From: Lianqing Yu <lqyu@nl...>  20060829 01:03:22

> When estimating the fundamental matrix, N equals to the number of > correspondences as each correspondence provides only one constraint. But > as you are estimating the homography, N equals to ***2 times the number > of correspondences*** as each provides two constraints. I can't agree with you on this point. Take estimating the fundamental matrix by minimizing geometric distance $\sum_i d(x_i, \hat{x}_i)^2 + d(x'_i, \hat{x'}_i)^2 $ for instance, if one regards $d(x_i, \hat{x}_i)^2 + d(x'_i, \hat{x'}_i)^2$ as one item, the number of residual is 1 for each point correspondence. This is what FMatrixComputeNonLinear has done. If one takes $d(x_i, \hat{x}_i)^2 + d(x'_i, \hat{x'}_i)^2$ as summation of two terms, the number of residual is 2 for each point correspondence. Furthermore, if ones take d(x_i, \hat{x}_i)^2 as a summation of coordinate differences in both x and y direction, the number of residual will be 4. For nonlinear estimation with Sampson error, I believe there is one residual for each correspondence because the error is a scaler $\epsilon^T (J J^T)^{1} \epsilon$. > Also, when providing vnl_levenberg_marquardt with the analytic Jacobian > matrix, it is a good practice to turn on > vnl_levenberg_marquardt::set_check_derivatives( int ), which will > compare the provided Jacobian with the one computed from numerical > differences. After verifying the two are equal within a small error > range, it is to be turned off for the consideration of speed. Thanks for this tip. Lianqing Yu  Original Message  From: "Gehua Yang" <yangg2@...> To: "Lianqing Yu" <lqyu@...> Cc: <vxlusers@...> Sent: Tuesday, August 29, 2006 3:27 AM Subject: Re: [Vxlusers] Why is the dimension of Jacobian in vnl_levenberg_marquardt is fixed in this way? > Hi, > > First of all, you'll need to understand the difference between > Homography and Fundamental matrix and the difference between their > associated error measures, which I am afraid we (VXL community) can > provide little help. > > I'll give this problem a shot based on my understanding of your > description. > > You are confused by "the number of residuals". In the context of the > LevenbergMarquardt, the objective function has a leastsquares form: > sum_i( f_i^2 ) (i=1...N). Thus, the input to vnl_levenberg_marquardt is > a vector of [f_1, f_2, ..., f_N] and the corresponding Jacobian is a > Ntimesk matrix (k is the degree of freedom). > > When estimating the fundamental matrix, N equals to the number of > correspondences as each correspondence provides only one constraint. But > as you are estimating the homography, N equals to ***2 times the number > of correspondences*** as each provides two constraints. > > Also, when providing vnl_levenberg_marquardt with the analytic Jacobian > matrix, it is a good practice to turn on > vnl_levenberg_marquardt::set_check_derivatives( int ), which will > compare the provided Jacobian with the one computed from numerical > differences. After verifying the two are equal within a small error > range, it is to be turned off for the consideration of speed. > > Gehua > > > Lianqing Yu wrote: > >> Hi, >> I've got an issue when I use vnl_levenberg_marquardt to refine the >> estimation of the 2D homography between calibration rig plane and >> image plane. >> My algorithm follows the gold standard algorithm proposed in [Hartley >> & Zisserman, 2000] (algorithm 3.3, p98) and I choose to use Sampson >> error for the sake of speed. For implementation, I write a class >> similar to FMatrixComputeNonLinear. To gain faster convergence, I >> provide a customized routine gradf(), in which the Jocabian is a 2N x >> 9 matrix (equation 3.31, p114), where N is the number of point >> correspondences between the two planes. >> Here comes the problem. For geometric minimization of Sampson error, >> the number of residuals is N, since there is one residual (the norm >> $\delta_x^2$, equation 3.12, p83) for each correspondence: >> HMatrix2DComputeSampson::HMatrix2DComputeSampson(int N) >> : vnl_least_square_function(9, N, use_gradient) {} >> However, if my class is initialized like this, I find in the >> vnl_levenberg_marquardt code that dimension of Jocabian matrix in >> gradf() is fixed as N x 9 and my program will crash. One workaround is >> to set the number of residuals to 2*N. >> HMatrix2DComputeSampson::HMatrix2DComputeSampson(int N) >> : vnl_least_square_function(9, 2*N, use_gradient) {} >> But this will introduce some minor problems (more RAM consumption, >> slower speed, unintuitive). >> In all, I'd like to know your opinions on this issue. >> I'd appreciate your patience to read this mail and thanks in advance. >> Lianqing Yu >> 8/28/2006 >> >> PhD Candidate >> Robot Vision Group >> National Laboratory of Pattern Recognition (NLPR) >> Institute of Automation, Chinese Academy of Sciences (CASIA) >> >> No. 95, Zhongguancun East Road, >> 100080,Beijing, China >> Email: lqyu@... <mailto:lqyu@...> >> >> >> >> >>Using Tomcat but need to do more? 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