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arXiv:1304.6530v1 [astro-ph.IM] 24 Apr 2013
PARABOLIC STRIP TELESCOPE
G.CHADZITASKOS
DEPARTMENT OF PHYSICS,
FACULTY OF NUCLEAR SCIENCES AND PHYSICAL ENGINEERING,
CZECH TECHNICAL UNIVERSITY, BŘEHOVÁ 7, CZ - 115 19 PRAGUE
E-MAIL: GOCE.CHADZITASKOS@FJFI.CVUT.CZ
Abstract. We present a proposal of a new type of telescopes using a rotating parabolic strip
as the primary mirror. It is the most principal modification of the design of telescopes from the
times of Galileo and Newton. In order to demonstrate the basic idea, the image of an artificial
constellation observed by this kind of telescope was reconstructed using the techniques described
in this article. As a working model of this new telescope, we have used an assembly of the
primary mirror—a strip of acrylic glass parabolic mirror 40 cm long and 10 cm wide shaped as
a parabolic cylinder of focal length 1 m—and an artificial constellation, a set of 5 apertures in a
distance of 5 m illuminated from behind. In order to reconstruct the image, we made a series of
snaps, each after a rotation of the constellation by 15 degrees. Using Matlab we reconstructed
the image of the artificial constellation.
Keywords: Instrumentation: high angular resolution, Techniques: image processing, Tele-
scopes.
1. Introduction
All known telescopic systems are of two basic types: the Galileo–Kepler using lenses (or lenses
with corrections of optical defects) as primary optical elements (refractors), or the Newton–
Cassegrain with spherical or paraboloid mirrors (reflectors) as the primary elements. The image is
observed using photographic cameras, CCD cameras or a light spectrometers. The images from the
CCD or the spectrometers are stored in computers for further elaboration. In all cases, the images
are exposed in the image plane. The primary elements of the reflectors and of the refractors collect
parallel rays of light in the focal plane, and other optical elements are used in different geometrical
configurations. For more details see [1].
As a rule, telescopes are supported by mountings which make it possible to point their optical
axes to the object to be observed. In the case of moving objects, the mounting makes the optical axis
able to follow the position of the object. There are several types of mountings, the most important
are parallactic, azimuthal, and four axes mountings. The azimuthal mounting has horizontal and
vertical rotational axes, and it is necessary to perform both rotations simultaneously during the
observation of moving objects. The parallactic mounting has two perpendicular axes, one axis is
parallel to the Earth’s axis, and a suitable rotation around this axis makes the telescope follow the
celestial objects. Most contemporary mountings are controlled by computers.
Up to now, telescopes have their primary elements of circular or regular polygonal form. The
angular resolution of such telescopes is almost the same in all lateral directions, and it is defined as
the smallest angle between two close points which can be distinguished. The angular resolution is
the ability of a telescope to render detail: the higher resolution, the finer details can be observed.
The angular resolution, together with the aperture size, are the two most important characteristics
of telescopes. In the case of circular primary mirror of diameter D, its area Pand angular resolution
1
δare
P=πD2
4, δ ≈1,22 λ
D
for a monochromatic light of wavelength λ[2].
There are two reasons for the construction of very large telescopes: to gain better angular
resolution and to collect more light [3]. In those cases improving the resolution and collecting more
light means increasing the diameter of the primary mirror.
2. Telescope with a rotating objective element
The basic idea of our new system has been inspired by X-ray computer tomography (CT). An
X-ray source is located on one side and an X-ray camera on the opposite side of a sample. The
integral absorptions of X-rays in different angles step by step during the rotation of the sample are
measured. The total absorption of all photons coming along different lines perpendicular to the
camera are registered as points of a one-dimensional picture. Finally, the inverse Radon transform
is used to reconstruct the image of absorption of X-rays in different points of the media.
Modern mathematical methods and software developed for CT involve gathering projection data
from multiple directions and feeding the data into a tomographic reconstruction software algorithm
processed by a computer [3]. Mathematical filters are used to improve the reconstructed image
[4]. The idea and the image processing of the technology of Single-Photon Emission Computer
Tomography (English 1996) is the same as for our parabolic strip telescope. This technology
is used in nuclear medicine where a patient is injected with a radiopharmaceutic which emits
gamma rays. The emitted gamma rays are collected by a gamma camera and the emitted image
is reconstructed [7].
The same approach can be used when a parabolic strip is the primary mirror (h, i) of a telescope,
following the schematic shown in Figure 1. The images of observed objects are lines. Each line
represents the integral intensity of light incoming from the ob ject or objects perpendicular to the
strip (parallel to the focal line) located inside the field of view which is guaranteed by the geometry.
Making a series of photos while rotating the telescope around its optical axis, one can use the inverse
Radon transform to reconstruct the image with the above mentioned angular resolution. It is also
possible to use other tools, for example Matlab [7]. Secondary optic elements can be used to focus
the lines from the focal plane into points.
The technological construction of the parabolic strip is simpler than planar or paraboloid surface,
because the stress of material helps to maintain the geometry. A precise parabolic bracket is
required to hold the parabolic strip made of elastic mirrors. It is also simpler to use adaptive
optics for the correction of defects. One can control the surface by a laser pointer located on one
side of the strip. By the detection of its reflected light on the other side of the strip during the scan,
the whole surface can be controlled and corrected by corrective elements. Of course, secondary
mirrors can be added, then we have a Newton–like or Cassegrain–like telescope. The area and the
angular resolution of a parabolic strip telescope are
P=LW, δL=λ
L,
where λis the wavelength of light, Lis the length of pro jection of strip on tangent plane at the
vertex line—in Figure 2 it is the length of x-axis projection of the strip—and Wis the width of
the strip.
The proposed telescopic system involves preferably parallactic mounting and an instrument for
the digitalization of the image connected to a computer.
The mounting, however, has to perform one more rotation than the usual mounting: a rotation
of the primary element or of the whole system around the optical axis is necessary, in order to
reconstruct the details of the object. The instrument for the digitalization of images has to be
located in the image plane of the telescope.
For refractors the new primary elements are cuts of parabolic cylinders and instruments for the
digitalization of the image are located in the image plane of the telescopes. Of course, this idea
can also be used for radio telescopes.
3. Advantages
We have proposed a fundamental modification of reflectors where the angular resolution can be
better than in the case of usual objectives of the same area. The reconstruction of images exploits
the discrete inverse Radon transform [6] or similar mathematical methods. The main advantages
of such telescopes are
•good angular resolution,
•low expenses,
•simple technological development,
•possibility to install a grid of large telescopes across the Earth,
•lower weight for use on satellites.
The only major complication is that one more rotational movement is needed to reconstruct the
image with the same angular resolution in all directions.
The good angular resolution can be used for direct observation of bright objects. The additive
rotation is not necessary for some purpose, for example when the plane of rotation of rotating
objects is known, and changes of orbits are observed. For example, the Jupiter-like planet around
close stars can be directly detectable by a 60 meter telescope, from point of view of the requirement
angular resolution.
The relatively low cost of parabolic strip telescopes makes it possible to cover the Earth from
North to South by a grid of telescopes. Each of them would observe, for example, 30 degrees
around the zenith. This would reduce the cost of mountings and multiply the observation time.
The cost of telescopes observing ±15 degrees will be reduced because of:
•high mounting not being necessary,
•low wind influence,
•low gravitational influence.
4. Secondary optics
The parabolic strip telescope offers the possibility to use secondary optical elements to focus light
from the primary mirror to short segments or points. But these secondary elements may introduce
errors or noise in resulting images. The most natural and easiest way is to use a "software"
secondary element, i.e. a computer process on images created by the primary mirror. This can be
explained on the following simple example. Let us observe two distinguishable objects lying on a
line parallel to the vertex line. The points represent objects whose structure is below the angular
resolution of a telescope in that direction. The image of each object is a segment of length W that
is parallel with the vertex line, and both segments lie on the same straight line. The centers of
the segments are located symmetrically to the optical axis in directions according to the reflection
law. Depending on the strip width W and the angle between the two objects, the two segments
are either separated or overlapping. If the segments are separated, their images are placed in the
centers of the segments. If the segments are overlapping, then a simple analysis gives the position
of images of the objects. One has to subtract the half width of the parabolic strip W
2from one
end of the overlapping segments and from the opposite end of the overlapping segments, and these
are the positions of objects on the image. In Figure 5 the end points of displaced segments are
shown. The segments are images of the artificial constellation. Angular resolution in determining
the position of a point in the direction of the vertex line is δW=λ
Wand is substantially less than
in the perpendicular direction δL=λ
L. This difference in angular resolution in two perpendicular
directions can serve for observation of some particular phenomena. For example, to monitor fine
movements of an object in the known direction, the telescope has to be adjusted so that in this
direction the best angular resolution is achieved. During computer processing all the photons
reflected from the primary mirror are recorded by pixels of a CCD camera in the image plane and
no one is lost. Using a secondary parabolic strip or a secondary parabolic lens from the Hartley
disc, the segments can be focused into points. Then the inverse Radon transform reconstruct the
image with appropriate angular resolutions.
5. The proof-of-principle experiment
In order to show that the principle works we have prepared a very basic experiment. For the
sake of simplification, the telescope was stationary. Figure 3 shows manufactured parabolic strips
of lengths 20 and 40 cm. In the experiment the 40 cm strip was used. The artificial constellation
was represented by a light shining through five apertures. The artificial constellations used are
shown in Figure 4. The constellation was then rotated in steps of 15 degrees and the images were
photographed by an ordinary digital camera. Figure 5 shows four photos for the constellation at
angles 0, 45, 90 and 135 degrees. Figure 6 shows the same photos after two steps of processing—
averaging and rotating. The reconstructed image is shown on Figure 7. For image processing,
Matlab was used.
The principle of the telescope was successfully tested. The resulting image can be compared
with the reconstructed images in [7]. With higher quality components and more measurements at
finer angle steps, the reconstructed image would be of better quality.
Acknowledgements
The support by the Ministry of Education of Czech Republic pro ject MSM6840770039 is ac-
knowledged.
References
[1] King H. C. 2003, The History of the Telescope, Dover Publication, New York
[2] Crawford F.S. Jr. 1968, Berkeley Physics Course 3. Waves (McGraw-Hill Book Co., New York).
[3] ESO 2012, The Very Large Telescope, https://www.eso.org/public/teles-instr/vlt.html
[4] Herman, G. T. 2009, Fundamentals of computerized tomography: Image reconstruction from projection, 2nd
ed., Springer,
[5] Jahne, B. 1995, Digital Image Processing, Springer–Verlag
[6] English, R. J.1996, Single–Photon Emission Computed Tomography: A Primer, Publ. of The Society of Nuclear
Medicine,
[7] Akram W., et al. 1996, Image Processing Using SPECT Analysis,
(http://www.clear.rice.edu/elec431/projects96/DSP/index.html)
[8] Chadzitaskos G., Tolar J. 2007, CZ Patent 298313
[9] Chadzitaskos G., Tolar J. 2004, in Proc. SPIE. 5487, Optical, Infrared, and Millimeter Space Telescopes 1137
doi: 10.1117/12.5546007(arXiv: astro-ph/0310064)
[10] Beylkin G. 1987, IEEE Transaction on Acoustics, Speech, and Signal processing, vol. ASSP-35, No. 2, 162
1
2
3
4
Figure 1. Principle of the new telescopic system. It consists of a parabolic strip
mirror 1, CCD camera 2 in the image plane, supported by mounting 3, and rotates
around the optical axis 4. The shots are stored in a computer, where the image is
reconstructed.
y
x
y
x
y
x
y
F
V
5
Figure 2. The paraxial beams 5 are reflected by strip on the focus line F, where
V is the vertex line of the strip.
Figure 3. Two parabolic strips of lengths 20 and 40 cm, for the proof-of-principle
experiment, the 40 cm strip was used
Figure 4. Two artificial constellations were used to demonstrate the principle.
For presentation we use the second one. The circle encloses the illuminated area.
Figure 5. The original images from camera made at angels 0, 45, 90, and 135 degrees
Figure 6. The same images after Matlab processing at corresponding angles
Figure 7. The image reconstructed from series of pictures made with angular
step 15 degrees