Next: Natural patterns
Up: Homework 5
Previous: Prerequisites
Suppose that we use the gradient operator for data interpolation:
|
(1) |
This approach roughly corresponds to minimizing the surface area and
represents the behavior of a soap film or a thin rubber sheet.
The corresponding inverse model covariance operator is the negative Laplacian
. The corresponding
covariance operator corresponds to the Green's function
that solves
|
(2) |
In 2-D, the Green's function has the form
|
(3) |
with some constant
.
To derive equation (3), we can introduce polar coordinates
around
with the radius
and note that the Laplacian operator for a radially-symmetric function
in polar coordinates takes the form
|
(4) |
Away from the point
, solving
|
(5) |
leads to
. To find the constant
, we can
integrate
over a circle with some small radius
around the origin and apply the Green's theorem
|
(6) |
Derive the model covariance function
which corresponds
to replacing equation (1) with equation
|
(7) |
and approximates the behavior of a thin elastic plate.
Next: Natural patterns
Up: Homework 5
Previous: Prerequisites
2022-11-02