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== Abstract ==

A method for discriminating features orders of magnitude finer than the pixel/voxel mesh.

A feature is a time or place where one value is adjacent to a different value.
Values of a given feature are spread over many voxels by a spread function.
The spreads of all features are superimposed so that precise and accurate location information seems lost.
Frequency transforms are limited to discriminate no more than &frac14; wave on original data spreads.

Recovering highly precise and highly accurate locations for features is illustrated.
Examples are given for discriminating at less than 1/10th the spread diameter.

== Ideas ==



== Images ==

{| class="wikitable" border="1"
|+ Image series with description
! Point !! Circle !! Identity !! Convolve 1 !! Convolve 2
|-
| [[File:Ddd.01.center.png|Center Point]]
| [[File:Ddd.02.circle.png|Center Point]]
| [[File:Ddd.03.identity.png|Center Point]]
| [[File:Ddd.04.convolve1.png|Center Point]]
| [[File:Ddd.05.convolve2.png|Center Point]]
|-
| colspan="5" | A bright point is shown in a dark field.
A circle is drawn around the bright point.

A second point is chosen on the circle.

A second circle is drawn around the second point.
One point on the second circle is coincident with the original point.
This is an identity function:
A point on the circle offset (x,y) around which a second circle is drawn is coincident with the origin at its (-x,-y).

N random evenly scattered second points are chosen around which are drawn N circles.
Each of the N circles coincides at the origin so that the count of overlapping circles at the origin is N.
Elsewhere, by both examination and calculation, the count of coincidence is not more than N/4.

For two origins, the combined operations leads to the same massing of coincidence at each origin.
|}

{| class="wikitable" border="1"
|+ Image series with description
! circle !! sparse disk !! triangle !! description
|-
| [[File:Ddd.circle.01.positive.png|Antenna]]
| [[File:Ddd.disk.01.positive.png|Antenna]]
| [[File:Ddd.triangle.01.positive.png|Antenna]]
| '''''INPUT DATA'''''<br>These are point spread patterns presented for convolution.
They are the inner edges of expected difference of two Gaussians of different radius.

The original sparse disk tests convolution on sparse input data.
|-
| [[File:Ddd.circle.02.negative.png|Convolution]]
| [[File:Ddd.disk.02.negative.png|Convolution]]
| [[File:Ddd.triangle.02.negative.png|Convolution]]
| '''''CONVOLUTION MASK'''''<br>This is the (-x,-y) inverse of the point spread patterns.
|-
| [[File:Ddd.circle.03.convolve.png|Result]]
| [[File:Ddd.disk.03.convolve.png|Result]]
| [[File:Ddd.triangle.03.convolve.png|Result]]
| '''''CONVOLUTION OUTPUT'''''<br>Four superimposed point spread patterns were presented.
This is the result of convolving the superimposed patterns with the inverse pattern.
|-
| [[File:Ddd.circle.04.switched.png|Denoised]]
| [[File:Ddd.disk.04.switched.png|Denoised]]
| [[File:Ddd.triangle.04.switched.png|Denoised]]
| '''''HIGH PASS FILTER'''''<br>This is the result of zeroing data below a threshold.
|}