Preconditioning for accelerated convergence

As usual we precondition by changing variables so that the regularization operator becomes an identity matrix. The gradient $\nabla$ in equation (21) has no inverse, but its spectrum $-\nabla'\nabla$,can be factored ($-\nabla'\nabla={\bf H'H}$) into triangular parts ${\bf H}$ and ${\bf H'}$ where ${\bf H}$ is the helix derivative. This ${\bf H}$ is invertible by deconvolution. The quadratic form $\bold h'\nabla'\nabla\bold h = \bold h'\bold H'\bold H \bold h$suggests the new preconditioning variable $\bold p =\bold H \bold h$.The fitting goals in equation (21) thus become  

$$\begin{array} {lllll} \bold 0 &\approx& \bold r_d &=& \bold... ... \bold 0 &\approx& \bold r_p &=& \epsilon \bold p \end{array}$$ (22)

with $\bold r_p$ the residual for the new variable ${\bf p}$.Experience shows that an iterative solution for ${\bf p}$ converges much more rapidly than an iterative solution for ${\bf h}$,thus showing that ${\bf H}$ is a good choice for preconditioning. We could view the estimated final map ${\bf h}={\bf H^{-1}p}$,however in practice, because the depth function is so smooth, we usually prefer to view the roughened depth $\bold p$.

There is no simple way of knowing beforehand the best value of $\epsilon$.Practitioners like to see solutions for various values of $\epsilon$.Practical exploratory data analysis is pragmatic. Without a simple, clear theoretical basis, analysts generally begin from ${\bf p=0}$ and then abandon the fitting goal $\bold 0 \approx \bold r_p =\epsilon \bold p$.Effectively, they take $\epsilon=0$.Then they examine the solution as a function of iteration, imagining that the solution at larger iterations corresponds to smaller $\epsilon$ and that the solution at smaller iterations corresponds to larger $\epsilon$.In all our explorations, we follow this approach and omit the regularization in the estimation of the depth maps. Having achieved the general results we want, we should include the parameter $\epsilon$ and adjust it until we see a pleasing result at an ``infinite'' number of iterations. We should but usually we do not.