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Astronomical Data Analysis Software and Systems VII
ASP Conference Series, Vol. 145, 1998
R. Albrecht, R. N. Hook and H. A. Bushouse, e
Ö Copyright 1998 Astronomical Society of the Pacific. All rights reserved.
ds.
Substepping and its Application to HST Imaging
Nailong Wu and John Caldwell
Dept. of Physics and Astronomy, York University, 4700 Keele St.,
North York, Ontario, M3J 1P3, Canada
Abstract. The substepping technique is used for the Hubble Space
Telescope (HST) imaging to cope with the problem of undersampling and
to improve resolution. In this paper, this technique is first introduced in
the language of signal/image processing. Then its application to HST:
FOS ACQ imaging and WFPC2 subpixel dithering, is described. Its
possible use for NICMOS is also discussed.
1. Introduction
The substepping technique is used to ameliorate the problem of undersampling
in data acquisition for an imaging system. For the HST, undersampling means
that the size of the pixel in a camera is larger than the critical value determined
by the optics of the telescope, and aliasing takes place.
Obviously, the problem of undersampling could be resolved if the pixel size
could be reduced. However, in many circumstances, the pixel size is fixed. Then,
substepping is the only solution to the problem. In essence, substepping means
that data are acquired in steps smaller than the pixel size, and then processed
to achieve resolution comparable to the step size.
2. Substepping in Data Acquisition and Reconstruction
In this section one­dimensional (1­D) notation and diagrams are used for clarity.
Results can be easily extended to the two­dimensional (2­D) case by changing
dimensional subscripts and operations to 2­D.
2.1. Image Formation and Sampling
The brightness distribution, a 0 (x), in the field of view is convolved with the point
spread function (PSF) of the optics of the telescope to form a continuous image,
a(x), which has lower resolution than a 0 (x) due to the convolution. Resolution
can by improved by deconvolution of a(x) with respect to the PSF to restore
a 0 (x).
Sampling is an operation to integrate a(x) within a pixel, and to assign the
result to this pixel (Figure 1A). Each pixel has its nominal position at the center
of the integral interval.
82

Substepping and its Application to HST Imaging 83
2.2. Substepping
To avoid aliasing, the sampling interval #x must be equal to (critical sampling)
or smaller than (oversampling) its critical value #x c , which is determined by the
highest frequency component in a(x). If #x > #x c (undersampling), aliasing
will take place.
In order to overcome the problem of undersampling (Figure 1A), we may
reduce the pixel size by a factor of N , such that 1/N#x # #x c (proper sampling,
including critical sampling and oversampling; Figure 1D for N = 2). We call
this normal sampling because both the integral and sampling intervals are equal
to the pixel size.
x
D
Integral interval
x
D
1/ 2
Sampling interval
x
D
Integral interval
x
D
Sampling interval
x
D
Integral interval
x
D
Sampling interval
( )
n
c
k
( )
n
k * ­1
Kernel
Normal sampling
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1 0
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0
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D
1/ 2
Integral interval
x
D
1/ 2
Sampling interval
( )
a x
Substepping (2 channels)
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01
Undersampling : Proper sampling :
Reduce pixel size to half
x x
x
x
1
1
Interlace
(B)
(A)
(C)
(D)
(E) ­1 0 x
+1
0 x
*
1 2
0 ­2 ­1 0 1 2 3
­1 0 1 2 3 4
Channel 1 :
Channel 2 :
2 channels :
Figure 1. Substepping (N=2) and normal sampling. The vertical
bar at a pixel's center represents the integral value. # and # -1 denote
convolution and deconvolution, respectively.
In the case where reducing the pixel size is impossible, we can use N integral­
sampling devices (channels) to acquire data. The resulting N sequences shifted
successively by 1/N#x (Figure 1A,B) are combined in a manner of interlacing
(C), so that the sampling interval becomes 1/N#x. We call this subsampling or
substepping because the sampling interval is equal to a subpixel size while the

84 Wu and Caldwell
integral interval is still equal to the pixel size. If N is su#ciently large such that
1/N#x # #x c , aliasing will be eliminated, and consequently resolution will be
improved.
2.3. Reconstruction Using the Substepped Sequence
Let us compare substepping (Figure 1C) with normal sampling (D), both with
the sampling interval 1/N#x. For the former, the integral interval is larger, and
the smoothing e#ect of integration is stronger. Therefore, its resolution is lower
than that of normal sampling.
The substepped sequence is a moving­sum of the normal­sampled sequence.
For instance, the pixel value at x = 1 in Figure 1C comes from the pixel value
at x = 1 in A, which is equal to the sum of the pixel values at x = 0, 1 in D.
The moving­sum operation is, in fact, a cross­correlation of the normal­
sampled sequence with the kernel k c (n) (Figure 1E). However, this cross­correlation
is equivalent to a convolution with the kernel k(n), k(n) = k c (-n). Therefore,
deconvolution of the substepped sequence with respect to the kernel k(n) can
be carried out to reconstruct the normal­sampled sequence. As a result, the
smoothing e#ect due to the moving­sum is eliminated and resolution is improved.
In summary, when the pixel size is too large, substepping in data acquisition
can be employed to eliminate aliasing. The resulting N sequences from N chan­
nels are interlaced. This operation alone can improve resolution. However, the
improvement in resolution is made mostly by deconvolution of the substepped
sequence with respect to the kernel k(n) to eliminate the moving­sum e#ect in
substepping, i.e., to weaken the smoothing e#ect of the integration operation.
Deconvolution with respect to the PSF (Sect. 2.1.), which can be used to
improve resolution (restore a 0 (x) from a(x)), is independent of reconstruction
of substepped data.
3. Application to HST Imaging
3.1. WFPC2 Subpixel Dithering
Undersampling with WFPC2 occurs because of the large pixel size of the CCD
chips. The substepping technique used to overcome this problem is named ``sub­
pixel dithering'' or simply ``dithering'' (Figure 2a).
During an observation, the pointing of the telescope is changed so that
successive images are shifted along each axis by subpixel amounts. Then, these
images are combined to obtain a single image having a smaller pixel size on a finer
grid, using the POCS­based method (Adorf 1995), ``Drizzling'' method (Hook &
Fruchter 1997), and deconvolution­based methods like acoadd, crcoad, mem,
and lucy in IRAF/STSDAS.
3.2. FOS ACQ Imaging
The undersampling problem with FOS arises from the large size of the diodes
used for data acquisition. In substepping (Figure 2b), the step of the diode's
motion is one quarter of its width in the X­direction, and one sixteenth of its
height in the Y­direction.

Substepping and its Application to HST Imaging 85
y
D
x
D
y
x
0
Pixel
Sub­pixel
(a)
y
D
x
D
y
x
0
Diode
Pixel
(b)
Figure 2. Substepping for HST imaging. Figure 3. An FOS ACQ
(a) WFPC2 dithering (N x = N y = 2). image (left) and its recons­
(b) FOS ACQ imaging (N x = 4, N y = 16). truction by mem (right).
The diode and the 64 (= 4 â 16) small areas on it would normally be
called the pixel and subpixels, respectively, according to standard substepping
terminology. However, in this particular case the small areas have commonly,
but inaccurately, become known as pixels.
The deconvolution with respect to k(n x , n y ) is called reconstruction. The
deconvolution task mem or lucy, or the direct inversion task tarestore, in
IRAF/STSDAS can be used for this purpose. tarestore results in high level
sidelobes (rings) and noise in reconstructed images; mem and lucy give much
better results (Wu & Caldwell 1997).
The smoothing e#ect due to the large diode size badly reduces resolution
in FOS ACQ images. Reconstruction can remove this e#ect and dramatically
improve resolution. Figure 3 shows an image, before and after reconstruction,
of a star behind the bar in an FOS barred aperture.
3.3. NICMOS Imaging
In NICMOS imaging, PSFs are critically sampled at # c = 1.0 and 1.75 µm for
Cameras 1 and 2, respectively. When a working wavelength is shorter than # c ,
the problem of undersampling occurs. The situation here is similar to WFPC2
in principle. Therefore, the substepping technique can be used.
References
Adorf, H.­M. 1995, in Astronomical Data Analysis Software and Systems IV,
A.S.P. Conf. Ser., Vol. 77, eds. R. A. Shaw, H. E. Payne & J. J. E. Hayes
(San Francisco: ASP), 456
Hook, R. N. & Fruchter, A. S. 1997, in Astronomical Data Analysis Software
and Systems VI, A.S.P. Conf. Ser., Vol. 125, eds. G. Hunt & H. E. Payne
(San Francisco: ASP), 147
Wu, N. & Caldwell, J. 1997, in Proceedings of 1997 HST Calibration Workshop,
Space Telescope Science Institute, ed. S. Casertano, et al. (in press)