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%\title{HESSI Imaging Capabilities}
%  \author{ Gordon J. Hurford, Edward J. Schmahl, Andrew Conway, Brian R. Dennis, 
%   Richard A. Schwartz,
%   Robert.P.Lin, Jim McTiernan, David Smith, Chris Johns-Krull, and
%   Andre Csillaghy, Markus J. Aschwanden and Thomas R. Metcalf}

\begin{titlepage}

 \begin{center}
              \huge{DRAFT}

              \huge{\today} 

\bigskip

 \Large{The Imaging Capabilities of HESSI}
\end{center}

\bigskip

 Gordon J. Hurford, Edward J. Schmahl, Andrew Conway, Brian R. Dennis, 
   Richard A. Schwartz,  

Lab for Astronomy and Solar Physics,

   NASA Goddard Space Flight Center,

   Greenbelt, MD 20771

\medskip
   Robert.P.Lin, Jim McTiernan, David Smith, Chris Johns-Krull, and
   Andre Csillaghy, 

Space Science Lab, University of California, 

Berkeley , CA 94720


\medskip
Markus J. Aschwanden and Thomas R. Metcalf, 

Lockheed Martin Advanced Technology Center, 

Solar \& Astrophysics Laboratory, Dept. L9-41

Palo Alto, CA 94304
\bigskip

{\bf Changes included:}

Completed references

GH: Ref to HXIS

ES: Subsection on Count rates and visibilities

ES: Added more on statistics

ES: Updated decimation scheme (thanks, DC)

MA: Make definitions consistent

\bigskip

{\bf TBD:}

GH: Figure for various slit-slat ratios and energy dep'ce

GH: Change (4) to "harmonic motion"

GH: More on statistics

MA: Show simulation results for Bproj, CLEAN, MEM, FwdFit

 \end{titlepage}

\tableofcontents
\newpage



\begin{abstract}

  The High Energy Solar Spectroscopic Imager (HESSI) will use
rotational modulation synthesis for imaging hard X-ray and gamma-ray
flares with spatial resolution of $2.3''$ and spectral resolution of
$\sim 1 $ keV. Like the Yohkoh/HXT hard X-ray telescope, HESSI
relies on Fourier methods to produce images, but HESSI has many
more sampled points in the Fourier plane, and is expected to
produce higher-resolution maps with greater dynamic range.

We summarize the methods to be used for HESSI imaging, touching
on crucial details of modulation such as the {\it Calibrated
Event List}, and describing the {\it Back Projection} method.
Beyond these, we outline four basic tools for image reconstruction:
{\it CLEAN}, {\it MEM}, {\it PIXONS}, and {\it Forward Fitting},
with simulated examples of each.

\end{abstract}


\section{Introduction}

The only practical method of obtaining $\sim$arc-second angular
resolution in hard X-rays and gamma-rays within the cost, mass, and
launch constraints of a small satellite is to use Fourier-transform
imaging. (See Prince {\it et al.} 1988 for a review of imaging
techniques.)  One of the most powerful of the Fourier family
techniques is rotational modulation synthesis, first proposed by Mertz
(1967) and implemented by Schnopper {\it et al.} (1968). Rotating
Modulation Collimators (RMC) were first constructed and used for solar
flare X-ray imaging by the Japanese Hinotori telescope (Makashima {\it
et al.}  1977, Ohki {\it et al.} 1982, Enome 1982), with angular
resolution of $28''$ in the 20-40 keV energy range.  The next hard
x-ray solar imager was the Hard X-ray Imaging Spectrometer (HXIS), not a
Fourier-transform imager, but important for developments in the $\sim
3-30$ keV range (Van Beek {\it et al.} 1980).  A later Japanese
telescope, HXT, on the Yohkoh satellite, used non-rotating Fourier
synthesis with angular resolution of $\sim 8''$ in the 20-100 keV
energy range (Kosugi {\it et al.}  1991).  A prototype of a rotation
modulation telescope---the High Energy Imaging Device (HEIDI)--- was
flown in 1993 on a balloon, without successful flare imaging, but with
a definitive test of a novel solar aspect system (SAS), proving
0.5-arcsecond performance at balloon altitudes. (Crannell {\it et al.}
1994)

\medskip

The High Energy Solar Spectroscopic Imager (HESSI) also uses
rotational modulation synthesis imaging. With a launch shortly 
after the peak of flare activity, HESSI has been designed to image
hard X-ray and gamma-ray flares with spatial resolution of $2.3''$
and spectral resolution of $\sim 1 $ keV (Lin {\it et al.} 1993, 1994, 1998,
Holman {\it et al.} 1997).  We summarize briefly the HESSI instrumentation
used to exploit Fourier synthesis imaging.
\medskip 

\subsection{The HESSI Imager}

HESSI uses nine bi-grid collimators, each consisting of a pair of
widely separated grids in front of an X-ray/gamma-ray detector. Each
grid consists of a planar array of equally-spaced, X-ray-opaque slats
separated by transparent slits.  The slits of each pair of grids are
parallel to each other and their pitches (p) are identical, so the
transmission through the grid pair depends on the direction of the
incident X-rays. (See Figure \ref{imager_schematic}) For slits and
slats of equal width, the transmission is modulated from zero to 50\%
and back to zero for a change in source angle to collimator axis
(orthogonal to the slits) of p/L where L is the separation between
grids (1550 mm). The angular resolution is then defined as p/(2L). For
HESSI, the transmission of the source photons through the grids is
modulated by the rotation of the spacecraft at about 15 revolutions
per minute. The detector records the arrival time and energy of
individual photons from anywhere on the Sun, allowing the modulated
counting rate to be determined as a function of rotation angle.


\begin{figure}
      \centering{
      \vspace*{-5.cm}
      \vbox{\psfig{figure=fig1:imager_schematic.eps,width=16cm,height=23cm}}\par
      }
      \label{imager_schematic}
\end{figure}

\medskip 

Note that the detectors have no spatial resolution and hence have
been optimized for high sensitivity and energy resolution.
The nine segmented Ge detectors (GeDs), one behind
each RMC, detect photons from $\sim$3 keV to 15 MeV. The GeDs are cooled
to $ \sim 75$ K by a space-qualified long-life mechanical cryocooler.
As the spacecraft rotates, the RMCs convert the spatial information
from the source into temporal modulation of the photon counting rates
of the GeDs. The instrument electronics amplify, shape, and digitize
the GeD signals, provide power, format the data, and interface to the
spacecraft electronics.

\medskip 


HESSI exploits one of the fundamental efficiencies of Fourier imaging:
precision spacecraft pointing can be traded for exact knowledge of
pointing.  With a Fourier imager, it is not necessary to stabilize the
spin axis to any better than a few arc minutes, as long as the system
gives sufficiently precise and high-bandwidth pointing information.
For HESSI, this information is provided by the {\it solar aspect
system} (SAS) --- a heritage of the HEIDI SAS --- and the {\it roll
angle system} (RAS).  The SAS consists of 3 linear diode arrays on
which the full solar image is projected.  The solar limb is determined
at 6 positions every 10 ms, giving pitch and yaw to $\sim 1.5''$.  The
RAS consists of a star-imaging linear photodiode, providing roll angle
to $2.7'$ every minute.

\medskip 

\subsection{Expected Imaging Capabilities}

In 2-3 years of mission life, HESSI is expected to obtain observations
of tens of thousands of microflares, thousands of hard x-ray flares,
and of order a hundred gamma-ray line flares.  Detected photons will
be tagged with $1\ \mu s$ time resolution up to a rate of $\sim 10^5$
photons/s.  The intense 3-150 keV X-ray fluxes that usually accompany
large gamma-ray line flares are absorbed by the front segment of the
detectors, so the rear segment will always count at moderate
rates. This is essential for gamma-ray line measurements with optimal
spectral resolution and high throughput.
\medskip 


Except in extreme circumstances, every photon is stored and then sent
down.  If an extended, extremely active period occurs, a decimation
scheme is enabled which digitally decimates the incoming photon event
stream, so that only a fraction of the events below an energy
threshold get sent to telemetry. Both the fraction and the energy
threshold are programmable.  There are decimators for both the front
and the rear segments of the detectors.  The front decimator
thresholds are controlled by software to avoid overflowing the
spacecraft recorders, and the rear decimators can be controlled from
the ground to limit the background rate.  \medskip

In addition to decimation, there are mechanical attenuators, which if
required, reduce the flux below the saturation level of $\sim 50,000$
events/s. Together, decimation and attenuation provide minimum loss of
imaging and no loss in gamma-ray spectroscopy.  

\medskip
It is expected that HESSI will provide detailed imaging spectroscopy
in each of ten energy intervals with $\sim 2 s$ time resolution for
one event every $\sim 5$ days with $\sim 10^4$ counts $s^{-1}$ above
15 keV.  With lower energy resolution ($\sim 100$ keV), crude imaging
information could be obtained in tens of milliseconds.
HESSI provides spatial resolution of 2.3 arcseconds at X-ray energies
below $\sim 200$ keV, 7 arcseconds to 400 keV, and 36 arcseconds for
gamma-ray lines and continuum above 1 MeV.
          


\section{Modulation Principles}
\smallskip

HESSI, like all RMCs, relies on temporal modulation by its grids to
provide spatial information about incoming photons.  The imaging
process may be thought of as a superposition of elemental processes of
photon passage through a sub-collimator.  Since the time of arrival of
each photon is determined to $1\ \mu s$, and the aspect system
determines the pointing of the telescope with arcsecond accuracy, the
known geometry of the subcollimators provides well-defined information
about the photon's origin.  \medskip

\subsection{One-dimensional Modulation}
\smallskip

We illustrate how this works by considering a subcollimator with two
identical grids, each parallel to the x,y plane, whose slats are, at a
given instant, parallel to the y axis, and whose spin lies
parallel to the z axis.  Let a photon travelling in the x-z plane
strike the plane of the front grid.  The incidence phase (measured in
the x direction) of the photon relative to the slats of the front grid
may be considered to be random since all photons arriving from a
single point at infinity lie on parallel lines with a continuum of
phases, and since the detectors have no spatial resolution, the modulation
profile is found by integrating over all incidence phases.

\medskip

The probability of passage of the photon through the front grid is
proportional to the projected area of the slits of that grid, as
viewed from the direction of the photon.  For an ideal grid of finite
thickness and infinite extent, this value is independent of the y
direction, but dependent on the photon direction in a fashion
characteristic of the grid thickness and the slit-slat ratio.
\medskip

In the case of a single thin grid with equal slits and slats, the
projected area, plotted as a function of incidence phase, is a
periodic, rectangular waveform with equally long maxima and minima.
For a thick grid, there is internal shadowing by the sides of the
slats, and the projected open areas are reduced from the thin case, so
the area waveform is still periodic, but with maxima and minima of
unequal lengths.  In the case of HESSI grids, the projected slit-slat
ratios are optimized to have equal projected slit and slat widths at
incidence angles of about $600''$. The aspect ratio of the slats
(x-width/z-thickness) is ~1:50. For photons with nearly normal
incidence, the effective slit/pitch ratio is about 0.6, and for
photons incident at angles of $415''$ in the x-z plane, the
effective slit/pitch ratio after shadowing is close to 1:2.  \medskip

After photons pass through the front grid, they have a finite
probability of passing through the rear grid.  If the incidence angle
is less than about $900''$, the probability of incidence on the rear
grid is almost unity.  Only for photons entering at angles of greater
than 1 degree does this probability become significantly less than 1
due to non-overlap of the front and rear grid areas.  The passage of
the photon through the second grid is, again, dependent on its
direction in exactly in the same way as for the front grid.  The
fraction of projected open area for an incident photon is therefore
the convolution of the transmission functions of each grid.  It is
easy to verify that in the ideal case, this is a triangle waveform.
For thin grids, the triangle function has pointed peaks and valleys.
But if the projected slit-slat ratio is larger or smaller than 1, the
valleys of the waveforms will be flattened.  The sharp corners of the
profile's peaks are also rounded by partial transmission effects.
Figure 2 shows examples of the equivalent area function for various
slit/pitch ratios, and also for (over-simplified) energy-dependent
effects that smooth the corners of the profile.

%\begin{figure} 
%\vspace{10 in} 
%\begin{picture} (300,400)(100,-625)
%\epsfig{figure=fig2:rel_area_vs_incid_angle.eps} 
%\end{picture} 
%\caption{Examples of the equivalent area function for various slit/pitch
%ratios, and also for (over-simplified) energy-dependent effects that
%smooth the corners of the profile.} 
%\label{slit_pitch_effect}
%\end{figure}

\begin{figure}
      \centering{
      \vspace*{-4.cm}
      \vbox{\psfig{figure=fig2:rel_area_vs_incid_angle.eps,width=16cm,height=23cm}}\par
      }
      \label{slit_pitch_effect}
\end{figure}



\medskip

It is useful and efficient to characterize the subcollimator
transmission functions in terms of the first few harmonics.  For
slit/pitch ratios equal to 1/2, and for energies low enough that the
slats are opaque ($E < 100$ keV), and for energies high enough that
diffraction effects are negligible ($E > 6$ keV), the effective
subcollimator area function is approximately a triangle waveform,
represented by the following function of
$\theta$, the photon's angle of incidence relative to the imager axis:

\begin{equation} 
 S_{triangle}(\theta)={A \over 2} \cdot ({1 \over 2}+
{4 \over {\pi^2}} cos(\theta)+
{4 \over {9 \pi^2}} cos(3\theta)  + 
{4 \over {25 \pi^2}}cos(5\theta) + \cdots) 
\end{equation} %(1)


\medskip
This is precisely the form used by Schnopper {\it et al.} (1970) for their
observations of galactic-center with a rocket-borne RMC.
But for HESSI, the even harmonic terms are usually not negligible, and the
Fourier coefficients are not those of a pure triangle waveform, so we
must use the more general form:

\begin{equation} 
S(\theta)=A \cdot (c_0+c_1\ cos[\theta-\theta_1]+c_2\ cos(2[\theta-\theta_2]) + c_3\ cos(3[\theta-\theta_3]) + \cdots)  
\label{count_1}
\end{equation} %(2)

where $c_0 A$ is the effective area averaged over $\theta$. In
practice, we keep terms only up to the 5th harmonic. In general, the
collimator transmission function is the convolution of the
transmission functions of the front and rear grids.  If the front and
rear grids of a subcollimator are geometrically identical (which is
very nearly true for HESSI), this means that the Fourier coefficients
of $S(\theta)$ are the squares of the transmission functions of a
single grid.  For the case of ideal grids of effective slit/pitch
ratio q, the Fourier coefficients of the grid transmission are

\begin{equation} 
 g_n = {1 \over 2\pi} \int_{-q\pi}^{q\pi} cos(n\theta) d\theta = sin[n\pi q]/(n\pi q) \qquad n=1,2,3, \cdots
\end{equation} %(3)

So the amplitudes $c_n$ are:

\begin{equation}  c_n = 4 sin^2[n\pi q]/(n\pi q)^2 \qquad n=1,2,3, \cdots
\end{equation} %(3)

When $q=1/2$, the case of a simple triangle profile (which occurs for
HESSI when $\theta=412''$), all the even amplitudes vanish, but for
$q=0.4$, (photons at normal incidence on the HESSI grids),
$c_2=0.1093$, and for $q=0.6$ (equivalent to angular incidence of
$824''$), $c_2=0.0486$.  In general, for real conditions, the relative
amplitudes $c_1,c_2, \cdots$ and the relative phases
$\theta_1,\theta_2,\cdots$ are dependent on the grid characteristics,
the roll angle, and the angle of incidence.  The latter dependence is
sufficiently small and slow that for sources on the Sun, that they can
be considered uniform within a map in a given time bin.  These parameters
have been computed from the grid characterizations, and are included
in the HESSI reconstruction algorithms.  \bigskip


\subsection{Rotational Modulation}
\smallskip

So far we have described the transmission only for one roll angle,
namely the instant at which the pattern projected on the sky from an
ensemble of sight lines from the detectors --- the so-called ``modulation
pattern'' --- has contours of constant phase perpendicular to the vector
on the plane of the sky from the source to the projected telescope
axis.  As the spacecraft spins, the projected modulation pattern spins
with it.  To the lowest order of approximation, if the grids are
infinitely wide and infinitely thin, and the spin axis does not move,
the modulation pattern would rotate unchanged.  Since, however, the
grids are thick, of finite extent, and the spin axis is not
necessarily constant during a rotation, the projected modulation
pattern changes with roll angle in a predictable way, which is
reflected in a slow, smooth change of the phase and amplitude
coefficients of the sub-collimator pattern.  \medskip


It is useful to take the telescope as our reference frame, and let a
point source move in a circle (roll angle $\alpha = 0-2\pi$) on the
celestial sphere around the spin axis.  Then, because no modulation
occurs in the y direction parallel to the slats, and modulation in the
x-direction follows equation (\ref{count_1}), the effective pitch changing in
proportion to $sec(\alpha)$.  During a rotation, the fundamental
coefficients $(c_1, \theta_1)$ change least, and the function S is a
simple function of roll angle $\alpha$.  For an ideal RMC (idealized
grids, steady spin), the argument $\theta$ in equation (\ref{count_1}) is
proportional to the radial distance R (arcsec) of the source and inversely
proportional to angular pitch p):

\begin{equation} \theta = 2\pi R cos(\alpha)/p
\end{equation} %(4)

The flux of photons on a HESSI detector from a point source a distance
R off axis is proportional to the effective area, which
leads to the modulation profile as a function of rotation angle $\alpha$:

\begin{equation}
P_0(\alpha) = A\cdot (1+c_1 cos(2 \pi R cos(\alpha)/p-\theta_1) + \cdots) 
\end{equation} %(5)

where p is the angular pitch (arc sec) of the grids in the
subcollimator.  For a given map center, the quantities ($c_n,
\theta_n$) are computed from the aspect system and the grid response
matrices, and are returned in the HESSI {\it calibrated event list}.
The above function is computable for any time interval and any point
source on the Sun. Therefore, since the the signal expected for a
superposition of sources is the superposition of point source signals,
any model source can be used to predict a modulation profile.  This is
the basis for all HESSI reconstruction algorithms.

\bigskip

\subsection{The HESSI Event List} 

The set of photon arrival times taken from the HESSI telemetry can be
histogrammed to yield a raw count rate profile.  The HESSI user can
select an almost arbitrary set of time bin sizes \{$\Delta t_i$ \} for
histogramming.  ($\Delta t_i $ is necessarily $ \ge 1\ \mu s$, and may be
as large as $\sim 1\ s$, but is more typically $\sim 0.5\ ms$). After the
time bin selection, the count/bin profile becomes a function of roll
angle $\alpha$, and one can proceed to calibrate the rates.

\medskip

For phase and amplitude calibration, one must select a phase center
for mapping.  This point can be found in several ways: (a) determine
the azimuth by the roll angle of fastest modulation, and the radius by
the number of modulation cycles per rotation; (b) make a rough map of
a large region (say a solar quadrant) and find the position of the
maximum; or (c) use context data or (d) previous flare position data.

\medskip

%\begin{figure}
%      \centering{
%      \vspace*{-2.cm}
%      \vbox{\psfig{figure=symbols_diagram.eps,width=10cm,height=15cm}}\par
%      }
%      \label{symb_diag}
%\end{figure}

Using the aspect system data, the distance of the line of maximum
transmission from map center is computed for each roll angle
$\alpha_i$ and each subcollimator pitch $p_k$. This gives the phase at
map center $\Phi_{ik}$. (See figure (\ref{symb_diag}.) 
For the case of a fixed spin axis,
$\Phi_{ik}=2\pi R cos(\alpha_i)/p_k-\theta_k$.  But in practice,
$\Phi_{ik}$ is computed without reference to $R$, using aspect data
and the subcollimator geometries alone.  One must incorporate the dead
time, a function $\tau_{ij}$ of roll angle and energy $E_j$, the
detector number k, and the subcollimator transmission $T_{ijk}$, as
well as the modulation amplitudes $c_n^{ijk}$ and modulation phases
$\Phi_n^{ijk}$ to determine the model modulation profile:

\begin{equation}
S_{ijk} =F_0 \cdot \lbrace \tau_{ijk} T_{ijk} [1 + \sum_n c_n^{ijk} cos[n(\Phi_{ik}-\Phi_n^{ijk})] \rbrace
\label{S_ijk}
\end{equation} %(6)

When the HESSI analysis software computes the quantities $ \tau_{ijk}, T_{ijk},
c_n^{ijk}$ and $\Phi_n^{ijk}$, they are returned in a structure called the
{\it calibrated event list}, the starting point for image reconstruction.
\bigskip

\subsection{Visibilities}

In radio astronomy, the quantity analogous to the modulation pattern
is the fringe pattern, which is the response of an interferometer to
point sources in the sky. This is the Fourier transform, and it is
represented by a sum of complex exponentials in the u,v plane:

\begin{equation}
 {\cal V}_i = f\ e^{2\pi i(u_i x + v_i y)}
\end{equation} %()

for each point source of flux $f$ at coordinate $(x,y)$.
The variables $u_i,v_i$ are the coordinates in the Fourier plane,
and for HESSI, $u_i=cos\phi_i/p$ and $v_i=sin\phi_i/p$, so the
points $(u_i,v_i)$ lie on circles in the Fourier plane.

The importance of visibilities to HESSI is that they represent an
intermediate between count-rate profiles and images that is
independent of the pointing.  The visibilities have all of the
spin-axis wobble, slit-shadowing effects, transmission and deadtime
accounted for.  Visibilities, unlike count-rate profiles, can be
added from 1/2 rotation to another.

\medskip
A significant simplification for imaging occurs when one changes
from the usual Cartesian coordinates to polar coordinates in both
the image plane and the Fourier plane:


\begin{eqnarray*}
  x\ =\ r\ cos\theta \quad & \quad u\ =\ k\ cos\phi\\
  y\ =\ r\ sin\theta \quad & \quad v\ =\ k\ sin\phi
\end{eqnarray*} %()

Since the visibilities are found on circles in the Fourier plane, the
radial integration drops out (${\cal V}$ is proportional to a delta function of
radius k), and after a little algebra, the Fourier transform for one
subcollimator of wavenumber k becomes:

\begin{equation}
  F(r,\theta)= k \int_0^{2\pi} {\cal V}(\phi) e^{ikr\ cos(\theta-\phi)} d\phi 
  = k {\cal V} \otimes e^{ikr\ cos(\phi)}
\end{equation} %()

The nice thing about this, is that for each radius, $F(r,\phi)$ is a
convolution of the visibility and a complex exponential.  Since
convolutions are orders of magnitude faster than matrix
multiplications, polar coordinate representations of HESSI data
provide a great increase in speed of ``backprojection'' (section 3.1)
or other reconstructions.

\subsection{Count Rates and Visibilities}

Given an incident flux of photons on a HESSI subcollimator, after the
selection of a map center, the predicted count rate ${\cal C}$ is
given by equation (\ref{S_ijk}) with subscripts $i,j,k$ dropped:

\begin{equation}
{\cal C}= F_0\ T\ \tau\ \cdot \lbrace 1\ +\ c_1\ cos[\Phi] + c_2\ cos[2(\Phi-\Phi_2)] \ +\ \cdots \rbrace
\label{c_series}
\end{equation} 

where $F_0$ is the incident\ photon\ flux\ on\ front\ grid,
 $T$  is the subcollimator transmission, $\tau$ is the livetime,
 $\Phi$ is the phase at map center,
 $c_n$ is the modulation amplitude, and
 $\Phi_n$ is the modulation phase.
(All these and subsequent variables are defined in the Appendix.)
\medskip

In general, the phase at map center depends on the results of the
aspect solution, the time binning, the subcollimator, and the position
of the map center.  The phase at map center $\Phi$ depends on the the
projection of the vector between the subcollimator axis
$(x_{coll},y_{coll})$ and the map center $(x_{map},y_{map})$:

\begin{equation}
  \Phi=2\pi \lbrace [x_{map}-x_{coll}]cos\alpha + [y_{map}-y_{coll}]sin\alpha)/p
\rbrace
\end{equation} 

where 
\begin{eqnarray*} 
 (x_{coll},y_{coll})  &  = & {\rm Subcollimator\ direction\ in\ inertial\ coordinates} \\
 (x_{map},y_{map}) & = & {\rm Map\ center\ in\ inertial\ coordinates }\\
 \alpha   & = & {\rm Azimuth\ angle\ of\ slits {\qquad\qquad\qquad\qquad}         }\\
  p  & = & {\rm Pitch\ of\ subcollimator\ grids  }
\end{eqnarray*}

In the simplest case, where,
the spin axis is at the origin and is stationary, the phase at map
center  is a function only of the azimuth $\phi$, the pitch $p$, and
the location of the map center $(R_0,\phi_0)$:

\begin{equation}
  \Phi = 2 \pi R_0 cos(\phi-\phi_0)/p  
\end{equation} 

\medskip

But in general, regardless of the position of the spin axis, the
predicted count rate is given by equation (\ref{c_series}).

The visibility of a point source at the map center is:

\begin{equation}
 V=F_0 \ e^{{\bf i} 2\pi  [x_{map}\ cos\alpha + y_{map}\ sin\alpha)]/p}
\label{vis_1}
\end{equation}

Note that the visibility does not depend on the subcollimator coordinates
$ (x_{coll},y_{coll})$, but the count rate is subject to phase shifts
caused by telescope motion.  The phase shifts due to this motion are:

\begin{equation}
\Delta\Phi=2\pi \lbrace x_{coll}\ cos\alpha + y_{coll}\ sin\alpha)/p
\end{equation}

And these are reflected in additional variations of the count rate
profile. (The quantity $\Delta\Phi$ is computed from the aspect solution.)
We can separate out this variation explicitly in the visibility:


\begin{equation}
 V=F_0 \ e^{{\bf i} [\Phi +\Delta\Phi]}
\label{vis_2}
\end{equation}

If we subtract out the mean of the count rate (equation \ref{count_1}),
and divide out the $T\tau$ factor, keeping only the fundamental term,

\begin{equation}
({\cal C}-F_0\ T\ \tau)/( T\ \tau c_1) = F_0\ cos\Phi 
\label{count_2}
\end{equation}

So, for this particular case of a point source at map center
the relation between the countrate profile and the visibilities are 
found by comparing equations \ref{vis_2} and \ref{count_2}:

\begin{equation}
Re\{ V  e^{-{\bf i}\Delta\Phi}\} = 
{ {{\cal C} -F_0 T \tau} \over { c_1 T \tau}}
 ={ {{\cal C} - <{\cal C}>} \over { c_1 T \tau}}
\label{V_count}
\end{equation}

It is important to note that, by equation \ref{V_count}, the
peak-to-peak variation of the visibility is several times larger than
the peak-to-peak variation of ${\cal C}$.  This is because the
visibility reflects the transform of the incident photons, while the
count rate is the modulated profile of only the detected photons.  In
fact, when the livetime is unity ($T=1$), and when the subcollimator
pattern is perfectly triangular, ($c_1\tau=2/\pi^2$), the
peak-to-peak variation of $Re\{ V \}$ is $\pi^2/2 = 4.9$ times that of
the count profile.

\medskip

For a more general case, where the source is an
arbitrary superposition of N point sources of flux $F_j$ and location
$(R_j,\phi_j)$, the visibility is:


\begin{equation}
 V= e^{{\bf i}  \Delta\Phi} \sum_{j=1}^N F_j \ e^{{\bf i}\Phi_j}
\label{vis_3}
\end{equation}

where the   $\Phi_j$ are the phases of the point sources at ($R_j,\phi_j$).

Then, the corresponding predicted count rate is:

\begin{equation}
{\cal C}= T\ \tau\ \cdot  \sum_{j=1}^N F_j\ \lbrace 1\ +\ c_1\ cos[\Phi_j)]\ +\ \cdots \rbrace
\label{count_3}
\end{equation}


Comparing equations \ref{vis_3} and \ref{count_3}, we find that
equation \ref{V_count} remains valid, providing that $F_0$ is defined
as the total flux of the N sources.  

\medskip

The generalization to higher harmonics is straightforward. The
visibility for the $n^{th}$ harmonic is given by:


\begin{equation}
Re\{ V^{(n)}  e^{-{\bf i}\Delta\Phi}\} = 
 {{\cal C}^{(n)} \over { c_n T \tau}}
\label{v_count_n}
\end{equation}

where  ${\cal C}^{(n)}$ is the $n^{th}$ term in the Fourier expansion
of the count rate (equation \ref{c_series}).  Note that equation
(\ref{v_count_n}) determines only a projection of the complex
visibility at each instant of time.  Since, however, each modulation
cycle contains phase as well as amplitude information, equation 
(\ref{v_count_n}) can be used to determine both components for each
modulation cycle. Each visibility profile so 
derived from equation \ref{v_count_n} then defines a portion of
the entire Fourier transform 
obtained on a circle of radius $n/p$ in the Fourier plane.

\section{Image Reconstruction} 

\subsection{Linear Methods: Backprojection and Fourier Transforms}

Back projection (Mertz, Nakano, and Kilner 1986) is the most
straightforward and basic method of image reconstruction. It is
equivalent to a Fourier transform (Kilner and Nakano, 1989.) A map
constructed by this method is called the "dirty map", being the
analogue of the radio astronomer's initial Fourier transform of the
observed visibilities.  Back projection is a perfectly linear process:
maps for arbitrary time intervals may be added together, and maps for
different pitches and harmonics may be summed, generally leading to
improvement of the image.  Further improvements to the image by CLEAN
or MEM (for example) do not share this property of linearity.
\medskip

The principle of back projection is to consider the quantities
$S_{ijk}$ in equation  (\ref{S_ijk}) as circular slices of the two-dimensional
modulation patterns $P_{im}$.  The modulation profile (\ref{S_ijk})
may be thought of as
the amplitude at an inertial position of a series of
two-dimensional modulation patterns that rotate around at the same
rate as the telescope.  The modulation patterns must shift in
phase and amplitude, depending on the relative position of the
telescope axis as a function of roll angle. 
\medskip

The brightness $D_m$ of each pixel (m) in the back-projection map is
defined by the following linear combination of the count rates $O_i$:

\begin{equation}
  D_m = \sum O_i W_{im} 
 \label{D_m}
\end{equation}

where the weights $W_i$ are linearly related to the modulation patterns 
$P_{im}$:

\begin{equation}
   W_{im} = (P_{im} - <P_m>)/(<P_m^2> - <P_m>^2)\cdot T_i \tau_i,
\label{W_im}
\end{equation}

and the averages $<P_m>$ and $<P_m^2>$ are taken over the range of
roll angles $\alpha_i$.  This particular normalization of the
backprojection weights $W_i$ was first derived by Durouchoux et al
(1983).  The explanation for the normalization of the weights $W_i$
derives from the fact that the predicted count rate $O_i$ for a point
source of strength $\cal S$ at sky position m is:

\begin{equation}
  O_i = {\cal S}\  T_i\tau_i\  P_{im}
 \label{O_pred}
\end{equation}

Inserting the last equation (\ref{O_pred}) into the equation for the
dirty map (\ref{D_m}) shows that the peak of the dirty map equals
the strength of the point source,

\begin{equation}
  D_m= {\cal S},
 \label{D_m_pred}
\end{equation}

since $\sum W_{im}  T_i\tau_i\  P_{im} =1$.  Although (\ref{D_m_pred})
is strictly true only for a single source, simulations show that it
produces nearly optimum back-projection maps for more complicated
sources.


\medskip

Figure \ref{bproj_fig} shows
an example of a back-projection image for a simulated double source
(Gaussian widths $=1 \times 1''$) and
the corresponding countrate profiles.


\begin{figure}
      \centering{
      \vspace*{-4.cm}
      \vbox{\psfig{figure=fig3.1:back-proj.eps,width=16cm,height=23cm}}\par
      }
      \label{bproj_fig}
\end{figure}



%\medskip
%
%\begin{eqnarray*}
%M_{ijk}(x,y) = F_0 \cdot \lbrace \tau_{ijk} T_{ijk} [1 &+& \\
%   \sum_n c_n^{ijk} cos[2\pi n
%[(x-x_c)cos\alpha &+& (y-y_c)sin\alpha]+\Phi_{ik}-\Phi_n^{ijk})] \rbrace
%\end{eqnarray*} %()

\medskip

Although it is possible to use only one modulation pattern per roll
angle to produce back-projection images, it turns out to be more
efficient to construct "universal" modulation patterns in the form of
cosine and sine pairs. Sinusoidal modulation patterns of arbitrary
phase can then be computed by appropriate linear combinations. This
formulation has the advantage that neighboring half rotations may use the
same "universal" set of patterns, even if there are different aspect
solution profiles.

\medskip

It is worth mentioning that in early versions of RMC design, there
were both sine and cosine subcollimators ({\it e.g.}, Mertz {\it et
al.} 1986), Murphy, 1990), but this turns out not to be necessary in
hardware, since with sufficient sampling of the modulation pattern,
relative amplitudes of the sine and cosine components can be
determined from the modulation profile itself.

\medskip

Each modulation pattern has contours of constant phase perpendicular
to the roll angle azimuth vector, and has the periods and amplitudes
used in equation (\ref{S_ijk}).  The modulation patterns so computed
for each angle bin, are added together with weights given by the count
rate divided by the product of the transmission factors and deadtimes.
The sum is the back-projection map.  For HESSI, simulations suggest
that summation of the 9-subcollimator back-projection maps gives a
dynamic range of order 10:1 for rates $\sim 10^5 counts^{-1} s^{-1}$.


\bigskip

\subsection{Nonlinear Reconstruction Methods}

All of the practical image reconstruction algorithms for improving a
back-projection (''dirty'') map are  nonlinear, 
since linear deconvolution algorithms such as Wiener filtering and 
inverse filtering are inapplicable to applications with incomplete sampling 
of the Fourier plane, as is the case for HESSI modulation synthesis.
A number of reconstruction methods already exist, or are being improved,
or are being considered for addition to the HESSI reconstruction suite.
We summarize several of these here.

\subsubsection{CLEAN (Cartesian, Polar)}
                    
CLEAN is an iterative algorithm which deconvolves a Point Spread
Function---the imager's response to a delta function source---from a
``dirty map''.  This algorithm is of fundamental importance in radio
astronomy, where it is used to create images of astronomical sources
obtained with interferometers.  The basic CLEAN method was developed
by H\"ogbom (1974). It was originally designed for point sources, but
it has been found to work well for extended sources as well when given
a reasonable starting model. The H\"ogbom CLEAN constructs discrete
approximations to the CLEAN Map in the plane from the convolution
equation

\begin{equation}                                                                             {\cal P} \otimes I_{source} = D
\end{equation} %()

where {\cal P} is the HESSI PSF for one or many
subcollimators and/or harmonics, D is the dirty map, and $\otimes$ denotes a
convolution.  The CLEAN algorithm starts with an initial approximation
$I_0=0$ to the residual map. At the $n^{th}$ iteration, it then searches 
for the largest value in the residual map.
A delta function is then centered at the location of the largest
residual flux and given an amplitude $\mu$ (the so-called ``loop gain'')
times this value.  ${\cal P}$ is
then subtracted from $I_{n-1}$ to yield $I_n$. 

\begin{equation}
  I_n = I_{n-1}\ -\   \mu  {\cal P}       
\end{equation} %()

Iteration continues until a
specified iteration limit is reached, or until the peak residual or
RMS residual decreases to some level. The resulting final map is
denoted $I_{final}$, and the position of each delta function is saved
in a ``CLEAN component'' table. At the point where component
subtraction is stopped, it is assumed that the residual brightness
distribution consists mainly of noise.  \medskip

To damp out high spatial frequency features which may be spuriously
created in the iteration, each CLEAN component is convolved with the
so-called CLEAN PSF, ${\cal P}_{clean}$ (the ``CLEAN beam'' in
radioastronomy), which is simply a suitably smoothed version or inner
portion of the PSF, ${\cal P}$.  A CLEAN map is produced when the final
residual map is added to the approximate solution,

\begin{equation}
    I_{clean}=I_{final} \otimes  {\cal P}_{clean}\ +\ \{D- {\cal P}\otimes I_{final}\}
\end{equation} %()

in order to include the noise.

\medskip
The main disadvantage of CLEAN is that it does not, at least in the
H\"ogbom version, compare the observed modulation profile with a model
modulation profile, to assess the ``goodness of fit'' during the iteration.
Two variants of CLEAN exist for HESSI.  One is for rectangular
coordinates, and the other is for polar coordinates.  In most
circumstances, the latter has a significant advantage of speed over
the former.
\medskip

Figure \ref{_fig} shows an example simulated source,
the ``dirty map'', the ``CLEAN map'', and the residuals.

\begin{figure}
      \centering{
      \vspace*{-2.cm}
      \vbox{\psfig{figure=fig3.2:clean.ps,width=17cm,height=17cm}}\par
      }
      \label{clean_fig}
\end{figure}


\subsubsection{Maximum Entropy Methods (MEM Sato, Polar MEM)}

The principle of Maximum Entropy is to find the map which is maximally
noncommittal with regard to the unavailable information ({\it i.e.}
the regions of the Fourier plane for which HESSI has no data).  The
image found is that which fits the data to within the noise level and
also maximizes the entropy.  In one view, the entropy is something
which, when maximized, produces a positive image with a compressed
range in pixel values.  (Cornwell 1984) Image entropy thus defined is
therefore not to be confused with a ``physical entropy''.  Several
kinds of "entropy" have been proposed to achieve this, but according
to Skilling (1984) and Sivia (1996) the only function which guarantees
that no untoward correlation is imposed is:

\begin{equation}
   {\cal H}\ = - \sum_{j=1}^M f_j \ ln(f_j) 
\end{equation} %()

The idea is to maximize the objective function Q:

\begin{equation}
   Q\ =\ {\cal H}\ - \lambda\ \chi^2 \ -\ \beta \sum_1^M f_j
\end{equation} %()

for some values of $\lambda$, where $\lambda$ is a parameter giving a
measure of the tug-of-war between maximizing entropy and minimizing
the $\chi^2$ statistic and $\beta$ is a Lagrange parameter multiplying
the flux constraint. At maximum,

\begin{equation}
   f_j =F-j\ e^{-\lambda \partial \chi^2/\partial f_j -\alpha}
\end{equation} %()

One may solve the equations (11) iteratively or by directed search of Q.
\medskip

One of the HESSI MEM reconstruction algorithms is {\it MEM\_Sato},
derived from the Maximum Entropy Method used for HXT imaging (Sato, 1998,
Sato,  Kosugi, and Makishima, 1999).


Figure \ref{mem_fig} shows an example simulated source, its count profile,
the ``MEM map'', and the residuals.

\begin{figure}
      \centering{
      \vspace*{-4.cm} %6
      \vbox{\psfig{figure=fig3.3:mem.ps,width=17cm,height=17cm}}\par
      }
      \label{mem_fig}
\end{figure}

\subsubsection{Forward Fitting (pixelized, non-pixelized)}

The forward-fitting method is based on models that represent a
spatial map by a superposition of multiple source structures, which
are quantified by circular Gaussians (4 parameters per source),
elliptical Gaussians (6 parameters per source), or curved ellipticals
(7 parameters), designed to characterize real solar flare hard X-ray
maps with a minimum number of geometric elements.

\medskip

In one realization of this scheme, maps of superimposed Gaussians are
created iteratively, at each step constructing a model modulation profile
from the map.  In this ``pixelized'' form, the map evolves into a best
fitting image.  A second form of forward-fitting is the unpixelized
version, where the parameters of the Gaussians are used to find the
``equivalent point source'', which is equivalent to an amplitude and phase 
at each roll angle.  The ``equivalent point source'' is then used to 
create the model modulation profile for comparison with the observed count
profile and continued iteration.  The unpixelized form requires at least
an order of magnitude less memory and computation than the pixelized form,
but the programs are more complex and their robustness is as yet unknown.
\medskip


\subsubsection{The PIXON Method}

The PIXON method is another technique removes the sidelobe
pattern of a telescope while mitigating the problems of
correlated residuals and spurious sources which are commonly seen in
Fourier deconvolution, chi-square fitting, and Maximum Entropy
approaches.
\medskip

The goal of the Pixon method is to construct the simplest,
i.e. smoothest, model for the image that would be consistent with the
data, i.e. have an acceptable chi-square fit. Being the simplest
model, the derived image would be artifact free, i.e. there would be
no spurious sources. In addition, the model is
necessarily be most tightly constrained by the data, and consequently
have the most accurately determined parameters.
\medskip


The PIXON method changes the global smoothness idea of MEM into a local
condition in which local maximum smoothness of the image is
imposed. From an information science point of view, one selects a
model with the minimum information content from the family
of multiresolution basis functions (pixons) and which
statistically fit the data. 
\medskip


Since the model has minimum complexity, spurious sources are unlikely
to arise.  Each parameter is determined using a larger fraction of the
data, hence it is determined more accurately. This usually results in
superior photometric and positional accuracy. And since the minimum
number of parameters are used, the data cannot be over fitted.
\medskip

\medskip

Figure \ref{clean_fig} shows an example simulated source, its count profile,
the ``PIXON map'', and the residuals.

\begin{figure}
      \centering{
      \vspace*{-2.cm} %-4, -8, -6
      \vbox{\psfig{figure=fig3.4:pixon.ps,width=17cm,height=17cm}}\par
      }
      \label{pixon_fig}
\end{figure}


\subsection{Undeveloped Techniques}

The available HESSI reconstruction algorithms in no way span
the universe of techniques.  A few of the likely methods to
be added to the existing suite of programs are outlined here.

\subsubsection{Wavelets}

HESSI has pitches at multiplicative intervals of $\sqrt{3}$, and so,
in a way it is a wavelet telescope, where the wavelet
scales are logarithmically spaced by  $\sqrt{3}$.  To date, no one has
exploited wavelet mathematics for HESSI imaging.
\medskip

\subsubsection{Non-Negative Least Squares}

A promising technique used for medical imaging (ref) is Non-Negative
Least Squares (NNLS). In general, linear Least Squares is not expected
to be useful for HESSI, but the enforcement of positivity in NNLS
provides an essential nonlinear constraint that may lead to good
images in the HESSI parameter domain.  

\medskip 

\subsubsection{Two-Dimensional Autocorrelation}

Mertz (1996) has suggested an autocorrelation technique for imaging
with RMCs.  It has the desirable properties of positivity and smaller
sidelobes and shows promise for fast mapping of single sources.
However, as Mertz points out, difficulties in processing complex
sources seem to limit its applicability.  It may, however, be useful
for centroid finding before final reconstruction.


\bigskip

\section{Statistical Limits}

Since HESSI is a photon-counting instrument, imaging involves
stochastic processes.  During the time during which the data for an
image are acquired, the incident photon rate $\lambda$ may be
considered a constant.  The number of photons per time bin is given by
Poisson statistics, where the probability density function
$p_{poi}(\lambda)$ is the expected number of photons per time bin
incident on the front grid.  The process of modulation is also
stochastic, since the incident phase relative to the collimator
fringes is random, and the probability of a photon detection is
proportional to the function (\ref{S_ijk}) normalized to unity at its
maximum.  The counts per bin is therefore given by the product of
these two probabilities:

\begin{equation}
  {\cal C}_i = p_{poi}(\lambda)\ S_{norm}
\label{ansatz}
\end{equation} %()

This is the {\it ansatz} used for simulations of HESSI count rate
profiles (Schmahl, 1998). Examination of equation (\ref{S_ijk}) for
$S_{ijk}$ shows that if its Fourier expansion is truncated at the
fundamental (as is often very useful), there are situations when S can
go negative.  This is not really a problem for backprojection or for
Fourier mapping, neither of which depend on positivity.  But it is of
importance for CLEAN, MEM, and Pixons.  Of singular importance, if it
is negative, the normalized version ($ S_{norm}$) of equation
(\ref{S_ijk}) fails to have any meaning as a probability.  

\medskip

\subsection{Fourier Coefficient Distribution}

The problem arises due to an expectation that  $S_{ijk}$ should
represent a count rate.  If, however, the Fourier coefficients
are considered as primary elements, then the statistics can be done with them,
regardless of their number and whether they produce a negative sum.
\medskip

Let us consider the fundamental only.  After choosing a map
center, at any given rotation angle, one can use the calibrated phase at
map center to determine the angular length of each modulation cycle. For the
fundamental, the Fourier coefficients can be computed from 4 angular bins,
assumed to be equally spaced within one cycle.  Let these bins have
counts \{$s_0,s_1,s_2,s_3$\}.  Then the DC, cosine and sine coeffcients,
derived either by the method of least squares or Fourier analysis,
may be written as:

\begin{eqnarray}
     c_0 &=& [s_0\ +\ s_1\ +\ s_2\ +\ s_3 ]/4  \\
     a_1 &=& [s_0\ -\ s_2]/2 \\
\label{cos}
     b_1 &=& [s_1\ -\ s_3]/2
\label{sin}
\end{eqnarray} %()


Then, since $\sigma^2 = \lambda$ for Poisson statistics, the standard
deviation for the DC coefficient is, as expected,
$\sigma_0=\sqrt{\lambda}/2$.  Given a value for $c_0$ (the mean cycle
flux), the allowed range of $a_1$ and $b_1$, due to the nonnegativity
of the countrates, is $-k/2$ to $k/2$, where $k$ is the number of
photons in the cycle.  The standard deviations for the sine and cosine
coefficients $a_1$ and $b_1$ are $ \sigma_0/ \sqrt{2}$.  

\medskip

$\cdots$

$\cdots$

$\cdots$

$\cdots$

\medskip A similar analysis can be done
for the higher harmonics $(n>1)$, where of course it is necessary to
have n bins per fundamental cycle for adequate sampling. 
%For the nth
%harmonic, the standard deviation of the sine and cosine coefficients
%is %$\sigma_n=\sqrt{\lambda/2n}$.  

\medskip 
\subsection{Regularization}

When the count rates are low, the time bins will contain very small
numbers of photons, perhaps all ones and zeroes, particularly for the
finest subcollimators, where the time bins must be small enough to
resolve the modulation cycles.  Even though the modulation cycles may
not be apparent to the eye, use can be made of the process called
``regularization'' to average cycles in phase. The binning that
regularizes the modulation profile is found by (1) sorting the
map-center phase so that it is increasing in time, (2) assigning
cycle boundaries whereever the phase is a multiple of $2\pi$, (3)
dividing each cycle into 4 bins, and (4) taking the mean count rate in
each bin.  This process is illustrated in Figure \ref{regularization} 

\begin{figure}
      \centering{
      \vspace*{-4.0cm}
      \vbox{\psfig{figure=fig:regularized_bins.ps,width=17cm,height=25cm}}\par
      }
      \label{regularization}
\end{figure}

\medskip 

After regularization, one may make in-phase averages of the
modulation profile with successively smaller ``super bins''.
Starting with $\sim 1/4$ the rotation period, one tests $c_0-c_2$ and
$c_1-c_3$ for significance, then steps down to smaller superbins,
testing at each level until one reaches the lowest timescale limited
by counting statistics.  Each of the differences so produced gives
a Fourier representation of the amplitude and phase of the
modulation profile for use in modeling ({\it e.g.} Forward Fitting)
the source.

\medskip 

It follows logically from the above scheme that data from the finer
grids contain as much information about point sources as the coarser
grids, regardless of the count rate (provided only that the grid
transparancies are the same).  This can also be seen by the following
argument.  Given one subcollimator of angular pitch $p_n, \ (n=1-9)$,
the modulation profile of two point sources separated by a distance
$\Delta x = p_n=4.3'' \times \sqrt(3)^{n-1}$ shows slow ($\it \Delta t 
\sim 1\ s$) amplitude modulation caused by ``beating'' of the sources.
During one half rotation, the amplitude has maxima at angles where the
modulation is fastest ({\it i.e.} where the phase at map center is
near zero), and minima at angles where the modulation is slowest ({\it
i.e.} where the phase at map center is near maximum).  By simple
geometrical scaling of both the instrument and the source geometry,
this statement is true for all $p_n$.  That is, for the finest
subcollimator, two point sources $4.3''$ apart produce the same
amplitude modulation as two point sources $4.3\times 3^4 = 348''$
apart would produce for the coarsest subcollimator. Extraction of the
imaging information for the finest grids is done automatically by the
backprojection, FT, CLEAN and MEM algorithms, so that even if
counts/timebin are of the order of unity and modulation is not
apparent to the eye, the algorithms make use of the information
contained in the data.  For Forward Fitting, the regularization 
scheme described above seems to be necessary.
$\cdots$ 

$\cdots$ 

$\cdots$

$\cdots$ 

$\cdots$ 

$\cdots$

\section{Figure Captions}

\begin{itemize}

\item[ ]{}
 Figure \ref{imager_schematic}

Schematic geometry of the HESSI subcollimators, showing representative
incidence of photons with respect to the collimator axis.  

\item[ ]{}
 Figure \ref{slit_pitch_effect}

Profiles of the equivalent area function for various
slit/pitch ratios, and also showing energy-dependent effects that smooth
the corners of the profile.

\item[ ]{}
 Figure \ref{symb_diag}
Diagram showing the relative positions of the spacecraft coordinates,
heliocentric coordinates and the mapped region.  Note that the
spacecraft rotates clockwise as viewed against the Sun ({\it i.e.} 
$\alpha$ increases with time). The projected distance $\Theta$ from
the line of subcollimator peak response to the map center is the
{\it phase at map center} used in the {\it calibrated event list}.
 
\item[ ]{}
 Figure  \ref{bproj_fig}

An example of a back-projection image for a simulated source and
the corresponding countrate profile.

\item[ ]{}
 Figure  \ref{clean_fig}

An example simulated source, its count profile,
the ``dirty map'', the ``CLEAN map'', and the corresponding 
count profile derived from the CLEAN map.


\item[ ]{}
 Figure \ref{mem_fig}

An example simulated source, its count profile,
the ``MEM map'', and the corresponding 
count profile derived from the MEM map.

\item[ ]{}
 Figure \ref{pixon_fig}

An example simulated source, its count profile,
the ``PIXON map'', and the corresponding 
count profile derived from the PIXON map.

\item[ ]{}
 Figure \ref{regularization}
The binning that regularizes the modulation profile.

\end{itemize}


\section*{References}

\begin{itemize}

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% Cornwell, T. J. , ``Is Jaynes' maximum entropy principle
% applicable to image reconstruction?'', in Indirect Imaging,
% J. A. Roberts, Ed., Cambridge University Press (Cambridge, England),
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\item[ ]{}
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\item[ ]{}
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\item[ ]{}
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\item[ ]{}
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\item[ ]{}
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% Hurford, Megeath, Palmer and Prince, SRL 86-15, Presented at Topical Meeting
% on Quantum-Limited Imaging and Image Processing; Honolulu, Hawaii, 
% sponsored by Optical Society of America, published in OSA Technical Digest, 
% 1987.

\item[ ]{}
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\item[ ]{}
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\item[ ]{}
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\item[ ]{}
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\item[ ]{}
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\section{Appendix}
\begin{eqnarray*} 
 F_0           &=& {\rm Incident\ photon\ flux\ on\ front\ grid} \\
 T             &=& {\rm Subcollimator\ transmission }\\
 \tau          &=& {\rm Livetime\ \qquad            }\\
 \Phi          &=& {\rm Phase\ at\ map\ center      }\\
 c_n           &=& {\rm Modulation\ Amplitude}, \quad n=1,2,3,4,5\\
 \Phi_n        &=& {\rm Modulation\ Phase}, \quad n=1,2,3,4,5\\
p              &=& {\rm angular\ pitch}\\
L              &=& {\rm separation\ between\ grids}\\
 p/(2L)        &=& {\rm angular\ resolution}\\
S_{triangle}   &=& {\rm effective\ subcollimator\ area\ for\ ideal\ thin\ grids}\\
\theta         &=& {\rm the\ photon's\ angle\ of\ incidence\ relative\ to\ the\ imager axis}\\
c_n            &=& {\rm modulation\ amplitude\ for\ harmonic\ n }\\
\theta_n       &=& {\rm modulation\ phases\ for\ harmonic\ n}\\
g_n            &=& {\rm Fourier\ coefficients\ of\ the\ grid\ transmission}\\
 q             &=& {\rm effective\ slit/pitch\ ratio}\\
\theta_n       &=& {\rm  relative\ phases\ for\ harmonic\ n}\\
\alpha         &=& {\rm  roll\ angle  }\\
P_0            &=& {\rm  modulation\ profile}\\
\Delta t_i     &=& {\rm  time\ bin sizes }\\
\Phi           &=& {\rm  phase\ at\ map\ center}\\
\tau           &=& {\rm  dead\ time}\\
 T             &=& {\rm  subcollimator\ transmission}\\
E              &=& {\rm  energy}\\
u,v            &=& {\rm  coordinates\ in\ the\ Fourier\ plane}\\
\phi           &=& {\rm  azimuthal\ coordinate\ in\ the\ Fourier\ plane}\\
k              &=& {\rm  radial\ coordinate\ in\ the\ Fourier\ plane}\\
{\cal V}       &=& {\rm  visibility}\\
{\cal C}       &=& {\rm  predicted\ count\ rate}\\
x_{coll}       &=& {\rm  subcollimator\ axis}\\
x_{map}        &=& {\rm  map\ center}\\
R_0,\phi_0     &=& {\rm  location\ of\ the\ map\ center }\\
\Delta\Phi     &=& {\rm  phase\ shift\ from\ telescope\ axis\ to\ sun\ center}\\
P_{im}         &=& {\rm  two-dimensional\ modulation\ patterns }\\
D_m            &=& {\rm  brightness\ of\ pixel\ m\ in\ the\ back-projection\ map }\\
O_i            &=& {\rm  predicted\ count\ rate\ in\ bin\ i }\\
{\cal P}       &=& {\rm  HESSI\ Point\ Spread\ Function }\\
\mu            &=& {\rm  the\ loop\ gain\ in\ CLEAN}\\
I_n            &=& {\rm  residual\ map\ intensity\ at\ nth\ iteration\ of\ CLEAN}\\
{\cal H}       &=& {\rm  Entropy}\\
\lambda        &=& {\rm  parameter\ in\ MEM}\\
p_{poi}        &=& {\rm  Poisson\ variate}\\
S_{norm}       &=& {\rm \ probability\ of\ a\ photon\ detection}\\
a_1, b_1       &=& {\rm  cosine\ and\ sine\ Fourier\ coeffcients}
\end{eqnarray*} 


\end{document}
\end



% RMCs that have flown in space:

% Ariel-V (Sanford 1975),

% SAS-3 (Mayer 1972) and 

% Hakucho (Kondo et al.Space Science Instr, 5, 211, 1981)

% Granat with 4 RMCs (Brandt et al. Adv Space Res 10, 239, 1990)

% Balloon RMCs:

% Theinhardt et al. 1984)

%                Results from the First Flight of the High Energy Imaging Device (HEIDI) Balloon Payload
% Authors:
%
%                Crannell, C. J.; Dennis, B. R.; Gaither, C. C., III; Hartman, C. N.; Orwig, L. E.; Lang, F. L.; Starr, R.; Schmahl, E. J.;
%                Greene, M. E.; Tan, H.; Hurford, G. J.; Johnson, W. N.
% Affiliation:
%
%                AA(NASA/GSFC), AF(CUA and NASA/GSFC), AH(UMd and NASA/GSFC), AI(Auburn U), AK(Caltech), AL(NRL)
% Journal:
%
%                American Astronomical Society Meeting, 183, #08.05
% Publication Date:
%
%                12/1993
% Origin:
%
%                AAS
% Abstract Copyright:
%
%                (c) 1993: American Astronomical Society
% Bibliographic Code:
%
%                1993AAS...183.0805C
