## A Tail of Two Cats

It is possible to reconstruct an image from the Fourier magnitudes
alone if we have a similar image to provide phase information. For
example, suppose we are trying to reconstruct the image of a cat, and
have the fourier magnitudes for it.

Since for this experiment we only have the magnitudes of the transform
it is represented in monochrome, and we cannot reconstruct the image.

In addition to these magnitudes, we have an image which we know is
similar to the missing cat. This image is of a Manx (tailless)
cat. Since we have the image, we can calculate both the Fourier
magnitudes and phases for the manx cat.

One simple method to try and restore the image of the cat is to simply
calculate an image using the known Fourier magnitudes from the cat
transform with the phases from the manx cat:

Despite the fact that the phases contain more structural information
about the image than the magnitudes, the missing tail is restored at
about half of its original weight. This occurs only when the phases
are almost correct. The factor of one half arises because we are
making the right correction parallel to the estimated phase, but no
correction perpendicular to the phase (and
<cos^{2}>=1/2). There is also some noise in the image.

This suggests a simple way to restore the tail at full weight: apply
double the correction to the magnitudes. An image is therefore
calculated with twice the magnitude from the desired image minus the
magnitude from the known image:

The resulting image shows the tail at full weight. However the noise
level in the image has also doubled.

### Crystallographic Interpretation:

Often in crystallography we have an *incomplete model*. Thus we
have observed structure factor magnitudes for a complete molecule, but
a model (from which we can calculate both magnitudes and phases) for
only part of it. In this case the missing potrion of the model may be
reconstructed by use of the appropriate Fourier coefficients.

The first attempt to reconstruct the cat's tail above corresponds to
an |F_{o}| map, the second to a
2|F_{o}|-|F_{c}| map.

In modern crystallography 2|F_{o}|-|F_{c}| maps have
been superceded by more advances map coefficients, such as
2m|F_{o}|-D|F_{c}| (sigmaa-a) and maximum-likelihood
map coefficients, but the principles are similar.

*Thanks to Eric Galburt for suggesting this page.*

More Fourier transforms.
Back to the index.