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A Wavelet Parallel Code for Structure
Detection
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Matthias Pruksch and Frank Fleischmann
OES-Optische und elektronische Systeme GmbH,
Dr.Neumeyer-Str.240, D-91349 Egloffstein, F.R. Germany
In two-dimensional computer experiments, star-like and planet-like objects are convolved with band-limited point-spread functions. Photon noise and read-out noise are applied to these images as well as to the point-spread functions being used for deconvolution. Why Richardson-Lucy like algorithms favor star-like objects and the difference in computational efforts are discussed.
I(x) = P(x) * O(x), | (1) |
In the case of band-limited point-spread functions or point-spread functions with incomplete coverage of the Fourier domain (interferometry), information is lost and therefore, deconvolution is not possible. Instead of one unique solution, a space of distributions solves (1) (Lannes, Roques & Casanove 1987). This raises the question of how the algorithms choose their reconstruction out of the space of possible solutions.
(2) |
(3) |
(4) |
Positive iterative deconvolution expands the deconvolution to a converging sum. In between iteration-steps the positivity constraint is applied such that the equation
(5) |
All pictures consist of 256 by 256 pixels and show Ida in full contrast (gamma correction 0.7). The smaller picture on the right side of each picture shows phase (no gamma correction) and modulus (gamma correction 0.25) of the corresponding spectrum. The spectrum is shown for positive horizontal frequencies only, since the values of the negative frequencies are the complex conjugate of the positive ones. The origin of the phase is located on the left center and the origin of the modulus in the center of the small picture.
Different band-limited point-spread functions have been applied to the object, but only one is discussed and illustrated in Figure 1 on the right. The point-spread function simulates wavefront aberrations of approximately . This is typical for a ground-based telescope that suffers from atmospheric turbulence. The band-limit is visible in the spectrum, since any value beyond the circle is zero (grey for phase and black for modulus).
The peaks in the point-spread function are best seen in Figure 2 on the left which shows the convolved image I(x). In order to avoid spurious information due to round-off errors, the point-spread function was calculated and convolved with the object in the Fourier domain. The contributions beyond the band-limit in Figure 2 on the left originate from noise that was added to the image after convolution.
All algorithms resolve the objects, as shown by the reconstructions by Richardson-Lucy in Figure 2 on the right, the algorithm by Müller in Figure 3 on the left and positive iterative deconvolution in Figure 3 on the right.
The Richardson-Lucy like algorithms seem to perform better in the reconstruction of stars, whereas positive iterative deconvolution shows more details for the planet-like object. This is due to the fact that positive iterative deconvolution uses only the positivity constraint, whereas the other algorithms allow statistical deviations. This is clearly seen by comparing the phase distribution of the reconstructions with that of the original object. Since the phase is the most important information, positive iterative deconvolution delivers in this sense the best solution.
Positive iterative deconvolution reconstructs the original data as far as it is available in the measured image and adds information only consistent to that data and the positivity constraint. Therefore, it shows equal quality for all objects. Since positive iterative deconvolution is also the fastest algorithm it is the algorithm of choice, if the reconstructions have to be consistent with measurements.
The work reported here has received financial support from the OES-Optische und elektronische Systeme GmbH.
Lannes, A., Roques, S., & Casanove, M. J. 1987, J.Mod.Opt., 34, 161
Lucy, L. B. 1974, AJ, 79, 745
Müller, R. 1997, private communication
Pruksch, M. & Fleischmann, F. 1997, accepted by Comput.Phys.
Richardson, W. H. 1972, J.Opt.Soc.Am., 62, 55
Next: Structure Detection in Low Intensity X-Ray Images using the Wavelet Transform Applied to Galaxy Cluster Cores Analysis
Up:
Astrostatistics and Databases
Previous:
A Wavelet Parallel Code for Structure
Detection
Table of Contents -- Index -- PS reprint -- PDF reprint