In both the forward and the adjoint operation,
each data coordinate is examined
and the nearest mesh point (the bin) is found.
For the forward operator,
the value of the bin is added to that of the data.
The adjoint is the reverse:
we add the value of the data to that of the bin.
Both are shown in two dimensions in module `bin2`.
bin2push data into bin
The most typical application requires an additional step, inversion.
In the inversion applications
each bin contains a different number of data values.
After the adjoint operation is performed,
the inverse operator divides the bin value
by the number of points in the bin.
It is this inversion operator that is generally called binning.
To find the number of data points in a bin,
we can simply apply the adjoint of `bin2` to pseudo data of all ones.
To capture this idea in an equation,
let denote the linear operator
in which the bin value is sprayed to the data values.
The inverse operation,
in which the data values in the bin are summed
and divided by the number in the bin, is represented by

(15) |

Figure 3

4/27/2004