2 2D continuous wavelet transform

This section is dedicated to present the continuous form of the 2D wavelet approach we are later using to analyse discrete CMB maps. As a difference with the multiresolution approach used in Sanz et al. ([1999]) the 2D wavelet method provides information on many more resolution elements than the former method. Moreover, this property is crucial for performing an efficient linear denoising preserving the Gaussianity of the underlying CMB field (as will be discussed in Sect. 3.2).

The continuous wavelet transform of a 2D signal f(x₁, x₂) is defined as

$$\int {\rm d}x_1\,{\rm d}x_2\,f(x_1, x_2) \times {\Psi } (R_1, R_2, b_1, b_2; x_1, x_2) \, ,$$ w(R₁, R₂, b₁, b₂) = (1)

$${\Psi }(R_1, R_2, b_1, b_2; x_1, x_2) \frac{1}{\sqrt{\vert R_1R_2\vert}} \times {\psi } \left(\frac{x_1 - b_1}{R_1}, \frac{x_2 - b_2}{R_2}\right) \, ,$$ = (2)

where w(R₁, R₂, b₁, b₂) is the wavelet coefficient associated to the scales R₁ and R₂ at the point with coordinates b₁ and b₂. The limits in the double integral are $-\infty$ and $\infty$ for the two variables. $\psi$ is the wavelet "mother'' function that satisfies the constraints

$$\int {\rm d}x_1\,{\rm d}x_2\,\psi = 0,\ \ \ \int {\rm d}x_1\,{\rm d}x_2\,{\psi }^2 = 1,$$ (3)

and the "admissibility'' condition (that allows to reconstruct the function f), i.e. there exists the integral

$$C_{\psi } \equiv (2\pi)^2\int {\rm d}k_1\,{\rm d}k_2\frac{{\vert \hat{\psi} (k_1, k_2)\vert}^2} {\vert k_1\,k_2\vert},$$ (4)

where $\hat{\psi} (k_1, k_2)$ represents the 2D Fourier transform of $\psi$ and || denotes the modulus of the complex number.

A reconstruction of the image can be achieved with the inversion formula

$$\frac{1}{C_{\psi}}\int \frac{{\rm d}R_1\,{\rm d}R_2}{{\vert R_1\,... ...rm d}b_2\,w(R_1, R_2, b_1, b_2) \times \Psi (R_1, R_2, b_1, b_2; x_1, x_2) \, .$$ f(x₁, x₂) = (5)

Next, let us introduce the scalogram of a 2D signal

$${\sigma}_{\rm w}^2(R_1, R_2)\equiv \langle w^2(R_1, R_2, b_1, b_2) \rangle$$ (6)

where $\langle \rangle$ means the average value calculated on the image.

Hereinafter, we shall consider 2D wavelets that are separable, i.e. $\psi (x_1, x_2) = \psi (x_1) \psi (x_2)$. In this case, ${\vert \hat{\psi} (k_1, k_2)\vert}^2 = {\vert \hat{\psi} (k_1)\vert}^2 {\vert \hat{\psi} (k_2)\vert}^2$. In particular, we are interested in the Haar, the Mexican Hat and the Daubechies 2D-transforms that can be generated in terms of the corresponding 1D wavelets. For the Haar case, we find

$${\vert \hat{\psi} (k)\vert}^2 = \frac{8}{\pi k^2}{\sin}^4\left(\frac{k}{4}\right),$$ (7)

with an absolute maximum at $k \simeq 4.7$, whereas for the Mexican Hat

$${\vert \hat{\psi} (k)\vert}^2 = \frac{4}{3{\pi }^{1/2}}k^4{\rm e}^{- k^2},$$ (8)

with a single peak at $k = \sqrt{2}$. The corresponding formulae for the Daubechies wavelets of order N can be found in Ogden ([1997]). The last wavelets form an orthonormal basis with compact support, increasing regularity with N and vanishing moments up to order N-1.

Just as an illustration we would like to present the scalogram for a CMB signal generated in a Standard Cold Dark Matter (SCDM) model. Let us assume that the image corresponds to a realization of a random field whose 2-point correlation function is homogeneous and isotropic: $\xi (r)$, $r^2\equiv x_1^2 + x_2^2$. This is equivalent to assume that the Fourier components $\hat{f}(k_1, k_2)$ satisfy

$$\langle \hat{f}(k_1, k_2)\,\hat{f}^*(k_1^{\prime}, k_2^{\prim... ... = P(k)\delta (k_1 - k_1^{\prime})\delta(k_2 - k_2^{\prime}),$$ (9)

where P(k), $k^2\equiv k_1^2 + k_2^2$, is the standard Fourier power spectrum and $\langle \rangle$ means average value over realizations of the field (the ergodicity of the field is assumed). So, taking average values and using Eqs. (1) and (6), one obtains the variance ${\sigma}_{\rm w}^2(R_1, R_2)$ of the wavelet coefficients or scalogram

$$\sigma_{\rm w}^2(R_1,R_2)= R_1R_2\int {\rm d}k_1\,{\rm d}k_2\,P(k){\vert \hat{\psi} (k_1R_1, k_2R_2)\vert}^2 .$$ (10)

For 2D white noise, i.e. P(k) = constant, one gets that the scalogram, $\sigma_{\rm w}^2$, is constant at any scale.

On the other hand, if the field f represents the temperature anisotropy of the CMB, $\frac{\Delta T}{T}$, one can obtain

$$\left< \left(\frac{\Delta T}{T}\right)^2\right> = \frac{1}{C... ...1\,{\rm d}R_2 \frac{{\sigma}_{\rm w}^2(R_1, R_2)}{R_1^2R_2^2}.$$ (11)

From the previous equation, ${\sigma}_{\rm w}^2/C_{\psi}R_1R_2$ represents the power per logarithmic scale. We remark that taking into account the homogeneity and isotropy of the field, the 2-scale dependence of the scalogram is redundant in this case. A more appropriate treatment in this continuous example would be one based on isotropic wavelets defined in terms of a single scale.

Figure 1: Top: scalogram for a SCDM model using the Haar wavelet. Bottom: scalogram along the diagonal, i.e., R1 = R2, for the SCDM (solid line), open CDM (dashed line) and flat-$\Lambda$ CDM (dotted line) models using the Haar (thin lines, bottom of the panel) and Mexican Hat (thick lines, top of the panel) wavelets

In the top panel of Fig. 1 we have represented the scalogram against the two scales R₁ and R₂ for SCDM using the Haar transform. The qualitative behaviour for the other transforms is similar. In the bottom panel we compare the scalogram along the diagonal for standard, open ( $\Omega = 0.3$) and a flat-$\Lambda$ ( $\Omega = 0.3 , \lambda = 0.7$) CDM models using the Haar and the Mexican Hat transforms. The qualitative behaviour for the two transforms is similar: there is a plateau for $R > 1^{\circ}$ and a maximum dependent on $\Omega$, corresponding to the first Doppler peak. Other secondary maxima appearing in the figure are related to the secondary peaks in the standard $C_\ell$ radiation power spectrum (the $C_\ell$ is given by $C_{\ell}\simeq P(k\simeq \ell )$). Therefore, the position and amplitude of the maxima that appear in the scalogram is model dependent, being this quantity tightly related to the $C_\ell$'s.

Figure 2: S/N ratio (open squares) in each box for a CMB map (SCDM model) with uniform noise at the level S/N = 1 using Daubechies 4. The three regions of the plot show the boxes that are kept unchanged ( $S/N \geq 1.5$), removed (S/N < 0.3) or treated with a soft thresholding technique (boxes in between). The soft thresholds (solid triangles) estimated using SURE are also plotted for the thresholded boxes