For the segmentation step, we first carried out a MLE reconstruction
stopped after 20 iterations. The stopping point was given by the
cross-validation test for the whole image. Then, we carried out the
wavelet decomposition into three planes plus the residual image.
Figure 3 shows wavelet plane and
Fig. 4 shows the residual image
.
Wavelet plane is used to identify the stars and other
prominent features. Figure 5 shows the extracted objects
obtained by setting the threshold to 22.8 counts. The threshold was
computed using the automatic method described in
Sect. 4.3. Finally, we carried out the segmentation using
the self-organizing network described in Sect. 4.4.
Figure 6 shows the final segmentation into 8 extended
regions plus an additional region for each bright object totaling 50
regions.
The regional hyperparameters were computed by the cross-validation
technique described in Sect. 4.5.2. Figure 7 shows
the cross-validation test curve for region No. 3 (corresponding to the
spiral arms of the galaxy aproximately) from which an optimum value of
was obtained. In the 50 cross-validation tests carried
out for this example, the following optimum values were obtained: for
the eight extended regions
respectively; for the other 42 regions (stars and bright objects) the
was set to the maximum of 300 in 29 cases while the other
13 objects received
values between 50 and 200.
Using the above set of hyperparameters, we carried out the final Bayesian reconstruction of data sets A and B with the FMAPE algorithm given by Eqs. (26), (23). After adding both reconstructions we obtained the final result, as shown in Fig. 8. A smooth background, a well reconstructed nebula, and sharp images of the stars are obtained. Noise amplification in the background and in the nebula was suppressed by the algorithm, while images of the bright objects were reconstructed to near Maximum Likelihood. Finally, Fig. 9 shows the image of the residuals in a linear scale, which exhibits no visible structure or correlation with the solution.
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Figure 8: Final reconstruction of NGC 4321 using the FMAPE algorithm with variable hyperparameters defined by the segmentation |
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Figure 9: Map of residuals of the reconstruction using the FMAPE algorithm with variable hyperparameters (linear scale) |
In order to compare the FMAPE method with variable hyperparameter with
the method with a single hyperparameter given by
algorithm (27) and with the Maximum Likelihood method given by
algorithm (28), we carried out also the reconstructions using
those methods. To make the results comparable, we reconstructed both
data sets A and B and added the results to obtain the final
reconstructions. The single hyperparameter for Bayesian
case (27), and the stopping point for the Maximum Likelihood
method (28) were also determined by cross-validation for the
entire image (Núñez & Llacer 1993a). We obtained a value of
for the Bayesian case and a stopping point of 30
iterations for the Maximum Likelihood method. Figure 10
and Fig. 11 show the results for the Bayesian
with single hyperparameter and Maximum Likelihood cases respectively.
If we compare the results of the three methods, we can observe than the reconstruction with a single hyperparameter (Fig. 10 left) is noisier both in the nebula and in the background than in the variable hyperparameter case. Also, the images of the stars are under-reconstructed. The MLE image (Fig. 10 right) is clearly under-reconstructed in the bright objects. If we increase the number of iterations to correct that effect, the noise in the lower intensity regions increases excessively, making the result unacceptable. The Bayesian method with variable hyperparameter appears to be the better method of reconstruction.
The second example is the reconstruction of a pixel real image
of the planet Saturn obtained with the WF/PC camera of the Hubble Space
Telescope, before the servicing mission. All images for this example are
shown in a linear grey scale, except where noted. Figure 12
shows the raw data to be reconstructed. The Point Spread Function
is shown in a logarithmic scale in Fig. 13.
Since the image of Saturn has no stars we segmented the image into 9 extended regions. For the segmentation step, we first carried out a MLE reconstruction stopped after 50 iterations. The stopping point was given by the feasibility test for the whole image. Then, as in the first example, we decomposed the image into three planes plus the residual image and carried out the segmentation by the self-organizing network described in Sect. 4.4. Figure 14 shows 9 regions resulting from the segmentation. In this example we do not have a second data set for cross-validation. In addition, the image was obtained from a CCD camera with readout noise, rendering the process of splitting the image into two halves by thinning questionable (Núñez & Llacer 1993a). Thus, in this example we have used the feasibility test described in Sect. 4.5.1 to determine the set of hyperparameters.
Figure 15 shows the result of the 9-region reconstruction. We have obtained a smooth background, a well reconstructed image of the planet and sharp divisions in the rings. By the use of the variable resolution FMAPE algorithm, we have avoided noise amplification in all regions of the image.
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Figure 14: Segmentation of the image in 9 regions of Bayesian hyperparameters using the method described in the text |
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Figure 15: Reconstruction of the image of Saturn using the FMAPE algorithm with the 9 channels defined by the segmentation |
Again, we compare the results of our new methodology with the constant
approach and with the MLE method. For that purpose, we used
the Saturn data for reconstructions by the FMAPE algorithm with
constant values of
and 500, and a reconstruction by
the Maximum Likelihood method at 100 iterations. The results show
characteristics similar to the ones described in the previous example:
the solution with variable hyperparameter is better reconstructed and
noise amplification is well controlled. In order to characterize the
solutions, we computed the mean normalized residuals for each of the 9
regions of the segmentation. Figure 16 shows the plot of the
mean residual as a function of region number. An ideal reconstruction
would give a straight line at a mean value of 1.0. The reconstruction
obtained by MLE has mean residuals that are increasing from below to
above the 1.0 value. The reconstruction with a single uniform
has mean residuals that are too high. With
most
of the values are too low but some are still too high. However, the
reconstruction with the variable
, although not perfect, is
much improved, with most of the mean residuals close to the ideal value
of 1.0.
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