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The idea of finding smooth solutions is to minimize a measure
of the gradient.
The first time I thought about doing this with *L*_{1},
I tried the wrong approach (and that put me off the track for 25 years).
The wrong approach is to take
the *L*_{1} norm of the *x*-component of the gradient
and add it to the *L*_{1} norm of the *y*-component of the gradient.
This is bad because it embeds the orientation
of the coordinate system.
Axiomatically,
in science we like solutions that are independent
of the human choice of a coordinate system.
Thus *L*_{1} appears to conflict with this basic requirement.
An approach independent of coordinate rotation and translation
on a grid is to minimize

| |
(8) |

where *u*=*u*(*x*,*y*) and where
the summation is over (*x*,*y*)-space.
Multivariate *L*_{1}-norm problems generally reduce
to a line search that is a weighted median.
Hoare's algorithm makes this very fast.
Unfortunately, this multidimensional generalization
of *L*_{1} does not seem to reduce to a weighted median
so Hoare's algorithm is irrelevant,
as might be other *L*_{1} experiences we have seen in 1-*D*.
I discussed the criterion for a while with Bill Symes.
We came up with this simple problem
where we would use zero side boundaries and seek the response
of an impulse in the medium.

0 0 0 0
0 a 10 0
0 b c 0
0 0 0 0

The free variables are *a*, *b*, and *c*.
We take the *x* derivative diagonally to the northeast
and the *y* derivative diagonally to the southeast.
| |
(9) |

A few manual calculations quickly convinced us that the solution is
*a*=*b*=*c*=0.
Thus multidimensional *L*_{1} does not look like the answer we seek.
It looks like the boundaries at infinite distance dominate the data
(in this case the 10).
Thus, the response to an isolated collection of spikes,
might simply be the spike values where they are given and zero elsewhere.

** Next:** Gaussian curvature
** Up:** Claerbout & Fomel: Gaussian
** Previous:** Curve through two points
Stanford Exploration Project

4/27/2000