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1 Daisuke Miyazaki, Katsushi Ikeuchi, "Inverse Polarization Raytracing: Estimating Surface Shae of Transarent Objects," in Proceedings of International Conference on Comuter Vision and Pattern Recognition, San Diego, CA USA, to aear htt://

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3 Inverse Polarization Raytracing: Estimating Surface Shaes of Transarent Objects Daisuke Miyazaki Katsushi Ikeuchi Institute of Industrial Science, The University of Tokyo, JAPAN htt:// Abstract We roose a novel method for estimating the surface shaes of transarent objects by analyzing the olarization state of the light. Existing methods do not fully consider the reflection, refraction, and transmission of the light occurring inside a transarent object. We emloy a olarization raytracing method to comute both the ath of the light and its olarization state. Our roosed iterative comutation method estimates the surface shae of the transarent object by minimizing the difference between the olarization data rendered by the olarization raytracing method and the olarization data obtained from a real object.. Introduction In the field of comuter vision, few methods have been roosed for estimating the shae of transarent objects, because of the difficulty of dealing with mutual reflection, which is the henomenon that the light not only reflects at the surface of the transarent object but also transmits into the object and causes multile reflections and transmissions inside it. In this aer, we use the term interreflection for such internal reflection. Raytracer simulates the interreflection, and renders the 2D image from 3D shae: Image = Raytracer(Shae) : () If an inverse function of raytracing were to exist, the 3D shae could be obtained straightforwardly from 2D data; however, there is no closed-form solution for the inverse roblem of raytracing. This aer resents a novel method for estimating the surface shae of transarent objects by numerically solving the inverse roblem of raytracing and, at the same time, by analyzing the olarization of transarent objects. An examle for alying the roosed method is shown in Figure. Figure (a) is the target object, and Figure (b) is the result of roosed method. Figure (c) is a rendered examle of the raytracing method by using the estimated shae. Polarization is a henomenon in which the light oscillates in one direction. Recently, research to estimate the Figure : Result for heart-shaed glass: (a) Target object, (b) result of roosed method, (c) raytracing image. shae of the object by using olarization has increased [ 5]. Saito et al. [6] and Miyazaki et al. [7,8] estimated the surface shae of transarent objects by means of olarization analysis. Unfortunately, because these methods do not consider interreflection, they do not rovide sufficient accuracy for estimating the shae of transarent objects. Other methods that estimate the 3D shae of transarent objects without using olarization have been roosed. Murase [9] estimated the shae of a water surface by analyzing the undulation of the water surface. Hata et al. [0] estimated the surface shae of transarent objects by analyzing the deformation of the light rojected onto the transarent objects. Ohara et al. [] estimated the deth of the edge of a transarent object by using shae-from-focus. Ben-Ezra and Nayar [2] estimated the arameterized surface shae of transarent objects by using structure-from-motion. Kutulakos [3] estimated both the deth and the surface normal of transarent objects by multile viewoints and multile light sources. These methods, however, do not estimate arbitrary shaes of transarent objects. There are other works that deal with transarent objects by using methods such as environment matting [4 9] and reflection searation [20 23]; however, they do not rovide enough information about the shaes of the transarent objects. We simulate the interreflection of transarent objects by using a method called olarization raytracing, and we use this method to estimate the surface shaes of transarent objects with arbitrary shaes. In this aer, a forward-facing surface of the transarent object is called a front surface, and an object surface facing away from the camera is called

4 a back surface. Our roosed method estimates the shae of the front surface by using olarization raytracing when the refractive index, the shae of the back surface, and the illumination distribution are given. The rest of the aer is organized as follows. In Section 2, we describe the theoretical background of the olarization raytracing method. In Section 3, we exlain our estimation method, which solves the inverse roblem of olarization raytracing method. Our measurement results are shown in Section 4, and our conclusions are resented in Section Polarization Raytracing The theoretical details of the rincile of olarization, which aears in this section, are resented in the literature [24, 25]. 2.. Conventional Raytracing A conventional raytracing method renders a 2D image from 3D geometrical shae data of transarent objects or other kind of objects. The algorithm of the conventional raytracing method can be divided into two arts. The first art is the calculation of the roagation of the ray. The second art is the calculation of the intensity of the light. Figure 2 describes the light reflected and transmitted between material and material 2. Materials and 2 may be, resectively, the air and the transarent object, and vice versa. Incidence angle, reflection angle, and transmission angle are defined in Figure 2. We assume that the surface of transarent objects is otically smooth; thus, the incidence angle is equal to the reflection angle. The transmission angle is related to the incidence angle as the following Snell s law: sin = n sin 0 (2) where is the incidence angle, 0 is the transmission angle, and n is the ratio of the refractive index of material 2 to that of material. In this aer, we assume that the refractive index of one object is a scalar value which is, at the same time, constant throughout any art of the object. The lane of incidence (POI) is a lane that includes the surface normal direction, the incident light direction, the reflected light direction, and the transmitted light direction. The intensity ratio of reflected light to incident light is called intensity reflectivity R, and the intensity ratio of transmitted light to incident light is called intensity transmissivity T. Subscrits k and? reresent the comonents arallel and erendicular to POI, resectively. Thus, arallel and erendicular comonents of intensity reflectivity are reresented as R k and R?, resectively, while those of intensity transmissivity are reresented as T k and T?, resectively. These values are defined as follows: R k = tan2 ( ; 0 ) (3) tan 2 ( + 0 ) Figure 2: Reflection, refraction, and transmission. R? = sin2 ( ; 0 ) sin 2 ( + 0 ) sin 2 sin 2 0 T k = sin 2 ( + 0 ) cos 2 ( ; 0 ) sin 2 sin 20 T? = sin 2 : (6) ( + 0 ) If an incidence angle is larger than the critical angle, then the light does not transmit and totally reflects. This henomenon is called total reflection and occurs when the incidence light is inside the object (namely, when material is the object and material 2 is the air). Critical angle C is defined as follows: sin c = n : (7) For the total reflection, we must use R k = R? = and T k = T? =0. The conventional raytracing method calculates the roagation of the ray by using the Snell s law (Equation (2)), and calculates the intensity of the light by using the total intensity reflectivity R and the total intensity transmissivity T, which are defined as follows: R = R k + R? Mueller Calculus T = T k + T? 2 (4) (5) : (8) In this aer, we call the raytracing method that considers the olarization effect the olarization raytracing method. The algorithm of the olarization raytracing method can be divided into two arts. For the first art, the calculation of the roagation of the ray, we emloy the same algorithm used in the conventional raytracing method. For the second art, the calculation of the olarization state of the light, there are three famous methods: Mueller calculus, Jones calculus, and the method that uses the coherence matrix. In this aer, we emloy Mueller calculus because of its simlicity of descrition, along with its ease of understanding and imlementation. These three methods have

5 Figure 3: Reflected and transmitted light observed by the camera. almost identical functions; thus, all discussions resented in this aer are also alicable to other calculi. Some researchers [26 32] also imlemented and imroved the olarization raytracing or imroved these three methods, and also, there is some commercial software [33 35] that uses olarization raytracing. In Mueller calculus, the olarization state of the light is reresented as Stokes vector s = (s 0 s s 2 s 3 ) T. The Stokes vector is a 4D vector. Its first comonent s 0 reresents the intensity of the light; its second comonent s reresents the horizontal ower of the linear olarization; its third comonent s 2 reresents the +45 -oblique ower of the linear olarization; and its fourth comonent s 3 reresents the ower of the right circular olarization. The Mueller matrix M, which is a 4 4 matrix, reresents how the object changes the olarization state of the light. The oeration of Mueller calculus is a linear oeration Mueller Matrices In this section, we resent an examle of calculation using Mueller calculus. Suose the geometrical setu when the reflected and transmitted light is observed from the camera is as described in Figure 3. In this figure, there are two kinds of coordinates sytems: x 0 y 0 z 0 coordinates and xyz coordinates. Here, the z 0 axis and the z axis are the same. x 0 is included in the POI and is facing to the same side as the surface normal is facing. The angle between x 0 axis and x axis is called the POI angle in xyz coordinates. In the case resented in Figure 3, observed light is a comosition of reflected light and transmitted light. The Stokes vector s 0 of the observed light is calculated as follows: s 0 = C()D( n)r( n)c(;)s r + C()T( n)c(;)s t : (9) Stokes vectors of the incident light are reresented as s r and s t, and where s r and s t reresent the lights that are set in the origin of the reflection and transmission, resectively. C is the rotation Mueller matrix and is given by: 0 C() = cos 2 ; sin sin 2 cos C A : (0) R and T are the reflection Mueller matrix and the transmission Mueller matrix, resectively, which are reresented as follows: 0 R = T = 0 (R k + R? )=2 (R k ; R? )=2 0 0 (R k ; R? )=2 (R k + R? )= Rk R? Rk R? (T k + T? )=2 (T k ; T? )=2 0 0 (T k ; T? )=2 (T k + T? )= Tk T? Tk T? C A () C A : (2) However, if the total reflection occurs, that is, if the incidence angle is larger than critical angle C, then R and T are set to be identity matrix and zero matrix, resectively. D is the retardation Mueller matrix and is given as: 0 D() = cos sin 0 0 ; sin cos C A (3) where is the amount of the hase shift (or retardation). The hase of the reflected light shifts when the total reflection occurs. Thus, for the total reflection, in the following equation is used. tan 2 = cos sin 2 ; n 2 : (4) sin 2 The hase of the reflected light inverts when the incidence angle is smaller than the Brewster angle B, which is defined as follows: tan B = n : (5) Thus, the value of is set as follows: 8 < Eq:(4) = : 2.4. Degree of Polarization C 80 B 0 otherwise : (6) The olarization state of the light is calculated by observing the object with a monochrome camera, which has a linear olarizer in the front. For a certain ixel, we denote the maximum intensity observed by rotating the olarizer as I max and the minimum as I min. The angle of the olarizer when the minimum intensity I min is observed is called the hase angle. This angle is defined as the angle from +x axis to +y axis in xyz coordinates (Figure 3). Because the linear olarizer is used in this research, the fourth arameter s 3 of the Stokes vector cannot be determined. The relationshi between the Stokes vector (s s s 2 ) T and I max, I min, is: 0 0 s 0 s A cos 2 ; sin 2 A@ I max + I min I max ; I min A : (7) 0 sin 2 cos 2 0 s 2

6 The degree of olarization (DOP) reresents how much the light is olarized and is defined as follows: s 2 ^ = + s2 2 + s2 3 : (8) s 0 However, linear olarizer can only calculate the following degenerated DOP: = I max ; I min s 2 = + s2 2 : (9) I max + I min s 0 For the remainder of this aer, we refer to the ratio calculated by Equation (9) as DOP Illumination Distribution In this aer, we assume that all light sources are unolarized. We also assume that the front surface of the object is uniformly illuminated with the same intensity in every direction, and that the back surface of the object is also uniformly illuminated with the same intensity in every direction but with a different intensity from the intensity that illuminates the front surface. 3. Inverse Polarization Raytracing In this section, we introduce our method for estimating the front surface shae of a transarent object using the DOP and the hase angle as inuts under the assumtion that the refractive index, the shae of the back surface, and the illumination distribution are given. Details of numerical algorithms are shown in the literature [36]. We denote the inut olarization data as I E. Polarization data are reresented as an image (2-dimensionally distributed data) where the DOP and hase angle are set for each ixel. The olarization raytracing exlained in Section 2 can render the olarization data from the shae of the transarent object by tracing the light ray and by Mueller calculus. We denote this rendered olarization image as I R. The shae of transarent objects is reresented as the height H, set for each ixel. Heights artially differentiated by x and y are called gradients, and are reresented as and q, resectively: = H q = H : (20) Surface normal n =(; ;q ) T is reresented by these gradients. The rendered olarization image I R deends uon height and surface normal, so it can be reresented as I R (H q). Our roblem is finding the best values to reconstruct a surface H that satisfies the following equation: I E = I R (H q) : (2) We call this equation the olarization raytracing equation from the analogy of image irradiance equation used in the shae-from-shading roblem. A straightforward definition of the cost function, which we want to minimize, can be as follows: ZZ E (x y)dxdy (22) where, E =(I E ; I R (H q)) 2 : (23) We will sometimes omit the variables (x y) in the subsequent discussions for the simlicity of descritions. I R deends uon, q, and H, while, q, and H deend uon each other with Equation (20). Thus, the cost function must be modified as follows: ZZ (E + E 2 ) dxdy (24) where, E 2 =(H x ; ) 2 +(H y ; q) 2 : (25) is a Lagrange undetermined multilier. Euler equations that minimize Equation (24) are derived as follows: = H x (26) q = H y (27) H = H ; 4 ( x + q y ) (28) where H is a 4-neighbor average of H. Each of the above equations can be decomosed into two stes: (k) = H (k) x (29) (k+) = (k) ; (30) q (k) = H (k) y (3) q (k+) = q (k) ; (k) H (k+) = H (k) ; 4 (k+) x + q (k+) y (32) (33) H () = H () ; () : (34) Here,, 2, and 3 are scalar values that are determined for each ixel and for each iteration ste. Suerscrit (k) reresents the iteration number. We do not write down the iteration number for Equation (34) because we do not use this equation due to the following reasons. One reason is that the cost function E deends uon the change of surface normal rather than on the change of height. Another reason is that the cost function E smoothly changes when the surface normal changes, but it does not smoothly change when the height changes. This fact was emirically roved in the reliminary exeriments. The algorithm goes as follows. First, we set initial values of the shae H for each oint of the front surface. Next, and q are calculated by Equations (29)(3). Then, we

7 solve Equations (30)(32). and 2 should be otimal values; thus, we use Brent s method to determine and 2, which minimizes the cost function E. After comuting and q at every ixel, we solve Equation (33) by the relaxation method [37,38] to determine the height H. We use the alternating-direction imlicit method to solve the relaxation roblem. To conclude, the front surface shae of the transarent object is estimated by an iterative comutation, where each ste of iteration solves Equations (29) (33), and the iteration stos when Equation (22) is minimized. There are two reasons why we use Equations (29) (33) instead of Equations (26) (28): () If we solve Equations (26) (28) simultaneously by setting an arbitrary value, a arameter tuning roblem will occur where must be set to an otimal value in order to stably solve these equations; (2) We can aly adequate numerical algorithms for each of Equations (29) (33). 4. Measurement Result 4.. Acquisition System For obtaining olarization data, we develoed an acquisition system, which we named Cocoon (Figure 4). The target object is set inside the center of the lastic shere whose diameter is 35cm. This lastic shere is illuminated by 36 incandescent lams. These 36 light sources are almost uniformly distributed satially around the lastic shere. The lastic shere diffuses the light that comes from the light sources, and it behaves as a sherical light source, which illuminates the target object from every direction. Note that the measurement is ossible for arbitrary illumination conditions though the intensity and the olarization state of the illumination distribution must be known. The target object is observed by monochrome camera from the to of the lastic shere, which has a hole on the to. Linear olarizer is set in front of the camera. We ut the target object on the black ie to make the incoming light from the back surface uniform and unolarized. The camera, object, and light sources are fixed. From four images taken by rotating the olarizer at 0, 45, 90, and 35,we calculate I max, I min, and (Section 2.4) Rendering Result Before estimating the shae of the transarent object, we analyze the rendered image of forward olarization raytracing (Section 2). From the sherical art, we observe a transarent acrylic hemishere, whose refractive index is.5 and diameter is 3cm. Obtained olarization image of the real object is shown in Figure 5(a). The figure reresents the DOP, where DOP 0 and DOP are reresented as black and white, resectively. The rendered olarization image of olarization raytracing is shown in Figure 5(b). The refractive index and the Figure 4: Acquisition System Cocoon. Figure 5: DOP image; (a) obtained from real object, (b) rendered by olarization raytracing, and (c) rendered by assuming that the internal interreflection does not occur. shae of the object are known. As for the illumination distribution, we have to obtain the ratio of the intensity of the light illuminating the front surface to that illuminating the back surface. Unfortunately, it is imossible to observe the light illuminating the back surface without moving the object out of the way. Therefore, we find the most aroriate value of the intensity of the back surface where the difference of the obtained DOP (Figure 5(a)) and the rendered DOP (Figure 5(b)) minimizes, by solving such minimization roblem. Figure 5(b) is calculated by using the intensity obtained from such minimization. For comarison, a rendered image with no interreflection is shown in Figure 5(c). This DOP image is rendered by assuming that the light reflected at the object s surface once is just observed and that the transmission does not occur. The root mean square (RMS) error between real data (Figure 5(a)) and DOP data of no interreflection (Figure 5(c)) was 0.48, while the RMS error between real data and olarization raytracing data (Figure 5(b)) was Simulation Result Here, we will show the result of estimating the 2D shae of a simulated object for evaluating the robustness of our algorithm. This virtual transarent object is a concave shae whose refractive index is.5. The object is reresented as a dotted line in Figure 6. We render the olarization data of

8 Figure 6: Simulation result of concave shae: (a) Initial state, (b)(c) result after 5, 20 loos, resectively. the object observed from the uer osition to the lower direction, and after that, we estimate the front surface shae of the concave object by using the rendered olarization data as inut data. Illumination is distributed uniformly from every direction with the same intensity. The light is not illuminated at the bottom of the shae but is illuminated on the front surface. Illumination distribution, the back surface shae, and the refractive index are given. The estimation result is illustrated in Figure 6. The dotted line is the true shae, and the solid line is the estimated shae. Figure 6(a) indicates the initial value, and Figure 6(b) and Figure 6(c) indicate the results after 5 and 20 loos of the roosed method. The shae, which is generated by scaling the true height by.2, is used as the initial state of the shae. The shae converged to the true shae at 20 loos. The average comutation time was 4.8[sec] for loo with 320 ixels by using Pentium4 3.4GHz. Here, the maximum number of tracings is 00 reflections or transmissions; however, if the energy of the light ray becomes less than a certain threshold, the tracing of the light ray is stoed Measurement Results of Real Object Hemishere For the first measurement result, we observe an acrylic transarent hemishere from the sherical art, which is also used in the rendering exeriment (Section 4.2). We assume that the refractive index and the back surface shae are known. We use the same value for the intensity of the light source, which is obtained in Section 4.2. The value is obtained by observing the same object as Section 4.2, and we will rovide a more objective result in the next section. The estimation result is shown in Figure 7. Figure 7(a) reresents the result of the revious method [6 8] and, at the same time, it reresents the intial value. Figure 7(b) is the result after 0 loos of our method. The average comutation time was 36[sec] for loo with 7,854 ixels. Here, the maximum number of tracings is 0 reflections or transmissions. More detailed evaluation is done in the 2D lane that is a cross section of the 3D object, which includes the center of the base circle and the line erendicular to that circle. A light ray that is inside this lane does not go out, and a light ray that is outside this lane does not come in. The roosed algorithm estimates the front surface shae, a semicircle, by Figure 7: Estimation result of hemishere: (a) Initial state (result of revious method), (b) result after 0 loos. Figure 8: Estimation result: (a) Initial state (result of revious method), (b)(c) results after 5 and 50 loos, (2a) initial state (true shae), (2b)(2c) results after 5 and 50 loos. using the olarization data of the 2D lane as inut data. The result of alying the roosed method is given in Figure 8(c) and Figure 8(2c). In Figure 8, the solid line reresents the estimated shae, and the dotted line reresents the true shae. For the estimated result shown in Figure 8(c), the result of the revious method (Figure 8(a)) is used for the initial state of the shae. For the estimated result shown in Figure 8(2c), the true shae, hemishere (Figure 8(2a)), is used for the initial state of the shae. Figures 8(b)(2b) and Figures 8(c)(2c) are the result after 5 and 50 loos, resectively. The shaes converge to the same shae even if the initial shaes are different. The reason why there is a rotruding noise at the to of the estimated shae might be the failure of considering the hole on to of the lastic shere. Although we have assumed that the object is illuminated uniformly with the same intensity from all directions, we now believe that this assumtion might not strictly hold for this hole. Therefore, considering such illumination distribution will be our future work. The value of the cost function (Equation (22)) er each iteration is lotted in Figure 9. The vertical axis in Figure 9 reresents the value of Equation (22), while the horizontal axis reresents the iteration number. A diamond mark is the value of the result whose initial state is the result of the revious method (Figures 8(a)(b)(c)). A square mark is the value of the result whose initial state is the true shae (Figures 8(2a)(2b)(2c)). The leftmost value is the value of the cost function of the initial state. Both the value and the shae did not change after around 8 loos. The average comutation time was 5.9[sec] for loo with 320 ixels.

9 (a) (b) (c) Figure : Estimation result: (a) Initial value, (b)(c) result after 5, 20 loos. Figure 9: Error for each loo: (square) Result when true shae is initial value, (diamond) result when result of revious method is initial value. Figure 0: Bell-shaed transarent acrylic real object. The RMS error between the estimated value and the true value is used to comare the accuracy between the roosed method and the revious method. The RMS error of the surface normal was 23:3 for revious method, 9:09 for our method when the initial state was the result of the revious method, and 8:86 for our method when the initial state was the true shae. The RMS error of the height was 2.70mm for the revious method, 0.672mm for our method when the initial state was the result of revious method, and 0.548mm for our method when the initial state was the true shae Bell-shaed Object Finally, we observe the transarent object shown in Figure 0. This object is made of acrylic and is a body-ofrevolution. Its refractive index is.5 and its diameter of the base is 24mm. The object is observed from the rojected area of the object. The front surface is a curved surface and the back surface is a disk. The camera is set orthogonally to the disk. We assume that the refractive index and the back surface shae are known. We use the same value for the intensity of the light source that is obtained in Section 4.2. This aer only concentrates on roosing a method to estimate the shaes of transarent objects, and obtaining the correct illumination distribution will be a future work. We estimate the shae of a cross-section of the object to analyze the recision of the roosed method. The crosssection includes the center of the base circle and the line erendicular to that circle. Figure (c) illustrates the estimated shae of the object. The solid curve reresents the obtained front height, and the dotted line reresents the given back height. The initial value is set to be a semi- circle shown in Figure (a). The estimated shae after 5 and 20 loos is illustrated in Figure (b) and Figure (c), resectively. RMS of the height was 0.24mm, where the true shae was obtained from the silhouette extracted manually by a human oerator from the hotograh of the object taken from the side. The average comutation time was 7.0[sec] for loo with 320 ixels. 5. Conclusion In this aer, we have roosed a novel method for estimating the surface shae of transarent objects by minimizing the difference between the inut olarization data taken by observing the transarent object and the comuted olarization data rendered by the olarization raytracing method. We estimated the shae of transarent objects by an iterative comutation. We used a uniform illumination in this aer; however, Hata et al. [0] estimated the shae of transarent objects by an iterative comutation where the object was illuminated by structured light. Ben-Ezra and Nayar [2] estimated the shae of a transarent object observed from many viewoints by an iterative comutation. To imrove the recision of measuring the surface shae of transarent objects, we should robably observe the target object from multile viewoints or under various tyes of illumination. In any case, the iterative comutation is considered to be necessary. Our aer rovides the technique for measuring the surface shae of transarent objects using iterative comutation, and this technique might be used as the basis for further develoments. Most of the artificial transarent objects have a lanar base that enables them to stand by themselves. Also, the material (refractive index) of the artificial transarent objects is known in many cases. Thus, the assumtion we adoted in this aer, back surface shae and refractive index are known, is effective in many cases. However, not all objects meet these conditions; thus, we intend to develo a method that can measure the back surface shae and refractive index at the same time as well as the front surface shae. Acknowledgments This research was suorted in art by the Ministry of Education, Culture, Sorts, Science and Technology. Daisuke Miyazaki was suorted by the Jaan Society for the Promotion of Science. The authors thank Joan Kna

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