sian distributed,and independent,they would be op Relative timally averaged using the inverse of the variances as Parameters Estimates errors weights [10]:wL =o2n/(+)=a2/(1+a2) fe(pixels】 857.3+1.3 0.2% and wR /()1/(1 +a2),where -0.199±0.002 1% a=VL/VR.Experimentally,we found that this dacm】 16.69±0.02 0.1% choice does not yield very good merged surfaces.It -0.0427±0.0003 makes the noisy areas of one view.interact too sig- 0.7515±0.0003 0.06% 0.6594±0.0004 nificantly with the clean corresponding areas in the other view,degrading the overall final reconstruc- e(degree8】 41.27±0.02 0.006% tion.This happens possibly because the random vari- ables Z and Z are not Gaussian.A heuristic solu- Lamp calibration.Similarly,we collected 10 images of the pencil shadow (like figure 3-top-right)and per- tion to that problem is to use sigmoid functions to formed calibration of the light source on them.See calculate the weights:wL =(1+exp{-BAV))1 section 2.1.Notice that the points b and t were man- and wR (1+exp(BAV))with AV =(V2- ually extracted from the images.Define S.as the co- V)/(V2 +Va)=(a2-1)/(a2 +1).The positive ordinate vector of the light source in the camera frame coefficient B controls the amount of diffusion between The following table summarizes the calibration results the left and the right regions,and should be deter- (refer to figure 5 for notation): mined experimentally.In the limit,as 3 tends to in- finity,merging reduces to a hard decision:Ze=Z if Parameters Estimates Relative VL>VR,and Ze=Z otherwise.Our merging tech- errors nique presents two advantages:(a)obtaining more -13.7±0.1 Se (cm) -17.2±0.3 coverage of the scene and (b)reducing the estimation ≈2% -2.9±0.1 noise.Moreover,since we do not move the camera be- 34.04±0.15 0.5% tween scans,we do not have to solve for the difficult hs (cm) E(degrees】 146.0±0.8 0.2% problem of view alignment (11,7,5].One merging Φ(degrees】 64.6±0.2 0.06% example is presented in experiment 3. Independently from local variations in accuracy The estimated lamp height agrees with the manual within one scan,one would also wish to maximize measure(with a ruler）of34±0.5cm. the global (or average)accuracy of reconstruction Our method yields an accuracy of approximately throughout the entire scene.In this paper,scanning is 3 mm (in standard deviation)in localizing the light vertical (shadow parallel to the y axis of the image). source.This accuracy is sufficient for final shape re- Therefore,the average relative depth error loz/Zel covery without significant deformation,as we discuss is inversely proportional to cos(see 3).The two in the next section. best values for the azimuth angle are then 0 and 4.2 Scene reconstructions =r corresponding to the lamp standing either to On the first scene (figure 6),we evaluated the accu- the right (=0)or to the left (=of the camera racy of reconstruction based on the sizes and shapes (see figure 5-top) of the plane at the bottom left corner and the corner object on the top of the scene (see figure 4-top). 4.Experimental Results Planarity of the plane:We fit a plane across the 4.1 Calibration accuracies points lying on the planar patch and estimated the standard deviation of the set of residual distances Camera calibration.For a given setup,we ac- of the points to the plane to 0.23 mm.This cor- quired 10 images of the checkerboard (see figure 1) responds to the granularity (or roughness)noise on and performed independent calibrations on them.The the planar surface.The fit was done over a sur- checkerboard consisted of approximately 90 visible face patch of approximate size 4 cm x 6 cm.This corners on a 8x9 grid.Then,we computed both mean leads to a relative non planarity of approximately values and standard deviations of all the parameters 0.23mm/5cm =0.4%.To check for possible global independently:the focal length fe,radial distortion deformations due to errors in calibration,we also fit factor ke and desk plane position IIa.Regarding the a quadratic patch across those points.We noticed desk plane position,it is convenient to look at the a decrease of approximately 6%in residual standard height da and the surface normal vector nd of II ex- deviation after quadratic warping.This leads us to pressed in the camera reference frame.An additional believe that global geometric deformations are negli- geometrical quantity related to nd is the tilt angle 6 gible compared to local surface noise.In other words (see figure 5).The following table summarizes the cal- one may assume that the errors of calibration do not ibration results (notice that the relative error on the induce significant global deformations on the final re- angle 6 is computed referring to 360 degrees): construction. 48sian distributed, and independent, they would be op￾timally averaged using the inverse of the variances as weights [lo]: WL = c&/(& -t riR) = a2/(1 + cx2) and WR = aiL/(uiL + giR) = 1/(1 + a2), where Q: = V'/VR, Experimentally, we found that this choice does not yield very good merged surfaces. It makes the noisy areas of one view interact too sig￾nificantly with the clean corresponding areas in the other view, degrading the overall final reconstruc￾tion. This happens possibly because the random vari￾ables 2: and 2: are not Gaussian. A heuristic solu￾tion to that problem is to use sigmoid functions to calculate the weights: WL = (1 + exp {-/?AV})-' , and WR = (1 + exp{PAV})-' with AV = (V," - lg)/(V," + Vi) = (a2 - 1)/(a2 + 1). The positive coefficient /? controls the amount of diffusion between the left and the right regions, and should be deter￾mined experimentally. In the limit, as ,8 tends to in￾finity, merging reduces to a hard decision: 2, = 2," if V' > V', and 2, I= 2: otherwise. Our merging tech￾nique presents two advantages: (a) obtaining more coverage of the scene and (b) reducing the estimation noise. Moreover, since we do not move the camera be￾tween scans, we do not have to solve for the difficult problem of view alignment [ll, 7, 51. One merging example is presented in experiment 3. Independently from local variations in accuracy within one scan, one would also wish to maximize the global (or average) accuracy of reconstruction throughout the entire scene. In this paper, scanning is vertical (shadow parallel to the y axis of the image). Therefore, the average relative depth error lczc /Z,I is inversely proportiQna1 to 1 cosEl (see [3]). The two best values for the azimuth angle are then E = 0 and < = T corresponding to the lamp standing either to the right (< = 0) or to the left (5 = T) of the camera (see figure 5-top). Parameters 4 ,Experimental Results 4.1 Calibration accuracies Camera calibration. For a given setup, we ac￾quired 10 images of the checkerboard (see figure l), and performed independent calibrations on them. The checkerboard consisted of approximately 90 visible corners on a 8 x 9 grid. Then, we computed both mean values and standard deviations of all the parametecs independently: the focal length fc, radial distortion factor IC, and desk plane position IId. Regarding the desk plane position, it is convenient to look at the height dd and the surface normal vector Ad of IId ex￾pressed in the camera reference frame. An additional geometrical quantity related to Ad is the tilt angle 0 (see figure 5). The following table summarizes the cal￾ibration results (notice that the relative error on the angle 0 is computed referring to 360 degrees): Relative errors Estimates kc , r f,. fuixels'l II 857.3 & 1.3 I 0.2% I -0.199 i 0.002 1% ' dd (cm) 0.06% 0.006% -0.0427 i 0.0003 0.7515 =k 0.0003 0.6594 !c 0.0004 41.27 & 0.02 - nd 6' (degrees) Lamp calibration. Similarly, we collected 10 images of the pencil shadow (like figure 3-top-right) and per￾formed calibration of the light source on them. See section 2.1. Notice that the points b and 5 were man￾ually extracted from the images. Define s, as the co￾ordinate vector of the light source in the camera frame. The following table summarizes the calibration results (refer to figure 5 for notation): 16&9 f 0.02 0.1% Parameters -13.7 i 0.1 1 3, (cm) 11 ( -17.2i0.3 ) 1 N 2% -2.9 * 0.1 I Relative errors Estimates hs (cm) E (degrees) d (degrees) The estimated lamp height agrees with the manual measure (with a ruler) of 34 & 0.5 cm. Our method yields an accuracy of approximately 3 mm (in standard deviation) in localizing the light source. This accuracy is sufficient for final shape re￾covery without significant deformation, as we discuss in the next section. 4.2 Scene reconstructions On the first scene (figure 6), we evaluated the accu￾racy of reconstruction based on the sizes and shapes of the plane at the bottom left corner and the corner object on the top of the scene (see figure 4-top). Planarity of the plane: We fit a plane across the points lying on the planar patch and estimated the standard deviation of the set of residual distances of the points to the plane to 0.23 mm. This cor￾responds to the granularity (or roughness) noise on the planar surface. The fit was done over a sur￾face patch of approximate size 4 cm x 6 cm. This leads to a relative non planarity of approximately 0.23mm/5cm = 0.4%. To check for possible global deformations due to errors in calibration, we also fit a quadratic patch across those points. We noticed a decrease of approximately 6% in residual standard deviation after quadratic warping. This leads us to believe that global geometric deformations are negli￾gible compared to local surface noise. In other words, one may assume that the errors of calibration do not induce significant global deformations on the final re￾construction. 34.04 ic 0.15 0.5% 146.0 i 0.8 0.2% 64.6 =k 0.2 0.06% 40