 007. [] CANNY, J. A comutational aoach to edge detection. 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