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We have thought of equation (1)
as a formula for finding from .Now consider the opposite problem, finding from .Begin by multiplying equation (2)
by the **transpose matrix**
to define a new quantity :
| |
(3) |

is not the same as ,but these two vectors have the same dimensionality
and in many applications
it may happen that
is a good approximation to .In general,
may be called an ``image'' of .Finding the image is the first step of finding itself.
Formally, the problem is
| |
(4) |

And the formal solution to the problem is
| |
(5) |

Formally, we verify this solution by substituting
(4) into
(5).
| |
(6) |

In applications,
the possible nonexistence of an inverse for the matrix is always a topic for discussion.
For now we simply examine this matrix for the interpolation problem.
We see that it is diagonal:
| |
(7) |

So, ; but
.To recover the original data,
we need to divide by the diagonal matrix .Thus, matrix inversion is easy here.
Equation
(5)
has an illustrious reputation, which
arises in the context of ``least squares.''
**Least squares**
is a general method for solving sets of equations
that have more equations than unknowns.

Recovering from using equation
(5)
presumes the existence of the inverse of
.As you might expect, this matrix is nonsingular when *stretches* the data,
because then a few data values are distributed
among a greater number of locations.
Where the transformation *squeezes* the data,
must become singular,
since returning
uniquely to the uncompressed condition is impossible.

We can now understand why an adjoint operator is often an
approximate inverse.
This equivalency happens in proportion to the nearness of the matrix
to an identity matrix.
The interpolation example we have just examined is one in which
differs from an identity matrix merely by a scaling.

** Next:** THE NORMAL MOVEOUT MAPPING
** Up:** INTERPOLATION AS A MATRIX
** Previous:** Looping over output space
Stanford Exploration Project

12/26/2000