Digital Signal Processing Reference
In-Depth Information
Table 10.1.
Error standard deviations after
denoising the synchrotron noisy map (additive
white Gaussian noise,
σ
= 5) by the wavelet, the
curvelet, and the combined denoising algorithm
Method
Error standard deviation
Noisy map
5
Wavelet
1.25
Curvelet
1.07
CFA
0.86
See Section 8.3 for the combined denoising algorithm.
transform filtered image and the residuals. On such data, exhibiting very anisotropic
features, the curvelets produce better results than wavelets.
The residuals after wavelet- and curvelet-based denoising presented in Fig. 10.16
show different structures. As expected, elongated features are better restored using
the curvelet transform, whereas isotropic structures are better denoised using the
wavelet transform. The combined denoising algorithm introduced in Section 8.3 can
obviously also be applied on the sphere to benefit from the advantages of both trans-
forms. This iterative method detects the significant coefficients in both the wavelet
domain and the curvelet domain and guarantees that the reconstructed map will
take into account any pattern that is detected as significant by either of the trans-
forms.
The results are reported in Table 10.1. The residual is much better when the
combined denoising is applied, and features can no longer be detected by eye in
the residual (see Fig. 10.17). This was not the case for either the wavelet- or the
curvelet-based denoising alone.
10.7.2 Morphological Component Analysis
The morphological component analysis (MCA) Algorithm 30 (see Section 8.5) was
applied to a decomposition problem of an image on the sphere using the transforms
developed earlier. The spherical maps shown in Fig. 10.18 illustrate a simple numer-
ical experiment. We applied the MCA decomposition algorithm on the sphere to
synthetic data resulting from the linear mixture of components that were sparse in
the spherical harmonics and the isotropic wavelet representations, respectively. The
method was able to separate the data back into their original constituents.
10.7.3 Inpainting
The inpainting algorithms described in Section 8.7 can also be applied on the
sphere. In Section 8.7, a total variation penalty is shown to enhance the recovery
of piecewise smooth components. Asking for regularity across the gaps of some lo-
calized statistics (e.g., enforcing the empirical variance of a given inpainted sparse
component to be nearly equal outside and inside the masked areas) yields other
possible constraints. In practice, because of the lack of accuracy of some digital
