Issue 
A&A
Volume 546, October 2012



Article Number  A114  
Number of page(s)  10  
Section  Astronomical instrumentation  
DOI  https://doi.org/10.1051/00046361/201118234  
Published online  17 October 2012 
Multichannel Poisson denoising and deconvolution on the sphere: application to the Fermi Gammaray Space Telescope
^{1} CEA, Laboratoire AIM, CEA/DSMCNRSUniversité Paris Diderot, CEA, IRFU, Service d’Astrophysique, Centre de Saclay, 91191 GifSurYvette Cedex, France
email: jeremy.schmitt@cea.fr
^{2} GREYC CNRSENSICAENUniversité de Caen, 6 Bd du Maréchal Juin, 14050 Caen Cedex, France
Received: 10 October 2011
Accepted: 10 February 2012
A multiscale representationbased denoising method for spherical data contaminated with Poisson noise, the multiscale variance stabilizing transform on the sphere (MSVSTS), has been previously proposed. This paper first extends this MSVSTS to spherical two and one dimensions data (2D1D), where the two first dimensions are longitude and latitude, and the third dimension is a meaningful physical index such as energy or time. We then introduce a novel multichannel deconvolution built upon the 2D1D MSVSTS, which allows us to get rid of both the noise and the blur introduced by the point spread function (PSF) in each energy (or time) band. The method is applied to simulated data from the Large Area Telescope (LAT), the main instrument of the Fermi Gammaray Space Telescope, which detects high energy gammarays in a very wide energy range (from 20 MeV to more than 300 GeV), and whose PSF is strongly energydependent (from about 3.5 at 100 MeV to less than 0.1 at 10 GeV).
Key words: techniques: image processing / methods: data analysis / gamma rays: general
© ESO, 2012
1. Introduction
1.1. Literature overview
The gammaray sky has been studied with unprecedented sensitivity and image capability thanks to the Large Area Telescope (LAT), the main instrument of the Fermi Gammaray Space Telescope (Atwood et al. 2009), in an energy range between 20 MeV to greater than 300 GeV. The detection of gammaray point sources is difficult for two main reasons: the Poisson noise and the instrument’s point spread function (PSF). The Poisson noise is due to the fluctuations in the number of detected photons. Moreover, the effect of Poisson noise is strong because of the weakness of the fluxes of celestial gamma rays, especially outside the Galactic plane and far away from intense sources. The PSF width is strongly energydependent, varying from about 3.5 at 100 MeV to less than (68% containment) at 10 GeV. Owing to largeangle multiple scattering in the tracker, the PSF has broad tails, for which the 95%/68% containment ratio may be as large as 3.
An extensive literature exists on Poisson noise removal and the interested reader may refer to Schmitt et al. (2011) and Starck et al. (2010) for a thorough review. Motivated by new Xray and gammaray data challenges, several restoration methods have been released in astrophysics that are based on wavelets (Movit 2009; Starck et al. 2009; Faÿ et al. 2011; Schmitt et al. 2010) or the Bayesian machinery (Conrad et al. 2007; Norris et al. 2010). Wavelets have also been used for source detection in Fermi data (Abdo et al. 2010), and a first Poisson denoising algorithm for spherical data was proposed in Schmitt et al. (2010). Starck et al. (2009) developed a denoising approach that effectively handles multichannel data acquired on a cartesian grid, and where the third dimension can be any physically meaningful index such as wavelength, energy, or time. While traditional techniques integrate over all the third dimension in order to improve the signaltonoise ratio (S/N) of the sources, their approach allows us to detect the sources while preserving all of their spectral information without sacrificing any the sensitivity. Nonetheless, none of these methods take into account the point spread function (PSF) of the instrument. Deconvolution can be very helpful in many situations such as source identification or flux estimation in the low energy bands where the resolution degrades severely. To the best of our knowledge, no general method for both multichannel denoising and deconvolving on the sphere has been developed in the literature.
1.2. Contributions
We propose a general framework for denoising and deconvolution that complies with all the above requirements, namely deals with

1.
multichannel data;

2.
acquired on the sphere;

3.
and contaminated by Poisson noise.
Our approach builds upon the concept of variance stabilization applied to the spherical wavelet transform coefficients (Starck et al. 2006). This approach gives a multiscale representation of the Poisson data with variancestabilized coefficients, which can be treated as if they were contaminated by a zeromean white Gaussian noise. The developed algorithms are validated on simulated Fermi HEALPix multichannel cubes (nside = 256) with energy bands ranging from 50 MeV to 50 GeV.
1.3. Paper organization
The rest of the paper is organized as follows. In Sect. 2, we present a new wavelet transform for multichannel data on the sphere, and we show how the variance stabilization transform can be introduced into the decomposition. Section 3 details the way that this transform can be used for Gaussian and Poisson noise removal. Section 4 describes our deconvolution algorithm on the sphere for both monochannel and multichannel spherical data. In Sect. 5, we finally draw some conclusions and give possible perspectives of this work.
1.4. Notations
For a real discretetime filter whose impulse response is h [i] , is its timereversed version. The discrete circular convolution product of two signals is written ⋆ , where the term circular represents periodic boundary conditions. The symbol δ [i] is the Kronecker delta.
For the wavelet representation, the lowpass analysis filter is denoted h and the highpass is taken as g = δ − h throughout the paper. We denote the upsampled version of h as h^{ ↑ j} [l] = h [l] if l/2^{j} ∈ Z and 0 otherwise. We define for and h^{(0)} = δ.
The scaling and wavelet functions used for the analysis (respectively, synthesis) are denoted φ (with ) and ψ (with and k ∈ Z) (respectively, and ).
2. Multiscale representation for multichannel spherical data with poisson noise
2.1. Fast undecimated 2D1D wavelet decomposition/reconstruction on the sphere
Our goal is to analyze multichannel data acquired on a sphere with a nonisotropic twoandonedimensional (2D1D) wavelet, where the two first dimensions are spatial (longitude and latitude) and the third dimension is either the time or the energy. Since the dimensions do not have the same physical meaning, it appears natural that the wavelet scale along the third dimension (energy or time) should not be connected to the spatial scale. Hence, we define the wavelet function as (1)where ψ^{(θφ)} is the spherical two dimensional (2D) spatial wavelet and ψ^{(t)} the onedimensional (1D) wavelet along the third dimension. Similarly to Starck et al. (2009), we consider only isotropic and dyadic spatial scales. We build the discrete 2D1D wavelet decomposition by first taking a spherical 2D undecimated wavelet transform for each k_{t}, followed by a 1D wavelet transform for each spatial wavelet coefficient along the third dimension.
Hence, for a given multichannel data set on the sphere Y [k_{θ},k_{ϕ},k_{t}] and after applying first the 2D spherical undecimated wavelet transform, we have the reconstruction formula (2)where J_{1} is the number of spatial scales, a_{J1} is the (spatial) approximation subband, and are the (spatial) detail subbands. To simplify the notations in the sequel, we replace the two spatial indices by a single index k_{r}, which corresponds to the pixel index. Equation (2)now reads (3)For each spatial location k_{r} and each 2D wavelet scale j_{1}, we then apply a 1D wavelet transform along t on the spatial wavelet coefficients w_{j1} [k_{r},·] such that (4)where J_{2} is the number of scales along t. The approximation spatial subband a_{J1} is processed in a similar way, hence yielding (5)Inserting Eqs. (4)and (5)into (3), we obtain the 2D1D spherical undecimated wavelet representation of Y: (6)In this expression, four kinds of coefficients can be distinguished:
2.2. Multiscale variance stabilizing transform on the sphere (MSVSTS)
Schmitt et al. (2010) proposed a multiscale variance stabilizing transform adapted for Poisson spherical data. This transform was dubbed the multiscale variance stabilizing transform on the Sphere (MSVSTS). The MSVSTS is a multiscale decomposition method designed for Poisson noise that is based on a variance stabilizing transform(VST). Poisson noise is indeed signaldependent, which complicates its removal. The aim of a VST is to get rid of this signaldependence by transforming a Poisson distribution into a Gaussian distribution of known variance. In a nutshell, the MSVSTS consists of plugging a VST into a multiscale transform – the isotropic undecimated wavelet transform on the sphere (IUWTS) – in order to realize (approximately) Gaussian zeromean multiscale coefficients with constant variance. The noise on coefficients can then be easily removed using Gaussian denoising methods such as wavelet shrinkage, as we see in the next section.
The MSVSTS scheme is defined recursively by inserting a (nonlinear) squareroot VST into the IUWTS steps, that is (11)where T_{j} is the VST operator on scale j(12)with the VST constants b^{(j)} and c^{(j)} that depend solely on the filter h and the scale level j. Zhang et al. (2008) showed that the MSVSTS detail coefficients d_{j} on locally homogeneous parts of the underlying intensity signal follow asymptotically a zeromean normal distribution with an intensityindependent variance that relies only on the filter h and the current scale j. Consequently, for a given h, both the stabilized variances and the constants b^{(j)} and c^{(j)} can be precomputed once and for all (Schmitt et al. 2010).
2.3. Multichannel MSVSTS
We now extend the MSVSTS machinery to the multichannel case. This amounts to plugging the VST into the spherical 2D1D undecimated wavelet transform introduced in Sect. 2.1. This again gives rise to four types of coefficients that take the following forms:

Approximationdetail coefficients (j_{1} = J_{1} and ): (14)

Detailapproximation coefficients ( and j_{2} = J_{2}): (15)

Approximationapproximation coefficients (j_{1} = J_{1} and j_{2} = J_{2}): (16)
In summary, all 2D1D wavelet coefficients are now stabilized, and the noise in all these wavelet coefficients is a zeromean Gaussian with known variance that depends solely on h on the resolution levels (j_{1},j_{2}). As before, these variances can be easily tabulated.
3. Application to multichannel denoising
We define X to be the noiseless data and Y their observed noisy version. In the case of the additive zeromean white Gaussian noise, we have , and for the Poisson noise . The main objective behind denoising is to estimate X from Y.
3.1. Warmup: Gaussian noise
We start with the simple and instructive case where the noise in Y is additive white Gaussian. As the spherical 2D1D undecimated wavelet transform described in Sect. 2.1 is linear, the noise remains Gaussian in the transform domain. Therefore, the thresholding strategies that have been developed for wavelet Gaussian denoising can be applied to the spherical 2D1D wavelet transform. Denoting the thresholding operator as TH, the denoised 2D1D estimate of X obtained by thresholding the wavelet coefficients in Eq. (6)reads (17)A typical choice of TH is the hard thresholding operator parametrized by the scalar threshold , i.e. (18)The threshold τ is typically chosen to be between three and five times the noise standard deviation.
3.2. Poisson noise
The treatment of Poisson noise is far more complicated than its Gaussian counterpart, the main reason being that the noise variance is equal to its mean. This is where the MSVSTS comes into play. The role of the MSVSTS is indeed to get rid of this dependence of the variance on the mean by ensuring that the transformed coefficients are Gaussian with constant variance (without loss of generality, this variance can be assumed to be equal to 1). In other words, after the MSVSTS, we are brought to a Gaussian denoising problem where standard thresholding approaches apply.
Nevertheless, denoising is not straightforward because there is no explicit reconstruction formula available because of the form of the nonlinear stabilization equations above. Formally, the stabilizing operators T_{j1,j2} and the convolution operators along the spatial and the third dimensions do not commute, even though the filter bank satisfies the exact reconstruction formula. To circumvent this difficulty, we propose to solve this reconstruction problem by advocating an iterative reconstruction scheme.
3.3. Iterative reconstruction
We define W to be the transform operator associated with the 2D1D IUWTS described in Sect. 2.1, and R to be its inverse transform. We define ℳ to be the multiresolution support, which is determined by the set of significant coefficients detected among WY at each scale (j_{1},j_{2}) and location (k_{r},k_{t}), i.e. (19)We define M to be the orthogonal projector onto ℳ, i.e. ∀d(20)Our goal is to seek a solution that preserves the significant structures of the original data by reproducing exactly the same wavelet coefficients as those of the input data Y, but only on scales and at positions where significant coefficients have been detected. Furthermore, as Poisson intensity functions are positive by nature, a positivity constraint is imposed on the solution. It is clear that there are many solutions satisfying the positivity and multiresolution support consistency requirements, e.g. Y itself. Thus, our reconstruction problem based solely on these constraints is an illposed inverse problem that must be regularized. Typically, the solution in which we are interested must be sparse by involving the lowest budget of wavelet coefficients. Therefore, our reconstruction is formulated as a constrained sparsitypromoting minimization problem over the transform coefficients d(21)and the intensity estimate is reconstructed as , where is a global minimizer of Eq. (21). We recall that ∥·∥ _{1} = ∑ _{j1,j2,kr,kt}  d_{j1,j2} [k_{r},k_{t}]  is the ℓ_{1}norm playing the role of regularization and is wellknown to promote sparsity (Donoho 2004). This problem can be solved efficiently using the hybrid steepest descent algorithm (Yamada 2001; Zhang et al. 2008), and requires about ten iterations in practice. Transposed into our context, its main steps can be summarized as follows:
where P_{+} is the orthogonal projector onto the positive orthant, and ST_{βn} is the entrywise softthresholding operator with threshold β_{n}, i.e. for x ∈ R, ST_{βn}(x) = max(0,1 − β_{n}/x)x.
The final multichannel MSVSTS Poisson noise removal algorithm is summarized in the following steps:
Fig. 1
Result of the multichannel Poisson denoising algorithm on simulated Fermi data over the energy band 220–360 MeV. Top: simulated intensity skymap. Middle: simulated noisy skymap. Bottom: denoised skymap. Maps are on a logarithmic scale. 

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Fig. 2
Result of the multichannel Poisson denoising algorithm on simulated Fermi data over the energy band 589–965 MeV. Top: simulated intensity skymap. Middle: simulated noisy skymap. Bottom: denoised skymap. Maps are on a logarithmic scale. 

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3.4. Experiments
The multichannel MSVSTS algorithm has been applied to a simulated Fermi data set, with 14 energy bands between 50 MeV and 1.58 GeV. Figures 1 and 2 depict the denoising results for two energy bands. The algorithm is able to recover most of the sources, even the faint ones, on each energy band. Even more importantly, the 2D1D MSVSTS denoising algorithm allows us to recover the spectral information for each spatial position, as can be seen from Fig. 3.
4. Deconvolution of spherical data with Poisson noise
We now introduce a wavelet deconvolution approach for monochannel and multichannel data on the sphere with Poisson noise. The main idea underlying the method is to apply the MSVSTS method described above. We first introduce the deconvolution problem and then describe how the MSVSTS can be used to solve the deconvolution problem.
4.1. Problem statement
Many problems in signal and image processing can be cast as an inversion of a linear system (22)where is the data to recover, is the degraded noisy observation, ε is an additive noise, and is a bounded linear operator that is typically illbehaved because it models an acquisition process that encounters loss of information. When H is the identity, it is just a denoising problem that can be treated with the previously described methods. Inverting Eq. (22)is usually an illposed problem, which means that there is no unique and stable solution.
Our objective is to remove the effect of the instrument’s PSF. In our case, H is the convolution by a blurring kernel (i.e. PSF) operator that causes Y to lack the high frequency content of X. Furthermore, since the noise is Poisson, ε has a variance profile HX. The problem at hand is then a deconvolution problem in the presence of Poisson noise.
We therefore need to both regularize the problem and handle the Poisson statistics of the noise. To regularize this inversion problem and reduce the space of candidate solutions, one has to add some prior knowledge of the typical structure of the original data X. This prior information accounts for the smoothness of the solution and can range from the uniform smoothness assumption to more complex knowledge of the geometrical structures of X.
In our LAT realistic simulations, the PSF width depends strongly on the energy, from at 50 MeV to better than at 10 GeV and above. Figure 4 shows the normalized profiles of the PSF for different energy bands.
4.2. Monochannel deconvolution
We first consider the singlechannel case. In the literature, several algorithms have been proposed to perform image deconvolution on a cartesian grid. The RichardsonLucy algorithm is certainly the most famous in astrophysics. In this paper, we propose a regularized RichardsonLucy algorithm to deconvolve data on the sphere data.
Fig. 3
Spectrum of a single gammaray point source recovered using the multichannel MSVSTS denoising algorithm. Top: single gammaray source from simulated Fermi data integrated along the energy axis (left: simulated source; middle: noisy source; right: denoised source). Bottom: spectrum of the center of the point source: intensity as a function of the energy band with 14 energy bands between 50 MeV and 50 GeV (black: true simulated spectrum; cyan: restored spectrum from our denoising algorithm. 

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The RichardsonLucy algorithm originates from a fixedpoint equation obtained by maximizing the Poisson likelihood with respect to X while preserving positivity. This entails a multiplicative update rule, starting at n = 0 and X^{(0)} = 1 and iterating (23)where ⊗ (respectively ) represents the elementwise multiplication (respectively division) between two vectors, and H^{T} is the transpose of H whose action on an image consists in convolving it with the timereversed version of the PSF associated with H. However, it is wellknown that owing to the lack of regularization, the RichardsonLucy algorithm tends to amplify the noise after a few iterations.
We define R^{(n)} as the residual at iteration n(24)where R^{(n)} can be written as the sum of its IUWTS detail subband and the last approximation subband a_{J}, that is, (25)The wavelet transform provides a means of extracting only the significant structures from the residual at each iteration. With increasing number of iterations, a large part of the residual becomes statistically insignificant. The regularized significant residual is then, for a location k_{r}(26)where ℳ is the multiresolution support defined in a similar way to Eq. (19). The regularized RichardsonLucy scheme then becomes (27)This algorithm is similar to Murtagh et al. (1995), except that the à trous wavelet transform is replaced by the undecimated wavelet transform. In the next subsection, we describe how the same algorithm can be extended to the multichannel case.
Fig. 4
Normalized profile of the PSF for different energy bands as a function of the angle in degree. Black: 50−82 MeV. Cyan: 220−360 MeV. Orange: 960−1.6 GeV. Blue: 4.2−6.9 GeV. Green: 19−31 GeV. 

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Fig. 5
Result of the multichannel deconvolution algorithm for different energy bands. Top: simulated (blurred) intensity skymap. Middle: blurred and noisy skymap. Bottom: deconvolved skymap. Energy band: 82–134 MeV. Maps are on a logarithmic scale. 

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Fig. 6
Result of the multichannel deconvolution algorithm for different energy bands. Top: simulated (blurred) intensity skymap. Middle: blurred and noisy skymap. Bottom: deconvolved skymap. Energy band: 220–360 MeV. Maps are on a logarithmic scale. 

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Fig. 7
Result of the multichannel deconvolution algorithm for different energy bands. Top: simulated (blurred) intensity skymap. Middle: blurred and noisy skymap. Bottom: deconvolved skymap. Energy band: 360–589 MeV. Maps are on a logarithmic scale. 

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Fig. 8
Result of the multichannel deconvolution algorithm for different energy bands. Top: simulated (blurred) intensity skymap. Middle: blurred and noisy skymap. Bottom: deconvolved skymap. Energy band: 589–965 MeV. Maps are on a logarithmic scale. 

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4.3. Multichannel deconvolution
As the PSF is channeldependent, the convolution observation model is now in each channel k_{t}, where H_{kt} is the (spatial) convolution operator in channel k_{t} with known PSF.
The same recipe as in the monochannel case applies with the notable difference that the spherical 2D1D MSVSTS is used instead of its monochannel counterpart. The multichannel multiresolution support ℳ is obtained after thresholding these coefficients.
We now define H to be the multichannel convolution^{1} operator, which acts on a 2D1D multichannel spherical data set X by applying H_{t} to each channel X [·,k_{t}] independently^{2}. The regularized multichannel RichardsonLucy scheme is then (28)where is the regularized (significant) residual (29)
4.4. Experiments
The algorithm was applied to the seven energy bands (50 MeV–1.58 GeV) of our simulated Fermi data set. Figures 5 to 8 display the deconvolution results for four energy bands. Figure 9 shows the performance of the multichannel MSVSTS deconvolution algorithm for a single point source. The deconvolution not only effectively removes the blur and recovers sharply localized point sources, but also allows us to restore all of the spectral information. To get a better visual impression of the performance of the deconvolution algorithm, Fig. 10 depicts the result of the algorithm on a single HEALPix face covering the Galactic plane. We find that the deconvolution is remarkably effective: our MSVSTS multichannel deconvolution algorithm manages to remove a large part of the blur introduced by the PSF.
5. Conclusion
This paper extends the MSVSTS framework to deal with monochannel deconvolution, multichannel denoising and multichannel deconvolution. Unlike the monochannel MSVSTS, the multichannel MSVSTS fully exploits the information in the 2D1D data set and allows us to recover the spectral information on the sources. As the PSF strongly depends on the energy, it is very important to have a multichannel method for deconvolution. Multichannel deconvolution using MSVSTS removes a large part of the PSF blur and significantly improves the sharpness of the spatial localization of point sources.
The software related to this paper, MRS/Poisson, and its full documentation will be included in the next version of ISAP (Interactive Sparse astronomical data Analysis Packages) via the ISAP web site^{3}.
Fig. 9
Profile of a single gammaray point source recovered using the multichannel MSVSTS deconvolution algorithm. Top: single gammaray point source on simulated (blurred) Fermi data (energy band: 360–589 MeV) (left: simulated blurred source; middle: blurred noisy source; right: deconvolved source). Bottom: profile of the point source (cyan: simulated spectrum; black: restored spectrum from the deconvolved source. 

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Fig. 10
View on a single HEALPix face. Result of an application of the deconvolution algorithm to the Galactic plane. Top left: simulated Fermi Poisson intensity. Top right: simulated Fermi noisy data. Bottom: Fermi data deconvolved with multichannel MSVSTS. Energy band: 360–589 MeV. Pictures are on a logarithmic scale. 

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Acknowledgments
Some of the results in this paper have been derived using the Healpix (Górski et al. 2005) and the MR/S software (Starck et al. 2006). This work was supported by the French National Agency for Research (ANR08EMER00901) and the European Research Council grant SparseAstro (ERC228261).
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All Figures
Fig. 1
Result of the multichannel Poisson denoising algorithm on simulated Fermi data over the energy band 220–360 MeV. Top: simulated intensity skymap. Middle: simulated noisy skymap. Bottom: denoised skymap. Maps are on a logarithmic scale. 

Open with DEXTER  
In the text 
Fig. 2
Result of the multichannel Poisson denoising algorithm on simulated Fermi data over the energy band 589–965 MeV. Top: simulated intensity skymap. Middle: simulated noisy skymap. Bottom: denoised skymap. Maps are on a logarithmic scale. 

Open with DEXTER  
In the text 
Fig. 3
Spectrum of a single gammaray point source recovered using the multichannel MSVSTS denoising algorithm. Top: single gammaray source from simulated Fermi data integrated along the energy axis (left: simulated source; middle: noisy source; right: denoised source). Bottom: spectrum of the center of the point source: intensity as a function of the energy band with 14 energy bands between 50 MeV and 50 GeV (black: true simulated spectrum; cyan: restored spectrum from our denoising algorithm. 

Open with DEXTER  
In the text 
Fig. 4
Normalized profile of the PSF for different energy bands as a function of the angle in degree. Black: 50−82 MeV. Cyan: 220−360 MeV. Orange: 960−1.6 GeV. Blue: 4.2−6.9 GeV. Green: 19−31 GeV. 

Open with DEXTER  
In the text 
Fig. 5
Result of the multichannel deconvolution algorithm for different energy bands. Top: simulated (blurred) intensity skymap. Middle: blurred and noisy skymap. Bottom: deconvolved skymap. Energy band: 82–134 MeV. Maps are on a logarithmic scale. 

Open with DEXTER  
In the text 
Fig. 6
Result of the multichannel deconvolution algorithm for different energy bands. Top: simulated (blurred) intensity skymap. Middle: blurred and noisy skymap. Bottom: deconvolved skymap. Energy band: 220–360 MeV. Maps are on a logarithmic scale. 

Open with DEXTER  
In the text 
Fig. 7
Result of the multichannel deconvolution algorithm for different energy bands. Top: simulated (blurred) intensity skymap. Middle: blurred and noisy skymap. Bottom: deconvolved skymap. Energy band: 360–589 MeV. Maps are on a logarithmic scale. 

Open with DEXTER  
In the text 
Fig. 8
Result of the multichannel deconvolution algorithm for different energy bands. Top: simulated (blurred) intensity skymap. Middle: blurred and noisy skymap. Bottom: deconvolved skymap. Energy band: 589–965 MeV. Maps are on a logarithmic scale. 

Open with DEXTER  
In the text 
Fig. 9
Profile of a single gammaray point source recovered using the multichannel MSVSTS deconvolution algorithm. Top: single gammaray point source on simulated (blurred) Fermi data (energy band: 360–589 MeV) (left: simulated blurred source; middle: blurred noisy source; right: deconvolved source). Bottom: profile of the point source (cyan: simulated spectrum; black: restored spectrum from the deconvolved source. 

Open with DEXTER  
In the text 
Fig. 10
View on a single HEALPix face. Result of an application of the deconvolution algorithm to the Galactic plane. Top left: simulated Fermi Poisson intensity. Top right: simulated Fermi noisy data. Bottom: Fermi data deconvolved with multichannel MSVSTS. Energy band: 360–589 MeV. Pictures are on a logarithmic scale. 

Open with DEXTER  
In the text 
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