- 3.1 2D wavelet method on a grid
- 3.2 Reconstruction of CMB maps and radiation power spectra
- 3.3 Comparison with other denoising methods

In general for a grid of
pixels, a discretization of the
parameters of the form:
*R*_{1} = 2^{n - j1}, *b*_{1} =2^{n - j1}*l*_{1},
*R*_{2} = 2^{n - j2}, *b*_{2} =2^{n - j2}*l*_{2} for integer-valued *j* and
*l* allows to introduce the 2D discrete wavelet function

= | (12) |

where *j*_{i} and *l*_{i} denote the dilation and the
translation indexes,
respectively, satisfying
.
The resolution level is defined by
,
corresponding to
.
We also introduce a scaling function
that allows to define a
complete basis to reconstruct discrete images,

(13) |

(14) |

(15) |

We will consider orthonormal discrete bases as the Haar and Daubechies ones. Denoting by any of the previous functions, the orthonormality condition reads:

(16) |

where (*f*,*g*) denotes the scalar product of two functions
in *L*^{2}(*R*^{2}).

The wavelet coefficients are now defined by

(17) |

The image is reconstructed using the following expression:

(18) |

A representation of the wavelet coefficients can be done by a square that contains small squares and rectangles associated to different levels of resolution. The first level, representing high-resolution, is

In the present work we have considered simulated maps
of size
square degrees, pixel
and
filtered with a 4.5' FWHM Gaussian beam for a standard CDM model
().
We have included non-correlated Gaussian
noise at different levels (*S*/*N* per pixel
between 0.7 and 3 at the pixel scale),
considering uniform and non-uniform noise.
This last case is introduced to account for the
non-uniform sampling of satellite observations.
As an extreme case we have simulated a noise map with two different
regions, one with *S*/*N*=3 (approximately one quarter of the map) and a
second one with *S*/*N*=0.7.

The purpose of the denosing of CMB maps is to reconstruct the original signal map as well as the radiation power spectrum .

Wavelet decompositions are performed with the
package *2D-W* developed by our group.
The procedure uses two scales,
*R*_{1}=2^{n-j1},
*R*_{2}=2^{n-j2},
*n*=9 in our case. High values of
*j*=(*j*_{1}+*j*_{2})/2 mean high resolution, i.e.,
small scales. We distribute the coefficients
*w*_{j1,j2,l1,l2}
in boxes corresponding to a
couple (*j*_{1},*j*_{2}), having a total of 81 boxes.
The coefficients related to the scaling function
are not included in the analysis and they are left untouched.
To perform denoising, the basic operation is the comparison between the
wavelet
coefficients dispersion of the signal in each box with the one of the noise.
The Gaussian white noise gives the same contribution in all boxes. Since the
signal is negligible at the highest resolutions, the noise dispersion can
be directly estimated from the map. Therefore, the signal dispersion can
also be estimated for each box.

In Fig. 2, we plot the *S*/*N* ratio (defined in
terms of the wavelet coefficients dispersions of signal and noise) for each
box for a CMB simulation with *S*/*N*=1 in real space.

For the case of uniform noise,
all boxes where the signal dominates (
)
are kept untouched,
whereas those with a high level of noise (*S*/*N* < 0.3) are removed.
On the other hand, the boxes in between are treated with a soft
thresholding
technique. Given a threshold
in terms of the noise coefficients
dispersion (
), the coefficients
are
rescaled as
(where the +, -
signs correspond
to negative and positive values of *w* respectively),
whereas the remaining
coefficients are set to zero. The threshold
for each box is chosen
using the SURE
method (Donoho & Johnstone [1995]).
The threshold is obtained by
minimization of an unbiased estimate of the expected mean squared
error of the estimation of the
signal wavelet coefficients (see for instance Ogden [1997]).
In Fig. 2,
the thresholds obtained with the SURE technique are plotted for a
CMB map with *S*/*N* = 1. As expected, lower
*S*/*N* levels are treated with higher thresholds, i.e., more coefficients
are removed.
Changing the range of *S*/*N* where the soft technique
is applied (providing is around *S*/*N* = 1), does not appreciably change
these results.
We have used Daubechies 4 but we obtain little or no
variations if we adopt higher order Daubechies wavelets.
However, the Haar transform gives worse results.
Table 1 shows the error in the map reconstructions for different
*S*/*N* ratios with Gaussian uniform noise.
The error improvement achieved with the denoising
technique applied goes from factors of 3 to 5 for
*S*/*N* = 3 - 0.7.
The four top panels of Fig. 3 show CMB maps with only signal
(SCDM), signal plus noise with a *S*/*N* = 1, the reconstructed map
using wavelets and the residual one.

S/N |
SURE | linear |

0.7 | 27.4 | 29.4 |

1.0 | 21.7 | 23.4 |

2.0 | 13.3 | 14.4 |

3.0 | 10.0 | 11.1 |

N.U. | 24.3 | - |

Regarding non-uniform (N.U.) noise, wavelet techniques
allow us to treat each location in the image separately.
At each fixed location and fixed (*j*_{1},*j*_{2}) we calculate the
dispersion of the corresponding noise wavelet coefficient
from 500 simulations of our non-uniform noise.
Since we consider non-uniform noise that is uncorrelated,
the average noise dispersion is the same for all the boxes,
as in the uniform noise case.
Therefore, we can get again the dispersion of the signal for each
(*j*_{1},*j*_{2}) pair as well as
the *S*/*N* ratio for each coefficient. Those coefficients with
*S*/*N* ratio in the
considered range (
)
are treated with a soft
thresholding technique, whereas the rest are either kept or removed
depending on their *S*/*N* ratio.
Since in presence of non-uniform noise, we cannot use the SURE
technique to estimate the optimal threshold
(as far as we know,
work is in progress to define a general threshold in the case of
non-uniform noise, Von Sachs & McGibbon [1999]),
we choose for all the thresholded coefficients .
This threshold is defined with respect to the noise
dispersion in each particular coefficient.
We have chosen this value of
because it gives
a good reconstruction when comparing with the original map,
but the results are not very sensitive to the choice of a
different threshold in the range
.
In Table 1 we present the error of the reconstructed map in the
presence of non-uniform noise as considered in this work.

Regarding the power spectrum, ,
the denoising method performs very
well. Figures 4 and 5 show the
reconstructed spectrum and the relative error
for
for different *S*/*N* ratios
(the power spectrum is obtained in the usual way,
averaging over the Fourier modes of the considered map at each *k*).
The relative
error is
for
in all the considered cases except for the map
with *S*/*N* = 0.7. In this last case the error increases to
for the same range of .

Finally, we have looked for possible non-Gaussian features introduced by the
non-linearity of the soft thresholding technique.
We have compared the probability density
function of the original and the reconstructed maps
using a Kolmogorov-Smirnov test.
Both distributions are compatible at similar or higher levels than
the original map and the corresponding linear reconstruction
obtained with the Wiener Filter technique.
Moreover, not significant change is observed in the skewness and
kurtosis of the original and reconstructed maps.
However, the application of soft thresholding to the
wavelet coefficients at a certain level, which are Gaussian distributed
for a Gaussian temperature field,
clearly changes their
distribution by removing all coefficients whose absolute values
are below the imposed threshold and shifting the remaining ones
by that threshold. Therefore, this technique is introducing a certain
level of non-Gaussianity that will depend on the threshold imposed and
that should be taken into account when analysing the data.
If we are mainly concern about preserving the Gaussian character
of the reconstructed image, denoising with wavelet techniques is
still possible.
Instead of using a soft thresholding technique, we can apply a
linear denoising method in wavelet space. In particular, we have
removed all wavelet coefficients at boxes with *S*/*N* < 1 and
left the rest untouched.
The reconstruction errors get only slightly worse with this
simple linear
technique as shown in Table 1. Regarding the reconstructed
power spectrum, this is at the same level than the SURE reconstruction
(see top-right panel of Fig. 5) for all the
considered cases. It is important to remark that the linear
denoising method based on 2D wavelets performs much better than the
Multiresolution one due to the larger number of boxes corresponding
to the product of the 2 scales considered.

A comparison between Wiener Filter (see for instance
Press et al. [1994])
and wavelet techniques has also been performed.
In relation to map reconstruction the error affecting the Wiener
reconstructed maps is comparable to that achieved with wavelet
techniques in all the considered cases. However, in order to apply
Wiener filter previous knowledge of the signal power spectrum is required.
In a real situation this may well not be possible.
The reconstructed and residual maps using Wiener Filter are shown in
Fig. 3.
In addition, when using Wiener Filter, the power
spectrum of the reconstructed image is clearly suppressed at high ,
giving much worse results than the
wavelet technique. For comparison, we have plotted in
Fig. 5 the
absolute value of the relative error of the
for the
reconstructions with Wiener Filter.
On the other hand, one could recover the 's of the original
signal by subtracting from
the power spectrum of the signal plus noise map the estimated
power spectrum of the noise, which is constant at each .
However, this method gives in general
worse results than the wavelet techniques
and besides cannot be used to reconstruct the image.
We have also applied a Maximum Entropy Method to the maps used in this
work, with the definition of entropy given by Hobson & Lasenby
([1998]).
This method provides reconstruction errors at similar levels as
the wavelet techniques. However, we remark that wavelet
techniques are computationally faster 0(*N*) and simpler to apply
(not requiring iterative processes) than Maximum Entropy Methods.

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