Detection/estimation of the modulus of a vector. Application to point source detection in polarization data
Given a set of images, whose pixel values can be considered as the components of a vector, it is interesting to estimate the modulus of such a vector in some localised areas corresponding to a compact signal. For instance, the detection/estimation of a polarized signal in compact sources immersed in a background is relevant in some fields like astrophysics. We develop two different techniques, one based on the Neyman-Pearson lemma, the Neyman-Pearson filter (NPF), and another based on prefiltering-before-fusion, the filtered fusion (FF), to deal with the problem of detection of the source and estimation of the polarization given two or three images corresponding to the different components of polarization (two for linear polarization, three including circular polarization). For the case of linear polarization, we have performed numerical simulations on two-dimensional patches to test these filters following two different approaches (a blind and a non-blind detection), considering extragalactic point sources immersed in cosmic microwave background (CMB) and non-stationary noise with the conditions of the 70 GHz Planck channel. The FF outperforms the NPF, especially for low fluxes. We can detect with the FF extragalactic sources in a high noise zone with fluxes Jy for (blind/non-blind) detection and in a low noise zone with fluxes Jy for (blind/non-blind) detection with low errors in the estimated flux and position.
keywords:polarization - methods: data analysis - techniques: image processing - cosmic microwave background - radio continuum: galaxies.
The detection and estimation of the intensity of compact objects (or small regions) embedded in a background plus instrumental noise is relevant in different contexts, e.g. astrophysics, cosmology, medicine, teledetection, radar, etc. Different techniques have proven useful in the literature. Some of the proposed techniques are frequentist, such as the standard matched filter (Nailong, 1992; Vikhlinin et al., 1995; Malik & Subramanian, 1997; Tegmark & de Oliveira-Costa, 1998; Sanz et al., 2001; Herranz et al., 2002a; Stewart, 2006), the matched multifilter (or multifrequency filter, Herranz et al., 2002b; Herranz et al., 2005) or the recently developed matched matrix filters (Herranz & Sanz, 2008; Herranz et al., 2009) that correspond to scalar, vector or matrix filters, respectively. Other frequentist techniques include continuous wavelets like the standard Mexican Hat (Vielva et al., 2001; Vielva et al., 2003; Barnard et al., 2004; Sanz et al., 2006) and other members of its family (González-Nuevo et al., 2006) and, more generally, filters based on the Neyman-Pearson approach using the distribution of maxima (López-Caniego et al., 2005a, b). All of these filters have been used in the literature, in particular, for the detection and estimation of the intensity of point-like sources (i.e. extragalactic objects) in Cosmic Microwave Background (CMB) maps. In addition, some of them have been applied to real data like those obtained by the WMAP satellite (López-Caniego et al., 2007) and simulated data (López-Caniego et al., 2006; Leach et al., 2008) for the upcoming experiment on board the Planck satellite (Tauber, 2005). Besides, Bayesian methods have also been recently developed (Hobson & McLachlan, 2003; Carvalho et al., 2009).
In some applications it is important to measure not only the intensity of the light (signal) but also its polarization. An example is the study of CMB radiation. The polarization is given by the Stokes parameters , and the total intensity of polarization is (Kamionkowski et al., 1997). For linear polarization, the previous expression reduces to . In such cases, three or two images are added quadratically followed by a square root.
Let us consider the case of linear polarization. In this case we have two images and different approaches can be used to deal with detection/estimation of point-like sources on these maps. On the one hand, one can try to get the polarization directly on the -map. In this approach, we will consider one filter, obtained through the Neyman-Pearson technique (NPF). On the other hand, we can operate with two matched filters, each one on and followed by a quadratic fusion and square root. We will call this procedure filtered fusion (FF). It is clear that from a formal point of view we are trying to ask which is the optimal way to find the modulus of a vector given the components. In the case we have only the map of the modulus of a vector and the components are unknown, the FF cannot be applied and the only possibility is the NPF.
We will develop the methodology for the cases of a 2-vector and a 3-vector because of the possible interesting applications to the 2-plane and 3-space . We will show the results when using numerical simulations on flat patches that are relevant for the component separation of sources (linear-polarized extragalactic sources) in CMB maps.
To develop our methodology, we shall assume that we have a compact source, located for simplicity at the centre of two images , and characterised by amplitudes , and a profile , immersed in noises and that are Gaussian and independently distributed with zero mean and dispersions and . Again, for simplicity, we will consider that , a condition that is verified in most polarization detectors. In general, we will consider that the noise is non-stationary. We remark that the previous assumptions can be easily generalised to different profiles and different types of noise in the two images but we will not consider it in this paper. We will assume a linear model for the two images
The -map, , is characterised by a source at the centre of the image with amplitude immersed in non-additive noise which is correlated with the signal.
2.1.1 Neyman-Pearson filter (NPF) on the -map
If the noise is distributed normally and independently, then at any point the integration over the polar angle leads to the 2D-Rayleigh distribution of in absence of the source (Papoulis, 1984)
whereas if the source is present, with amplitude , one obtains the Rice distribution (Rice, 1954)
where is the modified Bessel function of zero order. If our image is pixelised, the different data , , with the number of pixels, will follow the two distributions
being the value of in the pixel, , the null hypothesis (absence of source) and the alternative (presence of source), respectively, and the profile at the pixel. The log-likelihood is defined by
The maximum likelihood estimator of the amplitude, , is given by the solution of the equation
This equation can be interpreted as a non-linear filter operating on the data that we shall call the Neyman-Pearson filter (NPF).
2.1.2 Filtered fusion (FF)
In this case we use the same matched filter (MF) operating over the two images , , respectively, as given by (Argüeso & Sanz, 2008):
Then, with the two filtered images we make the non-linear fusion pixel by pixel.
Now, we shall assume that we have a compact source at the centre of three images , , characterised by amplitudes , , and a profile immersed in noise that is Gaussian and independently distributed with zero mean and dispersion . In general, we will consider that the noise is non-stationary. We will assume a linear model for the three images
The -map, , is characterised by a source at the centre of the image with amplitude immersed in non-additive noise correlated with the signal.
2.2.1 Neyman-Pearson filter (NPF) on the -map
If the noise is distributed normally and independently, then at any point the integration over the angles leads to the 3D-Rayleigh distribution in absence of the source, also called the Maxwell-Boltzmann distribution in Physics
whereas if the source is present, with amplitude , one obtains the distribution
If our image is pixelised, the different data , follow the two distributions
being , the null hypothesis (absence of source) and the alternative (presence of source), respectively and the profile at the pixel. The log-likelihood is defined by
The maximum likelihood estimator of the amplitude, , is given by the solution of the equation
This equation can be interpreted as a non-linear filter operating on the data that we shall also call the Neyman-Pearson filter (NPF).
2.2.2 Filtered fusion (FF)
In this case we use the same MF operating over the three images , , , respectively, as given by equation (8). Then, with the three filtered images , , we make the non-linear fusion pixel by pixel.
The European Space Agency (ESA) Planck satellite (Tauber, 2005), to be launched in 2009, will measure the anisotropies of the CMB with unprecedented accuracy and angular resolution. It will also analyse the polarization of this primordial light. It is of great interest the detection and estimation of polarized sources in CMB maps (The Planck Collaboration, 2006; Tucci et al., 2004; Tucci et al., 2005); since this radiation is linearly polarized (Kamionkowski et al., 1997), , we will apply the methods for the detection/estimation of the modulus of a 2-vector presented in section 2.1. However, some cosmological models predict a possible circular polarization of CMB radiation (see for example Cooray et al., 2003; Agarwal et al., 2008). Even if CMB is not circularly polarized, the extragalactic radio sources can indeed show circular polarization (Aller et al., 2005; Kirk & Tsang, 2006; Homan et al., 2006). Besides, circular polarization occurs in many other astrophysical areas, from Solar Physics (Tritschler et al., 2007; Reiner et al., 2007) to interestellar medium (Cox et al., 2007), just to put a few examples. In all these cases the results for the modulus of a 3-vector presented in section 2.2 could be useful.
In order to compare and evaluate the performance of the filters presented in section 2.1, we have carried out two-dimensional simulations with the characteristics of the 70 GHz Planck channel (The Planck Collaboration, 2006). The simulated images have pixels with a pixel size of arcmin. We simulate the and components of the linear polarization as follows: each component consists of Gaussian uncorrelated detector noise plus the contribution of the CMB and a polarized point source filtered with a Gaussian-shaped beam whose full width at half maximum (FWHM) is , (the FWHM of the 70 GHz Planck channel beam). So the source polarization components can be written as
where is the beam dispersion (size) and we assume that the source is centred at the origin. The CMB simulation is based on the observed WMAP low multipoles, and on a Gaussian realisation assuming the WMAP best-fit at high multipoles. We do not include other possible foregrounds, since we are doing a first approximation to the detection/estimation problem and we assume that we apply our filters to relatively clean areas far away enough from the Galactic plane. Alternatively, we could consider a case in which the majority of foregrounds have been previously removed by means of some component separation method.
We consider realistic non-stationary detector noise. We have simulated the noise sky pattern for a Planck flight duration of 14 months, assuming a simple cycloidal scanning strategy with a 7 degree slow variation in the Ecliptic collatitude of the -axis. This scanning strategy implies that the sky will be covered inhomogeneously. The simulations have the same characteristics as the ones used in Argüeso et al. (2006). In order to illustrate the effects of non-stationary noise we have chosen two representative zones of the sky: one zone of high noise but quite isotropic and another zone of low noise but more anisotropic. The first zone is located in a region far from the Ecliptic poles, where the noise pattern is quite uniform and the number of hits per pixel of the detector is small. The average r.m.s deviation of the first zone (high noise) in units of (thermodynamic) is (for each component and ) and its standard deviation is . For this particular scanning strategy approximately of the sky has this kind of noise pattern. The second zone is close to one of the Ecliptic poles, where the sky is scanned more times (low noise level) but the hit pattern is very inhomogeneous. It has an average r.m.s. deviation and its standard deviation is . Then, in the second zone the noise is lower but proportionally more anisotropic. For this scanning strategy, of the sky has a noise pattern with these characteristics. Other zones of the sky would be intermediate cases between those considered here.
We take values of the source fluxes (before filtering with the Gaussian beam) and ranging from Jy to Jy with a step of Jy. A flux of 0.1 Jy corresponds for the 70 GHz channel to , so that it is of the order of the detector noise r.m.s deviation. The fluxes also correspond to typical polarization fluxes (Tucci et al., 2004; Tucci et al., 2005). As we will see in section 4, for some low-noise cases it is necessary to simulate even lower flux sources in order to study the behaviour of the filters in the low signal to noise ratio regime. We will explicitly report on this in the appropriate cases in section 4. The number of simulations is for each combination of pairs of values of and . After carrying out the corresponding simulations for and , we add them quadratically and take the square root to calculate , the polarization modulus.
We study two different detection types: blind detection and non-blind detection. In the former case, we assume that we do not know the position of the source and then we place it at random in the image, in the latter case we know the source position and then we place it at the centre of the patch. We have considered images of pixels in order to do fast calculations. In order to avoid border effects, we simulate and filter pixel patches and, after the filtering step, we retain only the pixel central square. We have tested the case of larger patches but the obtained results do not change significantly.
We use two different filters: the filtered fusion (FF), which consists in the application of the matched filter to the images in and separately in a first moment, and then the calculation/construction of from the matched-filtered images and and the Neyman-Pearson filter (NPF), applied directly on , derived from the Neyman-Pearson lemma and presented with detail in section 2. These filters are suitable for the case of uncorrelated noise, but we apply them to simulations including the CMB, which is correlated. However, we have checked that our results are similar when we consider simulations with and without the CMB: the relative differences of the errors in the estimated fluxes and positions are at most of a few percent. This is due to the low value of the CMB r.m.s. deviation as compared to that of the detector noise. Hence, the methods derived in section 2.1 are also suitable for simulations including the CMB in the 70 GHz Planck case we consider.
In the blind case we apply these filters to each simulation, centering the filters successively at each pixel, since we do not know the source position. We estimate the source amplitude for the NPF, in this case we calculate the value of which maximises the log-likelihood, eqs. (6) and (7). For the FF we estimate separately and and obtain .
We compute the absolute maximum of in each filtered map and keep this value as the estimated value of the polarization of the source and the position of the maximum as the position of the source. Note that in the more realistic case where more than one source can be present in the images, it is still possible to proceed as described by looking for local peaks in the image. In the non-blind case we only centre the filters in the pixels included in one FWHM of the source centre (approximately of the total). In this way, we use the knowledge of the source position, then we calculate the absolute maximum of the estimated in these pixels. We also calculate the significance level of each detection. In order to do this, we carry out 1000 simulations with and , and we calculate the estimated value of the source polarization in this case for each filter. We consider the null hypothesis, , there is no polarized source, against the alternative hypothesis, , there is a polarized source. We set a significance level , this means that we reject the null hypothesis when a simulation has an estimated source amplitude higher than that of of the simulations without polarized source. We define the power of the test as , with the probability of accepting the null hypothesis when it is false, i.e. the power is the proportion of simulations with polarized source with an estimated amplitude higher than that of of the simulations without source. The higher the power the more efficient the filter is for detection. Note that the test can be performed in the same way in the blind and non-blind cases: in the second case we know the position of the source, but we do not know whether it is polarized or not.
4.1 Blind case
We carry out simulations in the blind case for the high noise and low noise zones as explained above. We apply the filters to the images and calculate the absolute maximum of for each filtered image. If the detection has a significance higher than we consider it as a real detection, otherwise it would be a spurious one. For this significance level, we calculate the power of the detection test for the different pairs of and values. We also calculate the estimated value of the polarization amplitude , convert it to the estimated flux in Jy, , and compare it with the real value . We plot against for the NPF and the FF with high and low noise in Fig. 1. We also compute the relative error of this estimation and its absolute value. For each pair of and values, we obtain the average and confidence intervals of the absolute value of the relative error, taking into account the performed simulations. These values are plotted against in Fig. 2 for the same cases shown in Fig. 1. We also calculate the estimated position of the detected source and obtain the position error expressed in terms of the number of pixels. The power and the average of the relative error in , of its absolute value and of the position error are presented in Table 1 for the two different types of noise. The rows in the Table are sorted in ascending order of . For the high noise case, we see in the Table that the power for the FF is higher than for the NPF. The position and polarization errors are also lower for the FF, this can also be seen in Figs. 1 and 2. For Jy, the FF and NPF perform similarly. For fluxes higher than Jy, the power with the FF is , the average relative error (bias) is , the average of the absolute value of the relative error is and the average position error is . From now on, we will use the flux limit for which the power is as a measure of the filter performance.
The results for the low noise case are logically much better, the FF also performs better than the NPF. The power of the two filters quickly reaches 100 for fluxes Jy and the errors in both flux and position remain stable from fluxes Jy. We therefore have cut the Table at Jy. In order to have a better sampling of the interesting signal to noise regime, we have simulated (using the same number of simulations as in the other cases) in addition the flux pairs , , , and Jy. This way the Table is much more informative. For fluxes higher than Jy, the power with the FF is , the average bias is , the average of the absolute value of the relative error is and the average position error is .
The FF is also much faster than the NPF. For instance the analysis of an image of 64x64 pixels in a personal computer takes 7 seconds with the FF and about 6 minutes with the NPF. This is due to the maximization involved. The computation time grows proportionally to the number of pixels, so that the NPF could be too slow if we want to analyze large images. This is the reason why we have considered small patches.
Finally, we show in Fig. 3 four images corresponding to a polarized source with embedded in high noise. For the sake of a better visualisation, we show pixel images instead of the sized images used in the simulations. We show the original image in including noise, CMB and source, the image of the source, the image filtered with the NPF and the image treated with the FF method. By simple visual inspection, it is possible to see that the the FF performs better than the NPF.
4.2 Non-blind case
We carry out simulations with a polarized source placed in the image centre and filter the images with the two different filters, centering them in the pixels at a distance from the source less than one FWHM. We simulate the same range of pairs as in the blind case (section 4.1). Then, as in the blind case, we calculate the maximum in , using only the selected pixels. This maximum is the estimated polarization of the source and the pixel where we find the maximum is the source position, we convert this amplitude to the estimated flux in Jy, . We have also performed a detection test, accepting as real sources only those detected with a significance lower than . In Table 2, we write the power of the test, the average of the relative polarization error, of its absolute value and of the position error. Our results are obtained from 500 simulations for each combination of pairs with high and low noise. The detection power is higher for the FF than for the other filter. For the high noise case, the improvement is very clear for Jy; we also obtain higher powers for the non-blind case than for the blind one. This could be expected, since we know the source position in the former case.
The position and polarization errors are also lower for the FF, this can also be seen in Figs. 4 and 5, which show the same quantities as Figs. 1 and 2 for non-blind detection. For fluxes higher than Jy, the power with the FF is , the bias is ,the average of the absolute value of the relative error is and the average position error is . The flux and position errors are lower in the non-blind case than in the blind one, especially for low fluxes.
The results for the low noise case are much better, the FF outperforms the NPF. From Jy on the results are quite similar. For fluxes higher than Jy, the power with the FF is , the average bias is , the average of the absolute value of the relative error is and the average position error is . Similarly to what happened with the low noise case of Table 1, we have cut the Table at flux Jy and we have added new cases in order to have a better sampling of the interesting signal to noise regime. In this case, we have simulated the pairs , , , , and Jy. The non-blind case gives better results than the blind one, especially for low fluxes.
In this paper, we deal with the detection and estimation of the modulus of a vector, a problem of great interest in general and in particular in astrophysics when we consider the polarization of the cosmic microwave background (CMB). The polarization intensity is defined as , where , and are the Stokes parameters. We consider the case of images in , and consisting of a compact source with a profile immersed in Gaussian uncorrelated noise. We intend to detect the source and estimate its amplitude in by using two different methods: a Neyman-Pearson filter (NPF) operating in and based on the maximisation of the corresponding log-likelihood and a filtered fusion (FF) procedure, i.e. the application of the MF on the images of , and and the combination of the corresponding estimates by making the non-linear fusion . We present the two filters in section 2 for two-dimensional, , and three-dimensional vectors, deriving the corresponding expressions for the estimation of the polarized source amplitude.
Since we are interested in applying these different filters to the CMB and this radiation is linearly polarized, we will only consider the two-dimensional vector case in our simulations.
Our goal is to compare the performance of the filters when applied to simulated images. Then, we carry out two-dimensional simulations with the characteristics of the 70 GHz Planck channel. The images have pixels with a pixel size of . We simulate the and components consisting of a compact source filtered with a Gaussian-shaped beam (FWHM of 14’) plus CMB and non-stationary detector noise. We consider two typical zones of the Planck survey: one with high noise and quite isotropic and another one with low noise but proportionally more anisotropic. These zones are extreme cases for the assumed scanning strategy we have chosen, and any other zone of the sky is an intermediate case between these two.
We study two types of detection: blind detection, in which we do not know the source position and non-blind, in which the position is known; we place the source at the centre of the image. We take values of the source fluxes in and , , and , ranging from 0.1 Jy to 0.5 Jy with a step of 0.1 Jy. Note that for extragalactic objects both and can take negative values, but since the sign of both components is irrelevant for the calculation of here we only simulate the positive case. We carry out 500 simulations for each pair and for all the cases of blind and non-blind detection with high and low noise.
We apply the filters to the simulated images, estimating the source amplitude as the maximum value of in the filtered images and the source position as the position of this maximum, then we convert the source amplitude to the source flux in Jy . We fix a significance for the detection and calculate a detection power for this significance, see section 3 for more details. We also calculate the relative error of the estimated flux, its absolute value and the position error (in number of pixels), these errors together with the detection power are written in Tables 1 and 2 for the blind and non-blind case. We also show the estimated fluxes and the absolute value of the relative errors in Figs. 1, 2, 4 and 5 for the different cases.
The FF performs better than the NPF (as can be seen in the Tables and Figures), especially for low fluxes and it is also much faster than the NPF. However NPF is still interesting in a case where only the modulus of a vector is known and not its components (for example, if we had a map of polarization but not the and maps separately). The powers are much higher and the errors much lower for the low noise case than for the high noise one. The filters also perform better in the non-blind case than in the blind one, especially for low fluxes.
We can detect extragalactic point sources in polarization images (at 100% power) with the FF in the high noise zone with fluxes Jy for (blind/non-blind) detection and in the low noise zone with fluxes Jy for (blind/non-blind) detection. The bias and the position error are very low and pixel, respectively for all these fluxes.
|High noise (,;)||pow||pow||err||err||err||err||pos||pos|
|Low noise (,;)||pow||pow||err||err||err||err||pos||pos|
|High noise (,;)||pow||pow||err||err||err||err||pos||pos|
|Low noise (,;)||pow||pow||err||err||err||err||pos||pos|
The authors acknowledge partial financial support from the Spanish Ministry of Education (MEC) under project ESP2004-07067-C03-01 and from the joint CNR-CSIC research project 2006-IT-0037. JLS acknowledges partial financial support by the Spanish MEC and thanks the CNR ISTI in Pisa for their hospitality during his sabbatical leave. FA also wishes to thank the CNR ISTI in Pisa for their hospitality during two short stays when part of this work was done. MLC acknowledges the Spanish MEC for a postdoctoral fellowship. Some of the results in this paper have been derived using the HEALPix (Górski et al., 2005) package.
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