case study: 리프로젝션 에러 실험 결과 조사

1. 실험실 환경에서의 결과 
In the case of experiments in a laboratory:  

(1) Yasutaka Furukawa & Jean Ponce, "Accurate Camera Calibration from Multi-View Stereo and Bundle Adjustment", CVPR 2008 
(* 영상의 크기와 리프로젝션 에러 사이의 관계를 수학적으로 설명)

For example, the robot arm (Stanford spherical gantry) used in the multi-view stereo evaluation of [18] has an accuracy of 0.01◦ for a 1m radius sphere observing an object about 15cm in diameter, which yields approximately 1.0[m]×0.01×π/180 = 0.175[mm] errors near an object. Even with the low-resolution 640×480 cameras used in [18], where a pixel covers roughly 0.25mm on the surface of an object, this error corresponds to 0.175/0.25 = 0.7pixels, which is not negligible. If one used a highresolution 4000× 3000 camera, the positioning error would increase to 0.7×4000/640 = 4.4pixels.  

The mean reprojection error decreases from 2-3pixels before refinement to about 0.25 to 0.5 pixels for most datasets after six iterations.  

In practice, we have found PMVS to be robust to errors in camera parameters as long as the image resolution matches the corresponding reprojection errors—that is, when features to be matched are roughly within two pixels of the corresponding 3D points.  

  

(2) Ivo Ihrke & Lukas Ahrenberg & Marcus Magnor, “External camera calibration for synchronized multi-video systems”, Journal of WSCG(the 12th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision), 2004                    

(3) Cheng-I Chen & Yong-Sheng Chen "Central Catadioptric Camera Calibration using Planar Objects", ICVS 2007

(4) Seitz, S. M., Curless, B., Diebel, J., Scharstein, D., and Szeliski, R. 2006. A Comparison and Evaluation of Multi-View Stereo Reconstruction Algorithms, CVPR 2006
: S. Seitz et al. Multi-view stereo evaluation web page at http://vision.middlebury.edu/mview/

Calibration accuracy on these datasets appears to be on the order of a pixel (a pixel spans about 1/4mm on the object). It is difficult to quantify the calibration accuracy because we don't have point correspondences in all views. However, when the images are reprojected onto the laser scanned mesh reconstruction and averaged, 1-2 pixel wide features are clearly visible. Some images and regions of the object are a bit out of focus, due in part to the limited depth of field afforded by the imaging configuration. This is probably a blessing in disguise, as it helps compensate for the lack of sub-pixel registration. While there is certainly room for improvement, we expect that this degree of accuracy will enable good results for most algorithms. Note that the images have been corrected to remove radial distortion.

2. 일반/극한 환경 (실내/실외)에서의 결과 
In the case of experiments in an exterior, interior or extreme circumstance:   



3. 리프로젝션 에러 최소화 관련 방법과 용어 및 문헌
methods or related terms to be used to minimize reprojection errors  

(1) sampson error

(2) LM(Levenberg-marquardt) algorithm
M. Lourakis. levmar: Levenberg-marquardt nonlinear least squares algorithms in C/C++. [web page] http://www.ics.forth.gr/~lourakis/levmar, July 2004.

(3) bundle adjustment

(4) iterative gradient descent (in Bouget's toolbox)


ref.
1) Gregorij Kurillo &  Zeyu Li & Ruzena Bajcsy, "Framework for Hierarchical Calibration of Multi-camera Systems for Teleimmersion", ICIT 2009

Figure 1 shows typical error distribution as obtained on the four cameras within the cluster. After the global optimization on all four cameras, the combined error distribution resembles Gaussian distribution with the mean value of 0.153 pixels and standard deviation of 0.091 pixels. Note that the reprojection error on the color camera (#4) is higher than on the grayscale cameras. The maximal error for this set of images was 0.8 though only a small number of points had errors in that range.

Bundle adjustment minimizes the following reprojection error:

Figure 4 shows the reprojection error on all cameras. The cameras whose position and orientation were obtained by indirect transformation path with the reference camera had no significantly different reprojection errors as compared to the cameras calibrated directly with the reference camera. The mean reprojection error between all the cameras was 0.3633 pixels with the standard deviation of 0.0486 pixels.

2) Kai Ide & Steffen Siering & Thomas Sikora, "Automating Multi-Camera Self-Calibration", WACV 2009

In practice the reprojection error is usually larger at image boundaries as radial distortions due to the optical system used in the cameras become more apparent. Somewhat surprisingly this is also true for the optics of the projector even though its high quality object lens should show no signs of radial distortion when projecting rectangular images onto a screen.

The reprojection error for our setup consisting of four cameras and one projector is shown shown in fig. 9. The total mean reprojection error is 0.29 pixels with a standard deviation of 0.20 pixels. It is important to mention that this result, despite being satisfactory, is not necessarily in itself meaningful as one has to compare this value with the distribution of points within the image space that have survived the RANSAC verification. An even spread throughout the image plane is desirable, a property that has been demonstrated in fig. 8. As we decrease radius and spacing to r = s = 1 we get results similar to the unmodified calibration sequence, yielding a relatively large reprojection error of 0.8 pixels in average.