Figure 1

Comparison of different filter algorithms in the one-dimensional test case. Each column corresponds to a different setting. The signal, drawn from a power-law power spectrum, is the same in each case and depicted in each panel with a solid line. The left column contains homogeneous noise, while in the middle column, the noise is suppressed in the left third of the interval and enhanced in the right third, and in the right column the noise is enhanced in some individual pixels. The first row shows the signal realization along with the data. The second row shows the reconstruction using the Wiener filter formula, assuming the correct power spectrum and under the assumption of homogeneous noise; the third row shows the critical filter reconstruction, assuming the power spectrum to be unknown, but still assuming homogeneous noise. The last row shows the extended critical filter reconstruction in which both the signal power spectrum as well as the noise variance are assumed to be unknown. The respective reconstructions are depicted by a dashed line that lies on top of the solid one in many cases. In the two right panels of the first row, some of the data points lie outside the area that is shown.Reuse & Permissions

Figure 2

Comparison of different filter algorithms in the spherical case. Each column corresponds to a different setting. The signal, drawn from a power-law power spectrum, is the same in each case. The left column contains homogeneous noise, while in the middle column, the noise is suppressed in the southern third of the sphere and enhanced in the northern third, and in the right column the noise is enhanced in some individual pixels. The first row shows the signal realization and the second row the data. The third row shows the reconstruction using the Wiener filter formula, assuming the correct power spectrum and under the assumption of homogeneous noise; the fourth row shows the critical filter reconstruction, assuming the power spectrum to be unknown, but still assuming homogeneous noise. The last row shows the extended critical filter reconstruction in which both the signal power spectrum as well as the noise variance are assumed to be unknown.Reuse & Permissions

Figure 3

Comparison of the different reconstructed angular power spectra for the spherical scenario. The solid line depicts the theoretical power spectrum, which is also used in the Wiener filter reconstructions. The dashed line corresponds to the power of the specific signal realization and the dash-dotted and dotted lines to the power spectrum reconstructed with the critical filter and the extended critical filter, respectively. Panel (a) shows the case with homogeneous noise, panel (b) the one in which the noise is enhanced and suppressed in one-third of the sphere each, and panel (c) the one in which the noise is enhanced in individual pixels.Reuse & Permissions

Figure 4

Absolute value of the pixelwise difference between the reconstructed maps and the signal realization for the spherical scenario. Each row shows the results for a different filter algorithm. As in Fig. 2, the left column shows the case with homogeneous noise, the middle column the one with enhanced noise in the northern third of the sphere and suppressed noise in the southern third, and the right column shows the case where the noise is enhanced in individual pixels. Note that the color bar differs from the one used in Fig. 2.Reuse & Permissions

Figure 5

Pixelwise uncertainty of the extended critical filter reconstructions in the spherical scenario. The left panel shows the case with homogeneous noise, the middle panel the case with enhanced noise in the northern third and suppressed noise in the southern third, and the right panel the case with enhanced noise in individual pixels.Reuse & Permissions

Figure 6

Pixelwise uncertainty of the extended critical filter reconstructions in the one-dimensional scenario. The left panel shows the case with homogeneous noise, the middle panel the case with enhanced noise in the right third and suppressed noise in the left third, and the right panel the case with enhanced noise in individual pixels. The dark curves represent $\pm \sigma$ and the light curve the difference between the reconstruction and the correct signal.Reuse & Permissions