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<H2><A NAME="SECTION001421000000000000000">
Patterns</A>
</H2>
Images BARS and OFFBARS, below, are of SCALE = 3; BARS30 and BARS60
have SCALE = 4; and RINGS and OFFRINGS have SCALE = 2. All of these
images have CUTS defined as 0, 100. Reference may be made to Figs.
16.1 and 16.2 for a reproduction of these images, produced using the
SET/SPLIT feature of MIDAS.
<P>
<DL>
<DT><STRONG>BARS.BDF</STRONG>
<DD><P>
consists, firstly, of three rectangular bars; each of
length 25 pixels; of width 1, 4, and 9 pixels; horizontally
positioned; and flush with pixel
boundaries. The uniform flux density in each pattern is 100 units per pixel.
The three rectangular patterns are also vertically positioned,
again flush with pixel boundaries. Note that because of this alignment of
images and pixel grids and because of the dimensions of the images being
in integer numbers of pixels -- which is not
likely to occur in reality -- no effect of the spatial discretisation
(i.e. of the necessarily finite pixel size) is visible.
<P>
<BR>
<DIV ALIGN="CENTER"><A NAME="image-bars"> </A><A NAME="9733"> </A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 9.1:</STRONG>
Images BARS, OFFBARS, BARS30 and BARS60.</CAPTION>
<TR><TD></TD></TR>
</TABLE>
</DIV>
<BR>
<P>
In order to illustrate the case of a point spread function which in
one dimension is extremely undersampled, a delta function is also
included (the lower left hand pattern in BARS). It is defined as:
<!-- MATH: $f(x,y) = 100$ -->
<I>f</I>(<I>x</I>,<I>y</I>) = 100 where <I>y</I> = <I>y</I><SUB>0</SUB> and
<!-- MATH: $x_{min} \leq x \leq x_{max}$ -->
<IMG
WIDTH="122" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img412.gif"
ALT="$x_{min} \leq x \leq x_{max}$">,
for
given <I>y</I><SUB>0</SUB>, <I>x</I><SUB><I>min</I></SUB> and <I>x</I><SUB><I>max</I></SUB>. Note that the one-pixel wide
pattern, flush with pixel boundaries, cannot be distinguished from the
delta function.
<P>
<DT><STRONG>OFFBARS.BDF</STRONG>
<DD><P>
has the same patterns as BARS, but offset to a small amount in <I>x</I> and
<I>y</I> relative to pixel boundaries. In the horizontal (x) direction,
the offset is 0.4 of a pixel side length, and in the vertical (y)
direction it is 0.2 of a pixel (both in the positive direction).
<P>
This yields for the rectangular patterns flux values of 32.0, 40.0,
and 8.0 in the top left corner, along the boundary, and in the top
right corner respectively; corresponding figures for the lower part of
each pattern are: 48.0, 60.0, and 12.0; finally the flux along the
left and right boundaries are 80.0 and 20.0. The comparison with BARS
shows the effect of discrete steps quite clearly.
<P>
The delta function in OFFBARS is defined on a different interval of
the x-axis. Its constant flux is apportioned to the pixels it runs
through in the horizontal direction, but in the y (vertical axis)
direction it has infinitesimal extension. In other words, shifting a
delta function perpendicularly to the axis to which it is parallel has
no effect while this is not true of the 1-pixel wide bar.
<P>
Note that flux conservation is at all times verified, both locally
(single pixel) and globally.
<P>
<DT><STRONG>BARS30.BDF</STRONG>
<DD><P>
rotates the horizontal patterns of BARS through -30 degrees.
<P>
In all cases the center of rotation is the pixel in the lower left
hand corner of the rectangular patterns. These pixel coordinates are
10, 20 and 34 pixel units distant from the delta function whose left
hand pixel has coordinates (42,52).
<P>
The delta function is again taken as a line of given starting point,
of given length, of given angle to the horizontal, and of
infinitesimal width. Note the difference between this and the 1-pixel
wide bar.
<P>
In order not to depend on any interpolation algorithms etc., creation
of these patterns was carried out by subdividing pixels, rotating the
subpixels, and accumulating flux contributions of these subpixels.
For a subdivision of
<!-- MATH: $299 \times 299$ -->
<IMG
WIDTH="53" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img413.gif"
ALT="$299 \times 299$">
the elapsed time may be a
clearer indicator of reconstruction time: BARS30 took approximately 30
hours on a VAX 11/785 machine.
<P>
An estimate of precision was obtained in the following way. For the
central portion of the larger rectangles, flux of 100 ought to
obtained. If pixel subdivision is coarse, values greater than or less
than 100 will be obtained. Hence, the sampling of pixels where the
value 100 is expected allows an estimate of accuracy, and should
additionally vary proportionally to <I>n</I><SUP>2</SUP> where pixel subdivision is
<!-- MATH: $n \times n$ -->
<IMG
WIDTH="12" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img414.gif"
ALT="$n \times n$">.
(The latter statement is based on empirically noting
that a small constant number of subpixels -- usually 1 -- were
``scattered'' in being projected. This <I>n</I><SUP>2</SUP> precision dependence was
empirically verified.) For <I>n</I> = 299, accuracy defined in this way
was determined as 0.003 units, i.e. in no pixel the residual from 100
units is larger than this value.
<P>
<DT><STRONG>BARS60.BDF</STRONG>
<DD><P>
was constructed in an analogous manner, with the same precision, as
BARS30. In this case, the angle of rotation was +60 degrees. The
coordinates of the left hand pixel of the delta function are again
(42,52), and the other patterns have lower left hand pixels distant by
10, 20 and 34 pixel units.
<P>
<DT><STRONG>RINGS.BDF</STRONG>
<DD><P>
relates to centered, circular patterns. The patterns here are a delta
function (infinitely thin) at radius 10 units, and rings of
thicknesses 1, 5 and 10 units at inner surface radii 22, 33 and 48
steps. Uniform flux density of 100 units/pixel is used. Additionally
flux 100 is located in the pixel at the center of the set of
concentric rings. Note again the difference between the delta
function and the ring of thickness 1: the former distributes flux (100
units) to any pixel which is overlapped by a mathematically-defined
circular curve; while the ring of thickness 1 distributes flux to
pixels proportionally to their overlap with the ring.
<P>
<BR>
<DIV ALIGN="CENTER"><A NAME="image-rings"> </A><A NAME="9742"> </A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 9.2:</STRONG>
Images RINGS and OFFRINGS.</CAPTION>
<TR><TD></TD></TR>
</TABLE>
</DIV>
<BR>
<P>
Total flux in the delta function was defined as
<!-- MATH: $2 \pi r \times 100$ -->
<IMG
WIDTH="54" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img416.gif"
ALT="$2 \pi r \times 100$">,
and subdivision was carried out in intervals of 0.025 degree.
<P>
Technically, the approach adopted in constructing these patterns was
to subdivide pixels, rotate (or otherwise project) the small
subpixels, and accumulate the flux (cf. BARS30 and BARS60). This
allows the precision of the annuli to be assessed by noting that the
approximation improves as the subpixels get smaller, and that the
typical flux excess or deficit in one output pixel amounts to the flux
content of one subpixel. Per output pixel, this gives a maximum error
of
<!-- MATH: ${100 \over n^2}$ -->
<IMG
WIDTH="60" HEIGHT="80" ALIGN="MIDDLE" BORDER="0"
SRC="img417.gif"
ALT="${100 \over n^2}$">
(where <I>n</I><SUP>2</SUP> subpixels correspond to the original
pixel with flux 100), and a mean error of half this value (where we
assume uniform statistical fluctuation; a histogram analysis of the
frames created confirms this). For <I>n</I> = 101, we therefore obtain a
maximum error estimate of less than 0.01 units.
<P>
RINGS is perfectly symmetric under rotation through multiples of
90 degrees.
<P>
<DT><STRONG>OFFRINGS.BDF</STRONG>
<DD><P>
is analogous to RINGS, except offset from the center of a pixel by
(0.3,0.25) units. Note the effect of this offset on the flux
distributed uniformly in a square of pixel dimensions at the center;
and note also that shifting the annuli as done in OFFRINGS necessarily
destroys the symmetry of the annuli. In other respects (time required
for construction, precision) OFFRINGS presents similar characteristics
to RINGS.
<P>
</DL>
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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