ON IMPROVING RHESSI VISIBILITIES BY FORCING THEM TO BE HERMITIAN

In radio interferometry, the complex visibilities naturally have a property 
called HERMITIAN (http://en.wikipedia.org/wiki/Hermitian_function) 
that enforces the desirable result that all maps are 
perfectly real, that is, the imaginary part is precisely zero. (The Fourier
transform of the visibilities is real valued if and only if the visibility is Hermitian.)
Radio visibilities always have the property that for every point (u,v) in the Fourier 
plane, radio visibilities always have a counterpart at (-u,-v) where the visibility is 
the complex conjugate. 

This is the definition of Hermitian:

Vis(-u,-v) = VIs(u,v)*

where the * means the sign of the imaginary part of Vis is reversed.


With RHESSI visibilities, the above is not precisely true, although in principle it should be. 
The radial tranfsformation from (u,v) to (-u,-v) corresponds to rotating the grids 180
degrees about the optical axis. The amplitudes should be unchanged (assuming that venetian
blinding has been compensated for) and the phases should shift by 180 degrees. This is
what the equation above is equivalent to. But the departure from Hermitian means that 
a dirty map made from visibilities can have a non-zero imaginary part. 

There are 3 ways in which RHESSI visibilities are not self-conjugate:

1. Noise or time variability causes uncorrelated relative errors in the Vis(u,v) and 
   Vis(-u,-v) values.
2. Sometimes one of the members of a Hermitian pair is flagged as bad, and is therefore
	absent in the edited visibilities.
3. The venetian blinding inherent in RHESSI's grids may not be completely compensated. 
   
   
WHY IS THIS IMPORTANT AND WHAT SHOULD WE DO ABOUT IT?

It is obviously desirable to make calibrated RHESSI visibilities as mathematically correct
as possible, and this means making them as similar as possible to radio astronomical 
visibilities, where the imaging routines have (usually) been honed to a high degree of 
mathematical perfection.  It is not acceptable to have the possibility of RHESSI dirty maps 
containing ignored imaginary parts.  (What would a visibility Clean algorith do with them?)

The absence of an exact Hermitian property in RHESSI visibilities is a fixable problem. 
And fixing it can lead to better Forward Fitting, better visibility back-projection, and 
possibly better MEM mapping.    The reasons for this are multifold:

* 	When visibilities are rigorously Hermitian, all linear combinations with Hermitian 
    (or real) coefficients are real. The imaginary terms all cancel out. Hence Fourier
    transforms (dirty maps) are real. Also, any maps made by a linear Hermitian process are real.

* 	Some missing visibilities (flagged as bad by the stacker, or dropped as outliers)  
	can be filled in by their Hermitian conjugate if it, by chance, has not been flagged.

* 	Enforcement of the Hermitian entails a simple point-by point average (arithmetic
	or geometric) of each component of the visibility at (u,v) with the complex conjugate of its 
	counterpart at (u,-v). This smooths the data and makes contours of elliptical sources 
	more elliptical.
	
*   In the enforcement of the Hermitian property, the estimated errors (sigamp in the visibility
    structure) are reduced by a pactor of sqrt(2). This comes without any penalty of re-binning
    that might reduce the visibility modulation as a function of position angle.

PROBLEMS

Testing shows that mem maps made with weak sources are "better" in many ways after Hermitian
forcing. Forward fit model amplitude profiles tend to have a smoother and more exact agreement
with the visibility amplitudes.

ejs 
2.19.2009