Re: [Vxl-users] Premise of vnl_levenberg_marquardt hold back Gold standard?

[Marc A. <yul...@ya...>]

Gehua,

Thanks for your reply!

As I use the second edition of H&Z book, the algorithm
on page 114 is 4.3.

As for the number of residuals, I still do not quite
understand why it's 4n. Since each point
correspondence gives 2 residuals, i.e the squared
distances in 2 images are d(x_i, xhat_i)^2 and
d(xpri_i, xhatpri_i)^2. 
it ought to be 2n.

below is the code segment of function f in my class
derived from vnl_least_squares_function:

void XXX:f(const vnl_vector<double>& x,
vnl_vector<double>& fx)
{
    ......
    for ( int i = 0; i < n; i++ ) {
        ...
        fx[2*i] = squared_distance(x1[i], x1_h[i]);
        fx[2*i+1] = squared_distance(x2[i], x2_h[i]);
    }
}

I will be very grateful if you would be patient to
give me a further explaination.

Marc.

--- Gehua Yang <ya...@rp...> wrote:

> First of all, Page 114, or Algorithm 3.7 is the Gold
> Standard Algorithm for 
> estimating an AFFINE transformation, not homography.
> 
> If homography is to be estimated, refer to Algorithm
> 3.3 on Page 98.  There 
> are two choices: Sampson error or Gold Standard
> error. For Sampson error, 
> there are only 9 parameters(or one can choose other
> parameterization). As 
> for Gold Standard error, there are 2n+9  variables.
> But there are 4n 
> residuals:
> sum d(x_i, xhat_i)^2 + d(xpri_i, xhatpri_i)^2
> Recall that LM takes a vector of residuals before
> squaring. for each d(.) 
> function, there are two residuals. Another way to
> look at it is that each 
> ideal point x hat brings in two parameters, but
> provides 4 constraints.
> 
> 
> Gehua
> 
> ----- Original Message ----- 
> From: Marc Anderson
> To: vxl...@li...
> Sent: Tuesday, November 16, 2004 9:58 PM
> Subject: [Vxl-users] Premise of
> vnl_levenberg_marquardt hold back Gold 
> standard?
> 
> 
> > Hi, all vxl guys!
> 
> > The class vnl_levenberg_marquardt checks if the
> number of parameters is 
> > less than that of residuals before carrying out
> the minimization process, 
> > otherwise it'll returns false. This requirement
> prevents the estimation of 
> > homography between two images using the Gold
> Standard algorithm(H&Z book, 
> > p114), in which case the number of parameters is
> 2n+9 while the number of 
> > residuals is n. ( where n is the number of point
> correspondences ). Would 
> > someone get the idea to figure this issue out?
> Thanks!
> 
> 



		
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