Mirror Localization for a Catadioptric Imaging System by Projecting Parallel Lights

Transcription

1 2007 IEEE International Conference on Robotics and Autoation Roa, Italy, April 2007 FrC2.5 Mirror Localization for a Catadioptric Iaging Syste by Projecting Parallel Lights Ryusuke Sagawa, Nobuya Aoki, Yasuhiro Mukaigawa, Toio Echigo, Yasushi Yagi Abstract This paper describes a ethod of irror localization to calibrate a catadioptric iaging syste. Even though the calibration of a catadioptric syste includes the estiation of various paraeters, in this paper we focus on the localization of the irror. Since soe previously proposed ethods assue a single view point syste, they have strong restrictions on the position and shape of the irror. We propose a ethod that uses parallel lights to siplify the geoetry of projection for estiating the position of the irror, thereby not restricting the position or shape of the irror. Further, we oit the translation process between the caera and calibration objects fro the paraeters to be estiated by observing soe parallel lights fro a different direction. We obtain the constraints on the projection and copute the error between the odel of the irror and the easureents. The position of the irror is estiated by iniizing the error. We also test our ethod by siulation and real experients, and finally we evaluate the accuracy of our ethod. I. INTRODUCTION Catadioptric iaging systes are often used to obtain various fields of view by observing rays reflected fro irrors. In particular, catadioptric onidirectional iaging systes [1], [2], [3] are widely used in various applications, such as robot navigation, surveillance, and virtual reality. There are two types of catadioptric iaging systes: central and noncentral. The forer has a single effective viewpoint, and the latter has ultiple viewpoints. Although central catadioptric systes have the advantageous feature that the iage can be transfored to a perspective projection iage, they have strong restrictions on the shape and position of the irror. For exaple, it is necessary to use a telecentric caera and a parabolic irror whose axis is aligned to the axis of the caera. Thus, a catadioptric syste ay not be a central due to possible isconfiguration. Several noncentral systes [4], [5], [6], [7] have been proposed for various purposes. For geoetric analysis with catadioptric systes, it is necessary to calibrate both the caera and irror paraeters. Several ethods of calibration have been proposed for central catadioptric systes. Geyer and Daniilidis [8] used three lines to estiate the focal length, irror center, etc. Ying and Hu [9] used lines and spheres to calibrate the R. Sagawa, N. Aoki, Y. Mukaigawa, and Y. Yagi are with the Institute of Scientific and Industrial Research, Osaka University, 8-1 Mihogaoka, Ibaraki-shi, Osaka, , JAPAN T. Echigo is with the Dept. of Engineering Inforatics, Faculty of Inforation and Counication Engineering, Osaka Electro- Counication University, 18-8 Hatsu-cho, Neyagawa, Osaka, , Japan echigo@a.sanken.osaka-u.ac.jp paraeters. However, since these ethods assue that the syste has a single viewpoint, they cannot be applied to noncentral systes. On the other hand, several ethods have also been proposed to calibrate noncentral iaging systes. Aliaga [10] estiated the paraeters of a catadioptric syste with a perspective caera and a parabolic irror using known 3D points. Strelow et al.[11] estiated the position of a isaligned irror using known 3D points. Micusík and Pajdla [12] fitted an ellipse to the contour of the irror and calibrated a noncentral caera by approxiating it to a central caera. Mashita et al. [13] used the boundary of a hyperboloidal irror to estiate the position of a isaligned irror. However, these ethods are restricted to onidirectional catadioptric systes. There are also soe approaches for calibrating ore general iaging systes. Swainathan et al. [14] coputed the paraeters of noncentral catadioptric systes by estiating a caustic surface fro known caera otion and the point correspondences of unknown scene points. Grossberg and Nayar [15] proposed a general iaging odel and coputed the ray direction for each pixel using two planes. Stur and Raalinga [16] calibrated the caera of a general iaging odel by using unknown caera otion and a known object. Pless [17] estiated the position of a ulticaera syste based on structure fro otion. Since these ethods estiate both the internal and external paraeters of the syste, the error of easureent affects the estiated result of all of the paraeters. In this paper, we focus on the localization of the irror in the calibration of catadioptric systes. The other paraeters areassuedtobeasfollows: The internal paraeters, such as the focal length and principal point of a caera, are known. The shape of the irror is known. Translation to the world coordinate syste is oitted fro the calibration. When we create a irror to use in a catadioptric syste, we know the shape of the irror with the accuracy of its cutting or olding achine. Therefore the knowledge of the irror shape is not a strong assuption. By oitting translation fro the calibration, we reduce the nuber of paraeters that are affected by the easureent error in the calibration. To oit translation fro the calibration, our ethod uses parallel lights as calibration patterns. Since a parallel light is regarded as a ray projected fro an object at infinite distance, such as the sun, we do not have to consider the translation fro a calibration pattern to the caera. In Section II, we describe the difference of the geoetry of projection of /07/$ IEEE. 3957

2 a point light source and a parallel light. Next, in Section III, we propose an algorith for irror localization using parallel lights. We test our ethod in Section IV and finally suarize this paper in Section V. Mirror surface x x Parallel light v II. GEOMETRY OF PROJECTION In this section, we explain the difference between the projection of a point light source and that of a parallel light. A. Projecting A Point Light Source If an object or feature point is projected to an iaging syste, it is regarded as the projection of a point light source at a finite distance. Thus, the calibration object is usually considered to be a point light source. The 3D position of the point light source is p. The ray fro the point light source is reflected by the irror at x. The reflected ray arrives at the viewpoint of caera O through iage point. Since the reflection angle at a irror is equal to the incident angle, p x p x = an R,t (x), (1) where N R,t (x) is the noral vector of the irror surface at point x. R and t are the rotation and translation of the irror, respectively, and a is a scale factor. By reoving scale factor a fro (1), we obtain two equations. We consider the following three cases of unknown paraeters: 1) The 3D positions of points are copletely unknown. 2) The relative positions between points are known, while their relative positions to the caera are unknown. 3) Their relative positions to the caera are known. The first case corresponds to structure fro otion that siultaneously estiates the position of the caera and the position of 3D points. If position p is unknown, then the paraeters to be estiated are R, t, R C, t C and p of each point, where R C and t C are the rotation and translation paraeters, respectively, of the caera for each iage. When n points are observed by k iages by changing viewpoints, the nuber of paraeters is 6+6k+3n, andthe nuber of equations is 2kn. Thus, k and n should satisfy 6+6k +3n 2kn. The iniu nuber of paraeters is 48 if k =3,n =8or k =4,andn =6. In this case, therefore, even though the paraeters that we really want to obtain are R and t only, the nuber of paraeters to be estiated is too great. The second case corresponds to calibration using a structured calibration object, such as a checker board, lines, and circles. If the relative positions of the 3D points are known, then the paraeters to be estiated are R, t, R C,andt C. Though the nuber of paraeters is reduced to 6+6k,where k is the nuber of iages, the nuber of constraints varies according to both the calibration object and the projection odel. Further, though the iniu nuber of paraeters is 12, since we do not assue single viewpoint projection odels, the actual nuber becoes larger than 12 in order to obtain constraints using a calibration object for general catadioptric iaging systes. Fig. 1. Iage plane O Caera N R,t (x) Projecting a parallel light to a catadioptric iaging syste. In the third case, the positions of the 3D points relative to the caera origin are easured a priori. If the position of point light source p is known, that is, if the position of the calibration object is known in the caera coordinate syste, then the paraeters to be estiated are only R and t. The nuber of paraeters is therefore reduced to 6. However, position p is difficult to easure in an actual case since it is in a position relative to caera origin O, which is unknown in the world coordinate syste. Hence, the accuracy of this approach is poor. B. Projecting a Parallel Light In this section, we explain the geoetry of projecting a parallel light and the reduction of paraeters to be estiated. Figure 1 shows the projection of a parallel light. The difference here copared to projecting a point light source is that the light source becoes a parallel light whose direction is v. Since a parallel light is equal to a light fro a distant point light source, v is regarded as sp x v = li s sp x. (2) Thus, the equation of projection becoes v = an R,t (x), (3) where N R,t (x) is the noral vector of the irror surface at point x. R and t are the rotation and translation of the irror, and a is a scale factor. Siilar to (1), we obtain two equations by reoving scale factor a fro (3). If we use a parallel light, the translation fro the light source to the caera can be reoved fro the paraeters. Since vector v is a relative direction in the caera coordinate syste, v actually consists of two paraeters. Therefore, when n different parallel lights are observed by k iages by rotating the caera, the nuber of paraeters is 6+3k+2n. A ethod we propose in Section III, which estiates the position of the irror, uses turntables to rotate the caera. While the translation is difficult to easure, the rotation paraeter of a turntable is accurately easured by using an encoder. Because the relative directions between parallel lights of different rotations are known, the other unknown paraeters are the rotation of the caera and the direction 3958

3 Turntable φ Mirror Caera Parallel light Turntable θ Fig. 2. Calibration syste using parallel lights: The caera and irror is rotated by two turntables. of a parallel light. Therefore, the nuber of paraeters is reduced to Moreover, our ethod independently coputes the direction of the parallel light fro an estiation of the irror position. Therefore, we separate the proble into two steps. The first estiates the direction of the parallel light and the second estiates the irror position. The nubers of paraeters for the two steps are 2 and 9. This is uch saller than that in the case of point light sources. III. MIRROR LOCALIZATION USING PARALLEL LIGHTS This section describes an algorith to estiate irror position. Our ethod consists of two steps: 1) Estiating the direction of a parallel light 2) Estiating the irror position We estiate the direction of a parallel light and the irror position separately. Figure 2 shows our calibration syste. Since the caera and irror are ounted on two turntables and rotated by the, they rotate around two axes that are perpendicular to each other. Thus, the relationship between the caera and the irror does not change with the rotation. To project a parallel light, we use a distant point object or a concave parabolic irror to generate a parallel light. If we use a parabolic irror, a point light source is set at the focal point of the parabolic irror, and the reflected light becoes a parallel light. The catadioptric iaging syste obtains parallel lights fro various directions while rotating. A. Estiating Parallel Light Direction When a rotating caera acquires iages of the parallel light, soe projected points are in the sae position even if the rotation paraeters are different. Naely, if two iages are obtained with the rotation paraeters of turntables R T1 and R T2 (R T1 R T2 ), then the projected points 1 and 2 ay becoe 1 = 2. This is because the rotation of the caera in this syste has abiguity concerning the rotation around the ray vector. In an actual experient, we findsuchapair,t1 and T2, by rotating the turntables. If 1 = 2, then the ray vector in the caera coordinate syste v is the sae for each rotation. Thus, R C R T1 v 0 = R C R T2 v 0, (4) where the ray vector in the world coordinate syste is v 0 and R C is the rotation of the caera relative to the turntables. Since a parallel light is projected, the translation fro the center point of rotation to the caera origin is oitted fro the equation. By odifying (4) as (R T1 R T2 )v 0 =0, (5) v 0 is coputed as the eigenvector of M T M associated with the sallest eigenvalue, where M = R T1 R T2. Though v 0 can be coputed by a pair of R T1 and R T2, if we obtain soe pairs of projected points, we can create M by stacking all rows of R T1 R T2 and solve Mv 0 =0siilarly. Our ethod observes the projected point of a parallel light by rotating the caera, and it finds the projected point where the parallel light is projected with different rotation paraeters. If the rotation atrices of two turntables are R θ and R φ, respectively, then a rotation paraeter is coputed as R T = R θ R φ. (6) The siplest way to copute the light direction is to set the incoing light parallel to the axis of turntable θ or φ. In this case, v 0 is easily obtained as the axis vector. If we use a distant point object as a parallel light source, we can find a point that does not ove in the iage while rotating by turntable θ or φ, and then use that point as the projection of a parallel light. B. Estiating Mirror Position We estiate the irror position by iniizing the following functional: N R,t (x) n 2, (7) where n is a noralized vector of v /, and N R,t (x) = 1. Since iniizing (7) is a nonlinear iniization proble, we estiate R, t, andr C by the Levenberg-Marquardt algorith. Our algorith becoes : 1) Set the initial paraeters of R, t, andr C. 2) Copute intersecting point x for each iage point. 3) Copute the error function (7). 4) Update R, t, andr C by the Levenberg-Marquardt algorith. 5) Repeat steps 2-4 until convergence. We explain coputing n and N R,t (x) in the rest of this section. Once the direction of parallel light v 0 is coputed, the ray vector v for each rotation paraeter R T in the caera coordinate syste is coputed as v = R C R T v 0. (8) Since we assue that the internal paraeters of the caera are known, / is coputed fro the projected pixel of 3959

4 TABLE I ERROR OF ESTIMATED PARALLEL LIGHT DIRECTION: THE MEAN ANGULAR ERROR AND THE STANDARD DEVIATION (STD.) OF THE ERROR (DEGREE). σ 1pair 6pairs (pixels) Mean Std. Mean Std the parallel light. Thus, n is coputed fro the observation of the parallel light and turntables. Since x is the intersecting point of viewing vector and the irror surface, noral vector N R,t (x) is represented as a function of R, t, and. Because we do not assue the shape of the irror, it is generally difficult to copute (x) by algebraic anipulation. Therefore, we copute N R,t N R,t (x) nuerically in this paper. To accoodate any irror shape, we approxiate the irror shape by a esh odel. Intersecting point x is coputed by projecting the esh odel onto the iage plane of the caera with R, t, and the internal paraeters of the caera. N R,t (x) is coputed as a noral vector of the esh odel at x. IV. EXPERIMENTS A. Estiating Accuracy by Siulation We first evaluate the accuracy of our ethod by siulation. In this siulation, we estiate the position of a parabolic irror relative to a caera. Since our ethod consists of two steps, we evaluate each step separately. For the first step, we easure the accuracy of the estiated parallel light direction. In this experient, the caera is a perspective caera that has a 40 field of view. The size of the iage is pixels. Radius h of the parabolic irror is 2.5, where the shape is defined as z = 1 2h r2, (r 2 = x 2 + y 2 ). The diaeter and height of the irror are 10 and 5, respectively. A esh odel of the irror is created by approxiating a paraboloid with triangles of length 0.05 on a side. The irror is shifted by (0.3, 0.6, 18.0) along each axis and rotated by (1.2, 0.8, 0.0) (degrees) around each axis. We easured the error of the estiated parallel light direction by adding a noise on the projected position of the parallel light in an iage. Table I shows the angular error of the estiated parallel light direction. We added a Gaussian noise of N(0,σ 2 ) to the projected position. The angular error is the angle between the vector of ground truth and the estiated result. We copared the results coputed using 1 pair and 6 pairs of projected points and found that the error can be reduced if we use 6 pairs. For the second step, we estiate the accuracy of the estiated irror position. In this experient, the caera is a perspective caera with an iage of pixels and focal length of 900 pixels. Radius h of the parabolic irror is 9.0. The diaeter and height of the irror are and 9.0, respectively. The irror is shifted by (0.0, 0.0, 50.0) along each axis. Thus, this catadioptric iaging syste is not a single viewpoint syste. To validate the accuracy of our ethod, we copared the three ethods by changing the nuber of paraeters, as shown in Table II. Since Method 1 and 2 use point light sources as feature points, they ust estiate the caera positions together with the irror position. Method 2 also estiates the 3D positions of the feature points, while Method 1 assues that the relative positions of the feature points are known. Therefore, Method 1 corresponds to calibration using a structured calibration object, and Method 2 corresponds to structure fro otion. In Methods 1 and 2, we iniized (7) by changing n to p x p x p x p x (9), where p = R C p + t C,andR C and t C are the translation and rotation paraeters, respectively, of the caera for each iage. In Method 2, the positions of 3D feature points p are also paraeters. We easured the error of the estiated irror position by adding Gaussian noises on the projected positions of the features, which were σ =0.0, 0.1, 0.5, 1.0 (pixels). For our ethod, we also added noises on the parallel light direction to copute the irror position. We use the ean error shown in Table I as the agnitude of noise when the light direction is estiated by 6 pairs of projected points. We assue that an initial value is known. Figure 3 shows the results of the error of the estiated irror position. We tested our ethod with 6 and 24 feature points to change the nuber of the constraints. We copute the RMS error of the translation between the ground truth and the estiated position in Figure 3(a). In Figure 3(b), we estiate the reprojected points of the incoing lights using the estiated position of the irror. Since the error of the proposed ethod is one-half and onequarter of that of Method 1 and Method 2, respectively, the result of our ethod is better than that of other ethods. B. Localizing Mirrors fro Real Iages To localize a irror fro real iages, we created an experiental calibration syste with two turntables, as shown in Figure 4. A caera and ultiple irrors are ounted on the syste and rotated by the turntables. In this experient, we coputed the irror positions of a catadioptric syste with copound parabolic irrors, which has been proposed in [18], [19], [20]. Figure 5 shows an exaple of the iage. The syste has 7 parabolic irrors, and the caera is a perspective caera. The caera is a PointGrey Scorpion, which has pixels and about a 22.6 field of view. The distortion of the lens is calibrated by the ethod [21], and the internal paraeters of the caera are coputed using OpenCV [22] as a preprocessing of calibrating the irrors. In this setup, the catadioptric syste is not a single viewpoint syste. The radii h of the center irror and side irrors are 9.0 and 4.5, respectively. The diaeter and height of the center irror are and 9.0, respectively, and the diaeter and height of the side irrors are 13.0 and 3960

5 TABLE II COMPARISON OF THREE METHODS FOR ESTIMATING THE MIRROR POSITION. METHODS 1 AND 2 USE POINT LIGHT SOURCES AS FEATURE POINTS. IN METHOD 1 THE RELATIVE POSITIONS BETWEEN POINTS ARE KNOWN. IN METHOD 2 THE POSITIONS OF POINTS ARE COMPLETELY UNKNOWN. Method # of irror paraeters # of caera positions # of features # of external paraeters # of para. of features total # of paraeters total # of constraints Our ethod Method Method Proposed Method 6 points σ=0.0 Proposed Method 6 points σ=0.1 Proposed Method 6 points σ=0.5 Proposed Method 24 points σ=0.0 Proposed Method 24 points σ=0.1 Proposed Method 24 points σ=0.5 Method1 Method Proposed Method 6 points σ=0.0 Proposed Method 6 points σ=0.1 Proposed Method 6 points σ=0.5 Proposed Method 24 points σ=0.0 Proposed Method 24 points σ=0.1 Proposed Method 24 points σ=0.5 Method1 Method2 RMS Error () RMS Error (pixels) Standard deviation of noise (pixels) Fig. 3. (a) Standard deviation of noise (pixels) Error in estiating the irror position: (a) RMS error of the translation. (b) RMS error of the reprojected points. (b) 4.5, respectively. The diaeters of the center and side irrors projected onto the iage are 840 and 450 pixels, respectively. We used a distant point as a parallel light source. We first chose a point in the iage that does not ove while rotating Turntable 2. Figure 6 shows the chosen point, which is a point of a building that is about 260 eters away fro the caera. Then, we rotated the two turntables and found the point in the iages anually. We obtained 13 lights fro various directions. We estiated the positions of the center and four side irrors independently. Table III shows the estiated results. Since soe of lights are occluded by the other irrors, the nuber of lights used for calibration varies depending on the irror positions. The side irrors are designed to be put at the Designed position relative the center irror. The Estiated positions are the positions relative to the caera coputed by the proposed ethod. The Relative positions are the estiated positions relative to the center irror. We copared the Designed and Relative positions. The errors along the x- and y-axes, which are parallel to the iage plane, were less than 1. Though the errors along the z- axis are about 1, we believe that this is because the field of view of the caera is narrow. To evaluate the accuracy in the iage space, we reprojected the lights to the iage using the estiated irror positions. The Reprojection errors are the distances of the reprojected points fro the input points. Since we chose the input points anually, the error of an input point is about 1 pixel. Therefore, it is reasonable to Fig. 4. Mirrors Caera Turntable2 Turntable1 Experiental calibration syste with two turntables. conclude that the reprojection error is about 1 pixel, and therefore that the proposed ethod works well. V. CONCLUSION This paper described a ethod of irror localization to calibrate a catadioptric iaging syste. In particular, we focused on the localization of the irror. Our ethod used parallel lights to siplify the geoetry of the projection. Since translation between the caera and the calibration objects is oitted fro the paraeters, the nuber of paraeters to be estiated is reduced. By observing soe parallel lights fro different directions, we could obtain the constraints on projection and copute the error between the odel of the irror and the easureents. Our ethod separately estiates the parallel light direction and the irror 3961

6 TABLE III THE ESTIMATED MIRROR RESULTS: THE SIDE MIRRORS ARE DESIGNED TO BE PUT AT THE DESIGNED POSITION RELATIVE THE CENTER MIRROR. THE ESTIMATED POSITIONS ARE THE POSITIONS RELATIVE TO THE CAMERA COMPUTED BY THE PROPOSED METHOD. THE RELATIVE POSITIONS ARE THE ESTIMATED POSITIONS RELATIVE TO THE CENTER MIRROR. WE COMPARED THE DESIGNED AND RELATIVE POSITIONS TO COMPUTE THE POSITION ERRORS. THE REPROJECTION ERRORS ARE THE DISTANCES OF THE REPROJECTED POINTS FROM THE INPUT POINTS USING THE ESTIMATED MIRROR POSITIONS. Mirror Nuber Position () Error of Lights Designed Estiated Relative Position () Reprojection (pixels) Center 13 (0, 0, 0) (-0.07, 1.41, 75.40) (0, 0, 0) N.A 2.24 Side1 7 (-15.00, 0, 9.22) (-15.46, 2.02, 85.99) (-15.39, 0.61, 10.59) (-0.39, 0.61, 1.37) 0.92 Side2 10 (-7.50, , 9.22) (-7.79, , 85.10) (-7.72, , 9.70) (-0.22, 0.37, 0.48) 1.67 Side3 10 (7.50, , 9.22) (7.50, , 85.52) (7.57, , 10.12) (0.07, -0.23, 0.90) 1.38 Side4 7 (15.00, 0, 9.22) (15.21, 1.56, 86.02) (15.28, 0.15, 10.62) (0.28, 0.15, 1.40) 0.76 Fig. 5. Side1 Fig. 6. Side2 Center Side3 Side4 Exaple of the iage of copound parabolic irrors. A distant point is used as a parallel light source. position. Finally, to validate the accuracy of our ethod, we tested our ethod in both a siulation and in real experients. In future studies, we will work on autoatically finding the parallel lights with the ai to iprove the accuracy of the irror localization. REFERENCES [1] S. Nayar, Catadioptric onidirectional caera, in Proc. 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