Illumination Distribution from Shadows

Transcription

1 Illumiatio Distributio from Shadows Imari Sat0 Yoichi Sat0 Katsushi Ikeuchi Istitute of Idustrial Sciece, The Uiversity of Tokyo Roppogi, Miato-ku, Tokyo , Japa { imarik, ysato, ki} 0iis.u-tokyo.ac.jp Abstract The image irradiace of a three-dimesioal object is kow to be the fuctio of three compoets: the distributio of light sources, the shape, ad rejlectace of a real object surface. I the past, recoverig the shape ad rejlectace of a object surface from the recorded image brightess has bee itesively ivestigated. O the other had, there has bee little progress i recoverig illumiatio from the kowledge of the shape ad rejlectace of a real object. I this papec we propose a ew method for estimatig the illumiatio distributio of a real scee from image brightess observed o a real object surface i that scee. More specijically, we recover the illumiatio distributio of the scee from a radiace distributio iside shadows cast by a object of kow shape oto aother object surface of kow shape ad rejlectace. By usig the occlusio iformatio of the icomig light, we are able to reliably estimate the illumiatio distributio of a real scee, eve i a complex illumiatio eviromet. 1 Itroductio The image irradiace of a three-dimesioal object is kow to be the fuctio of the followig three compoets: the distributio of light sources, the shape, ad reflectace of a real object surface. From the relatioship amog them, three kids of aalyses o the recorded image brightess are derived: recoverig the surface shape from the surface reflectace ad illumiatio of the scee, recoverig the surface reflectace from the surface shape ad illumiatio of the scee, ad recoverig illumiatio from the shape ad the reflectace of the surface. I the past, the first two kids of aalyses, the shape recovery ad the reflectace recovery, have bee itesively studied usig the shape from shadig method [7, 8, 9, 161 as well as through reflectace aalysis research [l, 12, 13, 15, 181. I cotrast, there has bee little progress o the subject of recoverig illumiatio from the kowledge of the shape ad the reflectace of a object surface. This is because real scees usually iclude both direct ad idirect illumiatio distributed i a complex way ad it is difficult to obtai correct illumiatio models to be used for the estimatio. Most of the previously proposed methods related with the first two kids of aalyses aimed to estimate illumiat directio ad color i a very specific illumiatio coditio such as a case where there would be oly oe direct light source i the scee. Accordigly, those methods caot be applied to the images take uder atural illumiatio eviromets. The purpose of this study is to preset our progress i recoverig a illumiatio distributio of a real scee from the kowledge of the shape ad reflectace of a real object. I the proposed method, we use radiace chages iside shadows rather tha appearace chages o the surface due to the 3D geometry of the surface ad the illumiat directio. More specifically, we estimate a illumiatio distributio of the scee by observig a radiace distributio iside shadows cast by a object of kow shape oto aother object surface of kow shape ad reflectace. Shadows i a real scee are caused by the occlusio of icomig lights as illustrated i Figure 1, ad thus shadows cotai various pieces of iformatio about the illumiatio of the scee. Nevertheless, i the past, shadows have bee used for determiig the 3D shapes ad orietatios of a object which cast shadows oto the scee [2, 11, 14, 191, while very few studies have focused o the the illumiat iformatio which shadows could provide. I the proposed method, we are able to reliably estimate a illumiatio distributio of a real scee by makig use of the occlusio iformatio of the icomig light. Also, our method is applicable to the images take uder a complex illumiatio eviromet such as images take i a ordiary room, icludig reflectios from the wall ad other objects i the scee. 1.1 Overview Before we describe the proposed method i detail, we should clarify the basic steps of our method. First, we take a image of the scee usig a color CCD camera so that shadows of a object appear i the image. I the rest of the paper, we refer to the image with shadows as the shadow image ad the object of kow shape, which cast shadows oto the scee, as the occludig object. (A typical example of a shadow image is show i Figure 3.) The, based o the radiace distributio iside shadows, $ IEEE 306

2 a illumiatio distributio of a real scee is estimated as a collectio of imagiary poit light sources distributed over the etire scee. The key idea of the proposed method is the discretizatio of the overall illumiatio distributio by usig the ode directios of a geodesic dome. I the proposed method, we assume that light sources i the scee are sufficietly distat from the objects ad thus all light sources project parallel rays oto the object surface. By substitutig a collectio of imagiary poit light sources for the etire illumiatio, we are able to derive a system of equatios i ukow radiace values of imagiary poit light sources from the image irradiace of the shadow image. We the solve for a solutio set of ukow radiace values which approximates the illumiatio distributio of the scee. The rest of the paper is orgaized as follows. I Sectio 2 ad Sectio 3, we explai how to estimate a illumiatio distributio of a real scee from the image irradiace of a shadow image. We first obtai a formula which relates a illumiatio distributio of a real scee with the image irradiace of the shadow image (Sectio 2). Secod, by assigig the image irradiace of the shadow image to the formula, we obtai a set of liear equatios with ukow illumiatio radiace values sampled at a equal solid agle. Fially, we solve the set of liear equatios for a illumiatio radiace solutio set which represets the illumiatio distributio of the scee (Sectio 3). Sectio 4 shows experimetal results of the proposed method applied to real images. To evaluate the accuracy of the illumiatio distributio estimated by our method, we superimpose a sythetic object with the same shape as that of the occludig object oto a image of the scee. I Sectio 5, we preset cocludig remarks. 2 Formula for Relatig Illumiatio Radiace with Image Irradiace I this sectio, we preset a formula which relates a illumiatio distributio of a real scee with the image irradiace of a shadow image. Based o the image formatio, the formula is obtaied as follows: 1. Illumiatio radiace to scee irradiace: fid a relatioship betwee the illumiatio distributio of a real scee ad the irradiace at a surface poit i the scee. 2. Scee irradiace to scee radiace: compute how much of the icomig lights are reflected from the surface toward a image plae. 3. Scee radiace to image irradiace: fid a relatioship betwee the reflected light from the surface ad the image irradiace at a correspodig poit o the image plae. (4 (b) Figure 1: Total irradiace: (a) without occludig object (b) with occludig object 2.1 From Illumiatio Radiace to Scee Irradiace First, scee irradiace is computed from the etire illumiatio of the scee. To take illumiatio from all directios ito accout, let us cosider a ifiitesimal patch of the exteded light source, of size Sei i polar agle ad Sq5i i azimuth as show i Figure 2. See from the ceter poit A, this patch subteds a solid agle Sw = sioisoisq5i. Let LO(Oi,q$) be the illumiatio radiace per uit solid agle comig from the directio (Oi, qbi); the the radiace from the patch is Lo(&, +i) si OiSQiS&[6], ad the total irradiace of the surface poit A is Lo(Bi,~i)~~~BisiBidBid~i (1) The occlusio of the icomig light by the occludig object is cosidered as E = S_:12 Lo(ei,~i)s(ei,~i)coseisieideid~i (2) where S(Bi, $i) are occlusio coefficiets; S(&, q5i) = 0 if Lo(&, &) is occluded by the occludig object; Otherwise = 1. qe%,d%) 2.2 From Scee Irradiace to Scee Radiace Some of the icomig lights at poit A are reflected toward the image plae. As a result, poit A becomes a secodary light source with scee radiace, which ca be computed from scee irradiace at poit A. The bidirectioal reflectace distributio fuctio (BRDF) f(&, 4%; Oe, &) is defied as a ratio of the radiace of a surface as viewed from the directio (ee,&) to the irradiace resultig from illumiatio from the directio (et, 4%). Thus, by itegratig the product of the BRDF ad the illumiatio radiace over the etire hemisphere, the scee radiace Rs(Oe,&) viewed from the directio 307

3 .. Figure 2: (a)the directio of icidet ad emitted light rays (b)ifiitesimal patch of a exteded light source) (e,, is computed as 2.3 From Scee Radiace to Image Irradiace Fially, the illumiatio radiace of the scee is related with image irradiace o the image plae. Sice what we actually observe is ot image irradiace o the image plae, but rather a recorded pixel value i a shadow image, it is also ecessary to cosider the coversio of the image irradiace ito a pixel value of a correspodig poit i the image. This coversio icludes several factors such as D/A ad A/D coversios i a CCD camera ad a frame grabber. Other studies cocluded that image irradiace was pro- portioal to scee radiace [6]. I our method, we calibrate a liearity of the CCD camera by usig a gray scale chart so that the recorded pixel values also become proportioal to the scee radiace of the surface. From Equatio 3 the pixel value of the shadow image P(Oe, q5e) is thus computed as 3.1 Approximatio of Illumiate Distributio by a Geodesic Dome First, the double itegral i Equatio 4 is approximated by discrete samplig over the etire surface of the exteded light source. Node directios of a geodesic dome are used for approximatig the illumiatio distributio of the scee as a summatio of illumiatio radiace sampled at equal solid agles. Nodes of a geodesic dome are kow to be uiformly distributed over the surface of a sphere. Therefore, by usig odes of a geodesic dome i a orther hemisphere as a samplig directio, the double itegral i Equatio 4 is approximated as a samplig at a equal solid agle 6w = 21~/. P(ee, de) = f(et14z; de, 4e)L(Oz, 4z)s(ot74%) ~0~0, 2= 1 (5) where L(&, 4,) is the illumiatio radiace per solid agle bw = 2/ comig from the directio (O,, 4,), which also icludes the scalig factor k betwee scee radiace ad pixel values. The umber of the odes ca be adjusted by chagig the samplig frequecy of a geodesic dome. It should be oted that the recorded pixel value P(e,, 4,) is computed as a fuctio of the illumiatio radiace Lo(&, 4,) ad the BRDF f(o,, 4,; O,, 4,) i Equatio 5. We thus take two differet approaches depedig o whether BRDF of the surface is give i the followig sectios. We explai the case where the BRDF is give i Sectio 3.2 ad Sectio 3.3, ad the other case where the BRDF is ot give i Sectio Kow Reflectace Properties: Lambertia Model Suppose the surface is a Lambertia surface; BRDF f(o,,~,;~,,~,) for a Lambertia surface is kow to be a costat. From Equatio 5, a equatio for a Lambertia surface is obtaied as S(O,, 4%) COS 8, si e,do,d4, (4) P(ee, d e) = KdL(Oz, ~ z)co~~zs(~zi where k is a scalig factor betwee scee radiace ad a 2=1 pixel value. Due to the scalig factor k, we are able to estimate ukow lo(^,, 4,)(i = 1,2,..., ) up to scale. TO obtai he scale factor k, we eed to perform photometric calibratio betwee pixel itesity ad physical uit (watt/m2) for the irradiace. 3 Estimatio of Illumiatio Distributio Usig Image Irradiace After obtaiig the formula which relates the illumiatio radiace of the scee with the pixel values of the shadow image, illumiatio radiace is estimated based o the recorded pixel values of the shadow image. 42) (6) where Kd is a diffuse reflectio parameter of the surface. From Equatio 6, a liear equatio is obtaied for each image Pixel ofthe shadow image as all1 + a2l2 + a3l all = P (7) where L, (i = 1,2,..., ) are ukow illumiatio radiace specified by ode directios of a geodesic dome. The coefficiets at(i = 1,2,..., ) represet Kd cos e,.!?, i Equatio 6; we ca compute these coefficiets from the 3D geometry of a surface poit, the occludig object ad the 308

4 am1ll + am2l2 + ~m3l aml = Pm(8) Therefore, by selectig a sufficietly large umber of image pixels, we are able to solve for a solutio set of ukow Li s [17]. Note that, sice each pixel cosists of 3 color bads (R, G, ad B), each bad of radiace L, is also estimated from the correspodig color bad of the image. 3.3 Kow Reflectace Properties: No-Lambertia Model Our method is ot limited oly to the Lambertia reflectio model; but it ca also be exteded to other reflectio models. As show i the previous case, our method requires a set of liear equatios with ukow illumiatio radiace. Hece, ay reflectio model is applicable to our method providig such a set of liear equatios is obtaied. Take a simplified Torrace-Sparrow reflectio model [ 15, 201 for example; the pixel value of shadow image p(@,, $e) is computed as 3.4 Ukow Reflectace Properties : Lambertia Model Eve i the case where the BRDF is ot give, we are still able to estimate a illumiatio distributio of a real scee if the surface is a Lambertia surface. The questio we have to cosider here is how to cacel the additioal ukow umber Kd i Equatio 6. A additioal image of the scee take without the occludig object is used to cacel Kd. We refer to the image as a surface image. The image irradiace of a surface image represets the surface color of the plae i the case where oe of the icomig light is occluded. From this, i the case of the surface image, the shadow coefficiets S(&, &) always become S(&, 4i) = 1. Therefore, usig Equatio 6, the image irradiace P (@,, of the surface image is computed as j=1 From Equatio 6 ad Equatio 10, the ukow Kd is caceled as qez, 4 a ) q e a, 4%) (9) where?(e,, 4,) is the agle betwee the surface ormal ad the bisector of the light source directio ad the viewig directio, Kd ad K, are costats for the diffuse ad specular reflectio compoets, ad U is the stadard deviatio of a facet slope of the Torrace-Sparrow reflectio model. From Equatio 9, we obtai a liear equatio for each image pixel where L(OZ7q5,)(z = 1,2,..., ) are ukow illumiatio radiace, ad (KdcosO, + -?(a,4 )2 K,&e 2h )S(Ot7q5,)(i= 1,2,..., )arekowcoefficiets. Agai, if we use a sufficietly large umber of pixels for the estimatio, we are able to solve for a solutio set of ukow illumiatio radiace L(@,, q5,)(i = 1, 2,..., ). We established the correspodece betwee the 3D world coordiate system i the scee ad the 2D image coordiate system by usig the camera calibratio algorithm proposed by Tsai [21]. From the calibratio process, a plae of z = 0 is also defied o the calibratio board, oto which the occludig object cast shadows. The questio of how to select image pixels to obtai a solutio set for ukow radiace values seems to be leadig to a iterestig research topic. For istace, a similar discussio o this subject ca be foud i [lo]. Fially, we obtai a liear equatio for each image pixel where rl (i = 1,2,..., ) are ukows, x,=, L(e,,&)cOs, cos@,s(@,, 4,) (i = 1,2,..., ) are computable coefficiets, ad is a right-had side quatity. Agai, if we use a sufficietly large umber of pixels for the estimatio, we are able to solve the set of liear equatios for a solutio set of ukow, L(e714 ) (i = 1,2,..., ). 2: 3=1 L(, r43 bjse, We should poit out that the estimated radiace from these equatios is a ratio of the illumiatio radiace i oe directio L(Oz74,) to scee irradiace at the surface poit E, =, L(O,, 4,)cos@,. Hece, without kowig the ratio of the scee irradiace amog color bads, there is o way to relate the estimated radiace over the color bads. Our method avoids this problem because of the iitial camera calibratio. Sice we use a white board with regularly spaced dots as a calibratio board, the recorded color of the board directly shows the ratio of the scee irradiace amog color bads. 4 Experimetal Results We have tested the proposed method by usig real images of idoor eviromets. To evaluate the accuracy of the illumiatio distributio estimated by our method, we 309

5 superimpose a sythetic object with the same shape as that of the occludig object oto a image of the scee take without the occludig object, ad compare the shadows of the sythetic object with those of the occludig object i the shadow image. Sectio 4.1 explais how to superimpose a sythetic object oto the real scee by usig the estimated illumiatio distributio. I Sectio 4.2, we describe experimetal results i the case where reflectace properties of a reflected surface are kow. The, i Sectio 4.3, we describe experimetal results i the case where reflectace properties of the surface are ukow. 4.1 Superimposig a Sythetic Occludig Object oto the Scee The ray castig algorithm is used to superimpose a sythetic object. If the ray geerated from camera projectio ceter through the image pixel itersects a sythetic object, we compute a color to be observed at the surface poit usig a simplified Torrace-Sparrow reflectio model from Sectio 3.3. From the model, a color to be observed at the surface poit Rs(B,, 4,) is computed usig the estimated illumiatio distributio of the real scee as Rsc(ee, de) = Kd,c i=l Lc(ei, 4i)cos& c = R, G, B where Lc(B,, q52) (i = 1,2,..., )are the estimated illumiatio radiace values. Otherwise, the ifluece of the sythetic object oto the real object surface is cosidered. I other words, we create shadows cast by the sythetic object oto the surface. First, we compute total irradiace El at the surface poit usig the estimated illumiatio distributio i the case where a sythetic object does ot occlude ay icomig light (Figure 1.a). Ei,C = 1 2= 1 Lc(Q,, 42)cOset c = R, G, B (13) where L(B,, q5z) (i = 1,2,..., ) are the estimated illumiatio radiace values. Secod, we compute total irradiace E2 at the surface poit i the case where the sythetic object occludes some of the icomig light (Figure 1.b). = ~,(e,, z= 1 q5,)cose2s(e2, q5z) c = R, G, B (14) where S(e,, q52) = 0 if the sythetic object occludes illumiatio radiace L(B,, q5z); otherwise, S(&, 4,) = 1. The, we compute the ratio of E2 to El, which represets how much of the irradiace at the itersectio would still be preserved if the sythetic object were placed i the scee. Fially, by multiplyig the ratio E2/E1 to the observed color of the image pixel I, we obtai the color I' that would be the color of the image pixel if there were a sythetic object i the scee. 4.2 Experimetal Results for Kow Reflectace Property A image of a surface with a occludig object called a shadow image was take uder the usual illumiatio eviromet i our office, icludig direct light sources such as fluorescet lamps ad widows to the outside, as well as idirect illumiatio such as reflectios from a wall (Figure 3). First, a illumiatio distributio of the scee was estimated usig the image irradiace iside shadows i the shadow image as explaied i Sectio 3.2. The a sythetic object with the same shape as that of the occludig object was superimposed oto a image of the scee take without the occludig object, called the suflace image. Sythesized results are show i Figure 4 (a), (b), ad (c). Also, we superimposed aother sythetic object of a differet shape oto the scee i Figure 4(d). The umber of odes of a geodesic dome used for the estimatio is show uder the resultig image. We foud through our experimets that, the larger umber of odes we used, the more the shadows of the sythetic object resembled those of the occludig object i the shadow image. Especially i the case of 521 odes, the shadows of the sythetic object are idistiguishable from those of the occludig object i the shadow image: this shows that the estimated illumiatio distributio gives a good presetatio of that of the real scee. Figure 5 umerically shows the improvemet of the accuracy by icreasig the umber of sampligs. The vertical axis represets average error i pixel values iside the shadow regios i the sythesized images compared with those i the shadow image. Here, the iitial differece i pixel values of shadow regios betwee the su@ace image ad the shadow image is set to 100 %. The horizotal axis represets the umber of odes of a geodesic dome used for the estimatio. From the plot i the figure, we ca clearly see that the accuracy improves rapidly as we use more imagiary poit light sources. Also the small pictures right ext to the plot shows error distributios iside shadow regios i the sythesized images. Darker color represets larger error i a pixel value i the shadow regios compared with the real shadows of the occludig object i the shadow image. 310

6 Ed ) ~..._ _._.. - _ _ _._... _._ average error loo 90 Figure 3: Iput images : (a) sugace image (b) shadow image (c) calibratio image l umber of odes (a) umber of odes : 89 (b) umber of odes : 193 Figure 5: Error Aalysis: kow reflectace property (c) umber of odes : 521 (d) umber of odes : 521 Figure 4: Sythesized images: kow reflectace property Also, the resultig images idicate that it is required to adjust the umber of odes of a geodesic dome depedig o the complexity of a scee to obtai a reasoably good estimatio for less computatioal cost. We are curretly extedig our work so that a appropriate umber of odes is automatically selected for the estimatio, depedig o the scee complexity. 4.3 Experimetal Results for Ukow Reflectace Property We also applied our method to the case where reflectace properties of a surface were ukow. The iput images used i this experimet are show i Figure 6. Sice the reflectace properties of the surface were ukow, the image irradiace of both the shadow image ad the suface image were used for estimatig the illumiatio distributio of the scee as explaied i Sectio 3.4. I the same way as i the previous case, a sythetic occludig object was superimposed oto the surface of the surface image. Sythesized results are show i Figure 7. Agai, i the case of 521 odes, the shadows i the resultig image strogly resemble those of the occludig object i the shadow image. This shows that the estimated illumiatio distributio gives a good represetatio of the characteristics of the real scee. We cocluded from our experimets that the proposed method is effective for providig a illumiatio distributio which ca be used as a substitutio for a real illumiatio distributio. 5 Coclusios I this paper, we have proposed a ew method for estimatig a illumiatio distributio of a real scee from a radiace distributio iside shadows cast by a real object of kow shape oto other object surface of kow shape ad kow reflectace. By usig the occlusio iformatio of the icomig light, we could estimate a illumiatio distributio of a real scee reliably eve for the images take i a complex illumiatio eviromet. There have also bee several methods proposed for measurig illumiatio of a real scee i the field of augmeted reality research [3,4,5]. However, those methods teded to measure the illumiatio directly from images of the scee ad therefore, they suffered from two techical problems: how to capture a wide field of view of the scee, ad how to record high dyamic rage of the scee. I the proposed method, sice we observe shadows ad ot the illumiatio itself, o effort to overcome these problems is required. To demostrate the effectiveess of the proposed method, we have successfully tested our method by usig sets of real images take i our office with differet surface materials of shadow regios. Refereces [ 11 R. Baribeau, M. Rioux, ad G. Godi, Color Reflectace Modelig Usig a Polychromatic Laser Rage Sesor IEEE IEEE Tras. PAMI, vol. 14, o. 2, pp , [2] J. Bouguet ad P. Peroa, 3D Photography o Your Desk, Itl. Coferece o Computer Visio, pp.43-50,

7 avera e error 130 ~ Figure 6: Iput images : (a) surface image (b) shadow image (c) calibratio image ::I,,,,,,,, umber of odes (a) umber of odes : 89 (b) umber of odes : 193 Figure 8: Error Aalysis: ukow reflectace property [I I] J. R. Keder ad E. M. Smith, Shape from Darkess: Derivig Surface Iformatio from Dyamic Shadows, Proc. Irl. Coferece o Computer Visio, pp , (c) umber of odes : 521 (d) umber of odes : 521 Figure 7: Sythesized images: ukow reflectace prop- P. E. Debevec, Rederig Sythetic Objects ito Real Scees: Bridgig Traditioal ad Image-based Graphics with Global Illumiatio ad High Dyamic Rage PhotOgraPhY. PmC. WX%W H 98, pp , July, G. Drettakis, L. Robert, S. Bougoux, Iteractive Commo Illumiatio for Computer Augmeted Reality Proc. 8th Eurographics Worhhop o Rederig, pp , A. Fourier, A. Guawa ad C. Romazi, Commo Illumiatio betwee Real ad Computer Geerated Scees, Proc. Graphics Iterjace 93, pp , B. K. P. Hor, Robot Visio, The MIT Press, Cambridge, MA., B. K. P. Hor, Obtaiig Shape from Shadig Iformatio, Chapter 4 i The psychlogy of Computer Visio, McGraw-Hill Book Co., New York, N.Y., B. K. P. Hor ad M. J. Brooks, The Variatioal Approach to Shape from Shadig, Computer Visio, Graphics, ad Image Processig, 33(2), pp , K. lkeuchi ad B. K. P. Hor, Numerical Shape from Shadig ad Occludig Boudaries, Artijicial Itelligece I7(1-3), pp , K. Ikeuchi ad T. Kaade, Automatic Geeratio of Object Recogitio Programs, Proc. IEEE(76). No. 8, pp , [12] G. Kay ad T. Caelli, Estimatig the Parameters of a Illumiatio Model usig Photometric Stereo, Graphial Models ad Image Processig, vol. 57, o. 5, pp , 1995 [ 131 J. Lu ad J. Little, Reflectace Fuctio Estimatio ad Shape Recovery from Image Sequece of a Rotatig Object, Proc. IEEE Itl. Coferece o Computer Visio 95, pp , [14] A. K. Markworth, O the Iterpretatio of Drawigs as Three- Dimesioal Scees, PhD thesis, Uiversiy of Sussex, [15] S. K. Nayar, K. Ikeuchi, ad T. Kaade, Surface reflectio: physical a d geometrical perspectives: IEEE Tras. PAMI, vol. 13, o. 7, pp , [16] A. P. Petlad, Liear Shape From Shadig, Irl. J. Computer Visio, 4(2), pp , [17] W. H. Press, B. P. Flaery, S. A. Teukolsky, W. T. Vetterlig, Numerical Recipes i C: The Art of ScietiJic Computig, Cambridge Uiversity Press, Cambridge, [18] Y. Sato, M. D. Wheeler, ad K. Ikeuchi, Object shape ad re- flectace modelig from observatio, Pmc. SIGGRAPH 97, pp , [19] S. A. Shafer ad T. Kaade, Usig Shadows i Fidig Surface Orietatios: Computer Visio, Graphics, ad Image Processig, 22(1), pp , [20] K. E. Torrace ad E. M. Sparrow, Theory for off-specular reflectio from rougheed surface: J. Optical Sociefy of America, vo1.57, pp i [21] R. Tsai, A Versatile Camera Calibratio Techique for High Accuracy Machie Visio Metrology Usig Off-the-shelf TV Cameras ad Leses, IEEE J. Robotics ad Automatio, vol. 3, o. 4, pp ,