New Geometric Interpretation and Analytic Solution for Quadrilateral Reconstruction

Transcription

1 New Geometric Interpretation an Analytic Solution for uarilateral Reconstruction Joo-Haeng Lee Convergence Technology Research Lab ETRI Daejeon, , KOREA Abstract A new geometric framework, calle generalize couple line camera (GCLC), is propose to erive an analytic solution to reconstruct an unknown scene quarilateral an the relevant projective structure from a single or multiple image quarilaterals. We exten the previous approach evelope for rectangle to hanle arbitrary scene quarilaterals. First, we generalize a single line camera by removing the centering constraint that the principal axis shoul bisect a scene line. Then, we couple a pair of generalize line cameras to moel a frustum with a quarilateral base. Finally, we show that the scene quarilateral an the center of projection can be analytically reconstructe from a single view when prior knowlege on the quarilateral is available. A completely unknown quarilateral can be reconstructe from four views through non-linear optimization. We also escribe a improve metho to hanle an off-centere case by geometrically inferring a centere proxy quarilateral, which accelerates a reconstruction process without relying on homography. The propose metho is easy to implement since each step is expresse as a simple analytic equation. We present the experimental results on real an synthetic examples. I. INTRODUCTION A new geometric framework, calle generalize couple line camera (GCLC), is propose to erive an analytic solution to reconstruct an unknown scene quarilateral an the relevant projective structure from a single or multiple image quarilaterals. We exten the previous approach, calle couple line camera (CLC), which moels a rectangular frustum of a pinhole camera using two line cameras [1], [2]. (A line camera in our context oes not refer to a capturing evice such as a line-scan camera. Rather, our geometric configuration is more relate to moeling approaches base on linear elements for camera calibration [3] or multi-perspective image [4].) Uner CLC configuration, geometric relation among the base rectangle, the image quarilateral an the optical center can be comprehensively escribe as simple equations of a compact parameter set. Hence, given a single image quarilateral, we can uniquely ientify the frustum by reconstructing the base rectangle an optical center using a close-form solution. The solution also contains a eterminant that tells if a image quarilateral is the projection of any rectangle prior to reconstruction. In the CLC-base reconstruction, no explicit form of camera parameters is involve since the formulation is base on pure geometric configuration of a pinhole projection. In application, an image quarilateral is represente by a set of iagonal parameters (i.e. relative lengths of partial iagonals an the crossing angle) rather than actual pixel coorinates. If require, unknown camera parameters such as the focal length can be compute subsequently using a stanar calibration technique [5], [6]. Generally the previous solutions require to reconstruct the camera parameters first [7]. For example, when we apply the IAC (image of the absolute conic) metho, the unknown focal length shoul be foun first [5], [8]. Another interesting feature of CLC-base reconstruction is geometric interpretation of the solution space, which leas to an optimize analytic solution [2]. For example, given an image quarilateral, two caniate line cameras are efine over two solution spheres. By the constraint of common principal axis, spheres are confine to two solution circles. Finally, the optical center is foun in the intersection of two solutions circles. We believe a similar geometric framework can be applie in other geometric computer vision problems such as investigating the solution space of n-view reconstruction. In this paper, we propose generalize couple line camera (GCLC) that inherits the key features of CLC an moels a projective frustum with a quarilateral base, which targets on a prospective application of projective reconstruction of an unknown scene quarilateral. While keeping the same centering constraint of CLC that the principal axis passes through the center of quarilaterals, we exten the moel with aitional parameters to escribe the lengths of all partial iagonals. In CLC, these parameters nee not be specifie since they cancel out ue to equilateral partial iagonals of a rectangle [1], [2]. The increase number of configuration parameters in GCLC, however, hiners to formulate a closeform solution for single view reconstruction. We investigate this property an propose an analytic solution that works for single view reconstruction uner special conitions, an a metho to approximate unknown iagonal parameters from multiple views. For practical application of CLC framework, we nee to hanle an off-centere case. In this paper, we also propose an improve metho compose of simpler operations base on geometric properties, not relying on constraine equation solving or explicit homography as in [1]. This paper is organize as follows. In Section II, we summarize the previous work on CLC [1], [2]. In Section III, we generalize CLC an escribe reconstruction solution incluing off-centere cases. In Section IV, we give experimental results on synthetic an real quarilaterals to emonstrate the performance. Finally, we conclue with remarks on future work.

2 p c v 0 v 3 u 2 l 2 s 2 y 2 y 0 l 0 q 0 v 2 m 2 vm m 0 u 0 s 0 v 0 (a) Camera pose when =1.7. v 2 v m v 0 c (b) Circular trajectory of p c for varying. f v m G v 1 v 2 (a) Scene rectangle G (b) 1st line camera C 0 (c) 2n line camera C 1 Fig. 1. An example of a canonical line camera: m 0 = m 2 =1, l 0 =0.6, l 2 =0.4, an =0.2. u 0 r u m u 3 II. PRELIMINARIES OF COUPLED LINE CAMERAS A. Line Camera Definition 1. A line camera captures an image line u i u i+2 from a scene line v i v i+2 where v i =(m i, 0, 0) an v i+2 = ( m i+2, 0, 0) for positive m i an m i+2. See Figure 1a. Definition 2. In a centere line camera, the principal axis passes through the center v m of the scene line v i v i+2 : v m =(v i + v i+2 )/2. (1) Definition 3. A canonical line camera is a centere line camera with two constraints for simple formulation: v m = (0, 0, 0) T an equilateral unit ivision: kv i v m k = kv i k = kv i+2 k =1. (2) For a line camera C i, let be the length of the principal axis from the center of projection p c to v m. Let i be the orientation angle of the principal axis measure between v m p c an v m v i. Definition 4. For a canonical line camera, its pose equation is expresse as follow: li l i+2 cos i = = i (3) l i + l i+2 where l i = ku i u m k is the length of partial iagonals. Let i be the line ivision coefficient of the canonical configuration i = l i l i+2 (4) l i + l i+2 Accoring to Eq.(3), we can observe the relation among i, an i. Note that when i is fixe, p c is efine along a circular trajectory or on a solution sphere of raius 0.5/. See Figure 1b. B. Couple Line Cameras Definition 5. Couple line camera is a pair of line cameras, that share the principal axis an the center of projection. By coupling two canonical line cameras, we can represent a projective structure with a rectangle base. See Figure 2. Definition 6. For couple line camera, we can erive a coupling constraint: = l 1 = tan 1 = sin 1 ( cos 0 ) (5) l 0 tan 0 sin 0 ( cos 1 ) () Coupling C 0 an C 1 (e) Projective structure u 1 u 2 (f) Projection of G to Fig. 2. Coupling of two canonical line cameras to represent a projective structure with a rectangle base. where is the coupling coefficient efine by the ratio of the lengths, l 0 an l 1, of two partial iagonals of. See Figure 2f. C. Projective Reconstruction Algorithm 1 (Single View Reconstruction with CLC). The unknown elements of projective structure, such as the scene rectangle G an the center of projection p c, can be reconstructe from a single image quarilaterals as in the below. First, the pose equation of Eq.(3) an the coupling constraint of Eq.(5) can be rearrange into a system of equations: = sin 0 cos 1 cos 0 sin 1 sin 0 sin 1 = cos 0 0 = cos 1 1 (6) Then, the length of the common principal axis can be compute from the system of equations in Eq.(6) as follows: = p A 0 /A 1 (7) where A 0 =(1 1 ) 2 2 (1 0 ) 2 an A 1 = 2 0(1 1 ) 2 2 (1 0 ) Once is compute, two orientation angles, 0 an 1, can be compute using Eq.(3). The base rectangle G can be reconstructe by computing its unknown shape parameter, the iagonal angle : cos = cos sin 0 sin 1 + cos 0 cos 1 (8) where is the iagonal angle of the image quarilateral. Finally, the projective structure can be reconstructe by computing the coorinates of a center of projection p c : p c = (sin cos 0, cos 1 cos cos 0, sin sin 0 sin 1 ) sin (9) D. Determinant Conition When Eq.(7) has a vali value, two conitions shoul be satisfie: (1) A 0 an A 1 have the same sign; an (2) the length of the common principal axis shoul not excee the iameter

3 u 2 l 2 s 2 q 0 p c y 2 y0 l 0 u 0 s 0 v1 G f v m v 0 v 2 v m vm m (a) Camera pose when =1.7. v 2 v m v 0 c (b) Trajectory of p c when is not fixe. v 2 v 3 (a) Scene qua. G (b) 1st line camera C 0 (c) 2n line camera C 1 Fig. 3. An example of a generalize line camera: m 0 =1, m 2 =1.4, l 0 =0.6, l 2 =0.4, an =0.2. u 0 r u m u 3 of each solution sphere: apple min(1/k 0 k, 1/k 1 k). These conitions can be combine into Boolean expressions: D = D 0 _ D 1 (10) 1 0 D 0 = ^ 1 apple 0 (11) D 1 = apple ^ 1 (12) where ^ an _ are Boolean an an or operations, respectively. Since 0, 1 an are the coefficients from a given image quarilateral, we can etermine if is an image of any scene rectangle before actual reconstruction. Once the eterminant D is satisfie, Algorithm 1 can be applie. E. Off-Centere Case CLC assumes the principal axis passes through the centers of the image quarilateral an the scene rectangle G. When hanling an off-centere quarilateral g, a centere proxy quarilateral shoul be foun first by solving equations that formulate ege parallelism between an g, centering constraint of, an a vanishing line erive from g [1]. Once is foun, the centere proxy rectangle G can be reconstructe using Algorithm 1. Since the inferre oes not guarantee congruency to g, the target scene rectangle G g shoul be reconstructe using a homography H between an G: G g = H g. In this paper, we propose a new metho to hanle an offcentere case. First, we erive a centere proxy quarilateral that is perspectively congruent to g. Then, we show that the target scene rectangle G g can be geometrically erive without relying on homography. See Section III-E. III. GENERALIZATION OF COUPLED LINE CAMERAS As a main contribution of this paper, we generalize a line camera to support a non-canonical configuration. Then, we show that a pair of generalize line cameras can be couple to represent a projective structure with a quarilateral base other than a rectangle. Finally, we escribe how we can reconstruct a projective structure from a single view with a sufficient prior knowlege to constrain the solution space. We also escribe how to hanle off-centere cases. () Coupling C 0 an C 1 (e) Projective Structure u 1 u 2 (f) Projection of G to Fig. 4. Coupling of two generalie line cameras to represent a projective structure with a quarilateral base. A generalize line camera C i is assigne for each iagonal of a scene quarilateral G. actual values of iagonal parameters. A. Generalize Line Camera Definition 7. In a general configuration of a line camera, the principal axis may not bisect the scene line: we may not consier the centering constraints of Eqs.(1)-(2). See Figure 3 where m 0 6= m 2. Accoringly, the pose equation of a canonical line camera in Eq.(3) shoul be generalize with two aitional parameters, m 0 an m 2. Assuming m 0 > 0 an m 2 > 0, the following geometric relation hols: l i : l i+2 = m i sin 0 : m i+2 sin 0 ˆi + ˆ (13) i+2 where ˆ 0 = m 0 cos 0 an ˆ 2 = m 2 cos 0. Definition 8. The generalize pose equation can be erive from Eq.(13): mi+2 l i m i l i+2 cos i = = g,i (14) m i m i+2 (l i + l i+2 ) where g,i is the generalize ivision coefficient g,i = m i+2l i m i l i+2 m i m i+2 (l i + l i+2 ). (15) For a fixe g,i, the center of projection p c is efine over a circular trajectory as in Figure 3b, or on a solution sphere [2]. B. Coupling Generalize Line Cameras By coupling two generalize line cameras, we can represent a projective structure with a quarilateral base G with vertices: v 0 = m 0 (1, 0), v 1 = m 1 (cos, sin ), v 2 = m 2 /m 0 v 0, an v 3 = m 3 /m 1 v 1 where m i s are the relative lengths of partial iagonals or iagonal parameters of G. See Figure 4. Definition 9. A generalize coupling constraint as follows: g = l 1 l 0 = m 1 sin 1 m 0 sin 0 ( m 0 cos 0 ) ( m 1 cos 1 ) g is efine (16)

4 C. Projective Reconstruction Using a trigonometric ientity an the pose equation of 2 Eq.(14), we can erive the equation for g by squaring both the sies of Eq.(16): sin 2 i =1 cos 2 i =1 2 g,i 2 (17) 2 g = m2 1(1 m 0 g,0 ) 2 (1 g,1 2 2 ) m 2 0 (1 m 1 g,1 ) 2 (1 g,0 2 2 ) (18) From Eq.(18), the length of the common principal axis can be expresse with GCLC parameters: s A g,0 = (19) A g,1 (a) Reference: G g an g (c) Reconstruction of G an G g (b) Inferring a centere in blue () Congruency of G an G g where A g,0 = m 2 0(1 m 1 g,1 ) 2 2 g m 2 1(1 m 0 g,0 ) 2 an A g,1 = m g,0(1 m 1 g,1 ) 2 2 g m 2 1(1 m 0 g,0 ) 2 2 g,1. Eq.(19) states that can be compute from known iagonal parameters, m i an l i, of a single pair of scene an image quarilaterals, not relying on their iagonal angles, an. Algorithm 2 (Single View Reconstruction with GCLC). Once the length of the common principal axis has been foun using Eq.(19) with prior knowlege on iagonal parameters, we can compute the orientation angles, 0 an 1, using the pose equation of Eq.(14). Then, the iagonal angle of a scene quarilateral an the center of projection p c can be compute using Eqs.(8) an (9), respectively. If we have no prior knowlege on iagonal parameters m i of G, we can infer them using multiple image quarilaterals j from ifferent views. By setting m 0 =1, the number of unknown iagonal parameters of G is reuce to three: m 1, m 2 an m 3. For each j, the crossing angle j of Eq.(8) is expresse with m 1, m 2 an m 3, an coefficients erive from known iagonal parameters l i,j of j. Since the reconstructe j s shoul be ientical regarless of views, the following ientity shoul hol: cos j = cos j+1. Hence, if we have four ifferent views, we can formulate three equations of three unknowns, m 1, m 2 an m 3 : cos 0 = cos 1 = cos 2 = cos 3 (20) The number of views are varying accoring to the egree of freeom in iagonal parameters. Although an analytic solution for Eq.(20) is not foun yet, the problem can be formulate as minimization of the following objective function: nx 1 f obj = k cos j cos j+1 k 2 (21) j=0 where n is the number of views. Generally, Eq.(21) can be solve using a numerical nonlinear optimization metho [9]. Since optimization may get stuck in a local minima, we may check the valiity using eterminant of Eq. 24. Algorithm 3 (n-view Reconstruction with GCLC). When Algorithm 2 cannot be applie ue to lack of knowlege on the scene rectangle G, but we have multiple image Fig. 5. Reconstruction of a synthetic quarilateral G g from an off-centere quarilateral g: m 0 =1, m 1 =0.75, m 2 =1.35, m 3 =1.4 an = Diagonal parameters m i an the vanishing line is given. quarilaterals j from n ifferent views, we can fin the unknown m i s by minimizing the objective function of Eq.(21). Then, we can apply Algorithm 2 for one of the views to reconstruct the projective structure. The number of views require in Algorithm 3 epens on the number of unknown m i s. For a general quarilateral of three unknown m i s except m 0 = 1, at least 4 views are require accoring to Eq.(20). For a parallelogram with known m 0 = m 2 =1an unknown m 1 = m 3, at least 2 views are require to fin m 1. See Section IV for real examples. D. Determinant Conition Similarly as in Section II-D, we can erive, from Eqs.(14) an (19), a conition D g that can etermine if is projection of a centere scene quarilateral G with known m i s. D g,0 = D g,1 = E. Off-Centere Case D g = D g,0 _ D g,1 (22) m 1(1 m 0 g,0) m 0(1 m 1 g,1) ^ 1 apple g,0 g,1 (23) m0 g,0) ^ 1 (24) apple m1(1 m 0(1 m 1 g,1) g,0 g,1 Let an off-centere image quarilateral g be projection of a scene quarilateral G g, which is also off-centere an unknown yet. See Fig. 5a. To apply Algorithms 2 an 3, we provie a metho to fin a centere proxy quarilateral that is an image of a centere scene quarilateral G. Specially, G is guarantee to be congruent to G g through parallel translation by t. We also show that the translation vector t can be compute in image space. Hence, we o not nee to compute homography H between G an to reconstruct G g as in CLC. See Section II-E an [1]. Algorithm 4 (Reconstruction from an Off-Centere uarilateral). An off-centere scene quarilateral G g can be reconstructe from its image g by aing extra steps to the GCLC methos presente in Section III-C. See Figure 5:

5 w0 w,0 w0 w,0 w1 w1 w,1 w,1 wm ug,2 ug,3 u3 um u2 ug,m us,1 vg,m vt,1 ug,0 ug,1 u0 om u1 g g u0 um us,0 u1 Gg v0 vm vt,0 v1 G Fig. 6. Derivation of a centere proxy quarilateral that is perspectively congruent to g. Assume the vanishing line w 0 w 1 is given. Fig. 7. Perspective-to-Eucliean vector transformation. p c 1) Infer a centere proxy quarilateral from g such that is projection of a centere scene quarilateral G that is congruent to the target quarilateral G g. See Algorithm 5; 2) Apply Algorithm 2 to to reconstruct the corresponing centere quarilateral G an the center of projection p c. If multiple g,j are available, apply Algorithm 3. 3) The target scene quarilateral G g can be compute as translation of G: G g = G + t where t can be compute from isplacement s = u m o m between centers of an g using Algorithm 6. Algorithm 5 (Centere Proxy uarilateral). Assuming a vanishing line w 0 w 1 is given, we can fin a centere proxy quarilateral by perspectively translating an off-centere quarilateral g. See Figure 6: 1) Fin the intersection points w,i between the vanishing line w 0 w 1 an each iagonal u g,i u g,i+2 of G g. 2) Fin the intersection point w m between the vanishing line w 0 w 1 an the line of translation o m u m. 3) Fin the intersection point u 0 between the line u g,0 w m an the line o m w,0. Similarly, fin u 2 from u g,2 w m an o m w,0. 4) Fin the intersection point u 1 between the line u g,1 w m an the line o m w,1. Similarly, fin u 3 from u g,3 w m an o m w,1. 5) The i-th vertex of is u i. Note that Algorithm 5 is compose of simple line-line intersections rather than geometric constraint solving as in [1]. Algorithm 6 (Perspective-to-Eucliean Vector Transformation). With GCLC efine with known an G (as in Fig. 4), we can project an image vector s to a scene vector t. First, we perspectively ecompose s along two iagonals of : 1) Fin the intersection points u s,0 between the line u 0 o m an the line u m w,1. Similarly, fin u s,1 from u 1 o m an u m w,0. 2) For each ecomposition coefficient s i of u s,i, compute the coefficient t i for v i using Eq ) The corresponing scene vector t can be expresse as a vector sum of two iagonal vectors, t 0 v 0 + t 1 v 1, of G assuming v m = 0. See Fig. 4b. Algorithm 6 is base on the following property of a generalize line camera. v t,2 u s,2 l u 2 2 l 0 u m u0 u s,0 q 0 v 2 m 2 v m m 0 v 0 v t,0 Fig. 8. Scaling transformation in a generalize line camera, which is explaine as a cross ratio between corresponing four points. Using projective invariance of cross-ratio [8], the following hols for two sets of collinear points, (v t,0, v 0, v m, v 2 ) an (u s,0, u 0, u m, u 2 ), in the scene an images lines, respectively: s i l i (l i + l i+2 ) l i (s i l i + l i+2 ) = t im i (m i + m i+2 ) (25) m i (t i m i + m i+2 ) where s i = ku s,i u m k/l i an t i = kv t,i v m k/m i. (See Fig. 8.) By solving Eq.(25) for t i, we get the following relation between t i an s i : s i m i+2 (l i + l i+2 ) t i = (26) s i m i+2 l i +((1 s i )m i + m i+2 )l i+2 Hence, if a line camera is efine, a scaling factor s i of image line can be mappe to t i of the scene, an vice versa. IV. EXPERIMENT We give experimental results on real an synthetic examples. All the experiments were performe in Mathematica implementations. We applie Algorithm 4 to real-worl quarilaterals foun in web images of moern architectures. We assume each image is inepenently taken by unknown cameras an not altere (by cropping). Each input quarilateral g,j is specifie in re lines in Fig. 9a an Fig 10a. To infer a centere proxy quarilateral j using Algorithm 5, we fin a vanishing line using patterns of parallel lines such as winow frames [10]. Once a set of centere quarilaterals j are foun, we estimate unknown iagonal parameters m i that minimize the objective function f obj of Eq.(21). In the experiment, we use NMinimize[] function of Mathematica for non-linear optimization [9]. With m i known, we can reconstruct the centere scene quarilateral G j which is congruent to the target scene quarilateral G g,j. See Fig. 9b an Fig 10b. The result of reconstructe 3D view frustum is omitte for the page limit.

6 #1 #2 #3 #4 of noise sources such as lens istortion or feature etection. When ae ranom noises of 1-pixel raius to vertices of j in image, the precision roppe with errors an in mi an, respectively. V. C ONCLUSION (a) Input: web images of Fountain Place in Dallas, Texas. Input (b) A reconstructe quarilateral with ifferent textures of given images. Two images from uncalibrate Fig. 9. Reconstruction of a quarilateral from fourcameras views using Algorithm 4. #1 #2 Output Reconstructe parallelogram! (a) Input: web images of the Docklan in Hamburg, Germany. m =2.87 (err 2.8%), phi=0.61 (err 0.7%), inc= (err 1.2%) using Ref-#2 1 (b) A reconstructe parallelogram with ifferent textures of given images. Fig. 10. Reconstruction of a parallelogram from two view using Algorithm 4. For a quarilateral case of Fig. 9, four images were use. The optimization converges when fobj with m1 = , m2 = , m3 = The mean of four j is with variance The optimization takes about 3 secons in 2.6 GHz Intel Core i7. Time for other reconstruction steps is trivial through evaluation of analytic expressions. For a parallelogram of Fig. 10, it converges when fobj with m1 = an = in 0.06 secon. We also applie Algorithm 4 to the synthetic quarilateral G of Fig. 4 with four ifferent views. The optimization for mi converges when fobj < in 3 secons. The mean error of reconstructe mi is Timing is similar to the real example of Fig. 9, but precision is much higher ue to absence We propose a novel metho to reconstruct a scene quarilateral an projective structure base on generalize couple line cameras (GCLC). The metho gives an analytic solution for a single-view reconstruction when prior knowlege on iagonal parameters is given. Otherwise, require parameters can be approximate beforehan from multiple views through optimization. We also provie an improve metho to hanle off-centere cases by geometrically inferring a centere proxy quarilateral, which accelerates a 2D reconstruction process without relying on homography or calibration. The overall computation is quite efficient since each key step is represente as a simple analytic equation. Experiments show a reliable result on real images from uncalibrate cameras. To apply the propose metho to a real-worl case with an off-centere quarilateral, a vanishing line shoul be available for each view. This conition can be easily satisfie in a specially texture quarilateral of artifacts [11]. Otherwise, we nee other types of prior knowlege to infer a centere quarilateral. For example, a preefine parametric polyheral moel can be a goo caniate [12]. Lastly, couple line projectors (CLP) [13] is a ual of CLC. We expect that generalize CLP can be combine with GCLC for a projector-base augmente reality application. R EFERENCES [1] J.-H. Lee, Camera calibration from a single image base on couple line cameras an rectangle constraint, in ICPR 2012, 2012, pp , 3, 4, 5 [2], A new solution for projective reconstruction base on couple line cameras, ETRI Journal, vol. 35, no. 5, pp , , 3 [3] Z. Zhang, Camera calibration with one-imensional objects, IEEE Trans. on Pattern Analysis an Machine Intelligence, vol. 26, no. 7, pp , [4] J. Yu an L. McMillan, General linear cameras, in Computer VisionECCV Springer, 2004, pp [5] P. Sturm an S. Maybank, On plane-base camera calibration: A general algorithm, singularities, applications, in CVPR 1999, 1999, pp [6] Z. Zhang, A flexible new technique for camera calibration, IEEE Trans. on Pattern Analysis an Machine Intelligence, vol. 22, no. 11, pp , [7] Z. Zhang an L.-W. He, Whiteboar scanning an image enhancement, Digital Signal Processing, vol. 17, no. 2, pp , [8] R. Hartley an A. Zisserman, Multiple View Geometry in Computer Vision, 2n e. Cambrige University Press, , 5 [9] J. A. Neler an R. Mea, A simplex metho for function minimization, The computer journal, vol. 7, no. 4, pp , , 5 [10] J.-C. Bazin, Y. Seo, C. Demonceaux, P. Vasseur, K. Ikeuchi, I. Kweon, an M. Pollefeys, Globally optimal line clustering an vanishing point estimation in manhattan worl, in CVPR 2012, 2012, pp [11] Z. Zhang, A. Ganesh, X. Liang, an Y. Ma, Tilt: transform invariant low-rank textures, International journal of computer vision, vol. 99, no. 1, pp. 1 24, [12] P. E. Debevec, C. J. Taylor, an J. Malik, Moeling an renering architecture from photographs: A hybri geometry-an image-base approach, in SIGGRAPH ACM, 1996, pp [13] J.-H. Lee, An analytic solution to a projector pose estimation problem, ETRI Journal, vol. 34, no. 6, pp ,