1. 1-D Polar covariance maps smoothing

When smoothing a 1-D {$n$}-pixel map {$m$}, we convolve it with a Gaussian kernel {$K_\sigma$}, with $\sigma$ being the bandwidth. The resulting smoothed map at pixel {$p$} becomes

{$ \tilde m_p = K_\sigma * m ~~~~~ \text{with} ~~~~~K_\sigma(p) \equiv \frac 1 {\sigma^2} \exp\left(\frac {p^2}{2\sigma^2}\right) $}

with * the convolution product. Since we work in pixel space, the convolution becomes {$ \tilde m_p = \frac 1{ n_{}}\sum_{p'}^{n_{}} K_\sigma(p-p') m_{p'} $}

From it, the covariance of the smoothed maps {$\tilde x$} and {$\tilde y$} then becomes

{$ Cov[\tilde x_p, \tilde y_p] = Cov[K_\sigma * x, K_\sigma * y] $}

{$ Cov[\tilde x_p, \tilde y_p] = K^2_\sigma * Cov[ x, y] $}

{$ Cov[\tilde x_p, \tilde y_p] = \frac12\frac1{\sigma^2 n_{}} K_{\sigma/\sqrt2} * Cov[ x, y]. $}

We conclude that the resulting covariance of two smoothed maps {$\tilde x$} and {$\tilde y$} by a Gaussian kernel {$K_\sigma$} is equivalent to the smoothing of the initial covariance map between {$ x$} and {$ y$} by a Gaussian kernel {$\displaystyle \frac{K_{\sigma/\sqrt 2}}{2\sigma^2 n_{}}$}.

As a more detailed calculation, let {$ \tilde m(p) = \frac1{n} \sum_i^n K_\sigma(p-p') m(p') $}

the value of the $n$-pixel smoothed map at pixel {$p$}, with {$\displaystyle K_\sigma(p) \equiv \frac 1 {\sigma^2} \exp\left(-\frac12 \frac{p^2}{\sigma^2}\right)$} the Gaussian smoothing kernel. Thus,

{$ Var[\tilde m (p)] = E [\tilde m(p)^2] - E [\tilde m(p)]^2 $}

{$ Var[\tilde m (p)] = E \left[\frac1 { n^2} \sum_i^n K_\sigma(p-p_i)m(p_i)\sum_j^n K_\sigma(p-p_j)m(p_j) \right] - \frac1 {n^2} \left( \sum_i^n K(p-p_i) \underbrace{E [m(p_i)]}_{=0} \right)^2 $}

{$ Var[\tilde m (p)] = \frac1 {n^2} \left(\sum_i^n K_\sigma^2(p-p_i) E[m(p_i)^2] + \sum_{i\neq j}^n K_\sigma(p-p_i)K_\sigma(p-p_j)\underbrace{ E[m(p_i)m(p_j)]}_{=0}\right) $}

{$ Var[\tilde m (p)] =\frac1 {n^2} \sum_i^n K_\sigma^2(p-p_i) Var[m(p_i)] $}

{$ Var[\tilde m (p)] =\frac 12 \frac1 {\sigma^2 n} \left(\sum_i^n K_{\sigma/\sqrt2}(p-p_i) Var[m(p_i)]\right). $}

Since

{$ K_{\sigma}(p)^2 = \frac 1 {\sigma^4} \exp\left(-\frac{p^2}{\sigma^2}\right) = \frac 12\frac 1 {\sigma^2} K_{\sigma/\sqrt2}(p) $}

This result is only valid for computation of the variance of smoothed noise maps.
Since gaussian smoothing correlates neighbouring pixels, the noise map cannot be represented by a diagonal pixel covariance matrix anymore, and the noise spectrum is not flat anymore.
Noise map generated from the new variance map are accurate only for small smoothing bandwidths.
Kernel must be discrete
Their is a factor 1/2 of difference when using the healpix smoothing, as if

{$ K_{\sigma}(p)^2 = \frac 1 {\sigma^4} \exp\left(-\frac{p^2}{\sigma^2}\right) = \frac 1 {\sigma^2} K_{\sigma/\sqrt2}(p) $}

2. 2-D Polar covariance maps smoothing

to do : investigate heapix smoothing implementation

3. Validation simulations

First set :
- Smooth cov maps as described above
- Generate noises maps
- compute new cov maps from MC
Seconde set :
- Generate noises maps
- Smooth noise maps
- compute new cov maps from MC
Compare :

First set	second set

pixels list comparison
cls noise comparison

observation
- altough pixel variance map is correctly reconsturcted (pixels list comparison), the spectra are very different.
- Full simulation (noise generation + smoothing) is tus advised.