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Optimal filters for the butterfly template constructed on the sphere in the setting consistent with WMAP observations (see text). (Functions/data defined on the sphere are illustrated here and subsequently using the Mollweide projection.)
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We derive optimal filters on the sphere in the context of detecting compact objects embedded in a stochastic background process. The matched filter and the scale adaptive filter are derived on the sphere in the most general setting, allowing for directional template profiles and filters. The performance and relative merits of the two optimal filter...
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Context 1
... white noise of constant variance 0.05(mK) 2 is added to reflect the noise in WMAP observations. We consider the construction in this setting of the MF and SAF for the directional butterfly template illustrated in Figure 2 (a) (the butterfly template is defined by the partial derivative in one direction only of a two- dimensional Gaussian on the sphere; see [18] for a definition). 2 2 The step of the butterfly template may be used to model the line-like discontinuity of Kaiser-Stebbins type cosmic string signatures [3]. ...
Context 2
... resultant MF and SAF are illustrated in Figure 2 (b) and (c) respectively. Optimal filters are constructed here in the context of L 2 -norm preserving dilations, i.e. for p = 2. Furthermore, in practice the template profiles are assumed to be band-limited at ℓ max , in which case all expressions and summations involving ℓ are computed up to ℓ max only. ...
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... Differentiating E ms with respect to (Ξ(·; j)) m,k and setting the result equal to zero, we obtain a linear system which, using (19) and (20), can be cast in the matrix form in (18). ...
... From the relation between Wigner-D functions and spherical harmonics in (9), it can be seen that (22) can be obtained from (16) by setting m = 0 and (Ξ(·; j)) m,k = (1/c ) (Ξ(·; j)) m δ m,k . Hence, by setting k = k = m in (19) and (20), spectral coefficients of the filter in (21) can be directly obtained from (18) as ...
We present a framework for the optimal filtering of spherical signals contaminated by realizations of zero-mean anisotropic noise processes. Filtering is performed in the wavelet domain given by the scale-discretized wavelet transform on the sphere. The proposed filter is optimal in the sense that it minimizes the mean square error between the filtered wavelet representation and wavelet representation of the noise-free signal. We also present a simplified formulation of the filter for the case when azimuthally symmetric wavelet functions are used. We demonstrate the use of the proposed optimal filter for denoising of an Earth topography map in the presence of uncorrelated, zero-mean, White Gaussian noise. The proposed filter is found to be superior to the hard thresholding method, particularly in the high noise regime.
... Many noise removal techniques, with different assumptions and constraints, have been proposed in the literature [19], [21]- [23]. These methods process signals in either spatial or spectral domain and assume the signal and/or noise to be a realization of an isotropic random process on the sphere. ...
We present a joint SO(3)-spectral domain filtering framework using directional spatially localized spherical harmonic transform (DSLSHT), for the estimation and enhancement of random anisotropic signals on the sphere contaminated by random anisotropic noise. We design an optimal filter for filtering the DSLSHT representation of the noise-contaminated signal in the joint SO(3)-spectral domain. The filter is optimal in the sense that the filtered representation in the joint domain is the minimum mean square error estimate of the DSLSHT representation of the underlying (noise-free) source signal. We also derive a least-square solution for the estimate of the source signal from the filtered representation in the joint domain. We demonstrate the capability of the proposed filtering framework using the Earth topography map in the presence of anisotropic, zero-mean, uncorrelated Gaussian noise, and compare its performance with the joint spatial-spectral domain filtering framework.
... Given this decomposition and the assumption that the noise term is homogeneous and isotropic with zero mean, the spherical harmonic coefficients of the optimal matched filter are uniquely determined (Schäfer et al. 2006;McEwen et al. 2008) to be is the total power spectrum of the noise field. Figure 3 shows the optimal matched filters in each λ v bin, designed for the template profiles obtained in Section 3. ...
... ,tot 0 MF 2 , is minimized. The power of the optimal matched filter can be quantified in terms of the maximum detection level (McEwen et al. 2008) for a single isolated void, ...
We report a detection of the gravitational lensing effect of cosmic voids from the Baryon Oscillation Spectroscopic Data Release 12 seen in the Planck 2018 cosmic microwave background (CMB) lensing convergence map. To make this detection, we introduce new optimal techniques for void stacking and filtering of the CMB maps, such as binning voids by a combination of their observed galaxy density and size to separate those with distinctive lensing signatures. We calibrate theoretical expectations for the void lensing signal using mock catalogs generated in a suite of 108 full-sky lensing simulations from Takahashi et al. Relative to these templates, we measure the lensing amplitude parameter in the data to be A L = 1.10 ± 0.21 using a matched-filter stacking technique and confirm it using an alternative Wiener-filtering method. We demonstrate that the result is robust against thermal Sunyaev–Zel’dovich contamination and other sources of systematics. We use the lensing measurements to test the relationship between the matter and galaxy distributions within voids and show that the assumption of linear bias with a value consistent with galaxy clustering results is discrepant with observation at ∼3 σ ; we explain why such a result is consistent with simulations and previous results, and is expected as a consequence of void selection effects. We forecast the potential for void CMB lensing measurements in future data from the Advanced ACT, Simons Observatory, and CMB-S4 experiments, showing that, for the same number of voids, the achievable precision improves by a factor of more than 2 compared to Planck .
... where we have split it into an amplitude term κ 0 (λ v ) ≡ κ template (0; λ v ), and a normalised shape function k(θ) defined by the spherical harmonic coefficients k L0 . Given this decomposition and the assumption that the noise term is homogeneous and isotropic with zero mean, the spherical harmonic coefficients of the optimal matched filter are uniquely determined (Schäfer et al. 2006;McEwen et al. 2008) to be: ...
... is minimized. The power of the optimal matched filter can be quantified in terms of the maximum detection level(McEwen et al. 2008) for a single isolated void, ...
We report a $5.3\sigma$ detection of the gravitational lensing effect of cosmic voids from the Baryon Oscillation Spectroscopic (BOSS) Data Release 12 seen in the $Planck$ 2018 cosmic microwave background (CMB) lensing convergence map. To make this detection, we introduce new optimal techniques for void stacking and filtering of the CMB maps, such as binning voids by a combination of their observed galaxy density and size to separate those with distinctive lensing signatures. We calibrate theoretical expectations for the void-lensing signal using mock catalogs generated in a suite of 108 full-sky lensing simulations from Takahashi et al. (2017). Relative to these templates, we measure the lensing amplitude parameter in the data to be $A_L=1.10 \pm 0.21$ using a matched-filter stacking technique, and confirm it using an alternative Wiener filtering method. We demonstrate that the result is robust against thermal Sunyaev-Zel'dovich contamination and other sources of systematics. We use the lensing measurements to test the relationship between the matter and galaxy distributions within voids, and show that the assumption of linear bias with a value consistent with galaxy clustering results is discrepant with observation at $\sim 3\sigma$; we explain why such a result is consistent with simulations and previous results, and is expected as a consequence of void selection effects. We forecast the potential for void-CMB lensing measurements in future data from the Advanced ACT, Simons Observatory and CMB-S4 experiments, showing that, for the same number of voids, the achievable precision improves by a factor of more than two compared to $Planck$.
... These constraints are satisfied by choosing (Schäfer et al. 2006;McEwen et al. 2008) ...
We present a new method for detection of the integrated Sachs-Wolfe (ISW) imprints of cosmic superstructures on the cosmic microwave background, based on a matched filtering approach. The expected signal-to-noise ratio for this method is comparable to that obtained from the full cross-correlation, and unlike other stacked filtering techniques it is not subject to an a posteriori bias. We apply this method to Planck CMB data using voids and superclusters identified in the CMASS galaxy data from the Sloan Digital Sky Survey Data Release 12, and measure the ISW amplitude to be $A_\mathrm{ISW}=1.64\pm0.53$ relative to the $\Lambda$CDM expectation, corresponding to a $3.1\sigma$ detection. In contrast to some previous measurements of the ISW effect of superstructures, our result is in agreement with the $\Lambda$CDM model.
... In order to address this issue, we propose the following recipe to determine the optimal ordering of samples in a vector θ, which we refer to as the condition number minimisation method. For a given L and samples in θ given by (19), the vector θ is constructed as follows: ...
... farthest from the poles, which is a natural choice for the ring of 2L − 1 samples along φ. • For each m = L − 2, L − 3, . . . , 0, choose θ m from the set Θ, given in (19), which minimizes the condition number of the matrix P m . Such a placement of samples along co-latitude ensures the robust inversion of the system given by (14), thus resulting in an accurate computation of spherical harmonic coefficients by the proposed forward SHT. ...
... We discuss the computational complexity of the proposed transforms later in the paper. We note that Θ in (19) is composed of equiangular samples along co-latitude, which is simple, but not the only choice to place iso-latitude rings in the proposed sampling scheme. ...
We develop a sampling scheme on the sphere that permits accurate computation
of the spherical harmonic transform and its inverse for signals band-limited at
$L$ using only $L^2$ samples. We obtain the optimal number of samples given by
the degrees of freedom of the signal in harmonic space. The number of samples
required in our scheme is a factor of two or four fewer than existing
techniques, which require either $2L^2$ or $4L^2$ samples. We note, however,
that we do not recover a sampling theorem on the sphere, where spherical
harmonic transforms are theoretically exact. Nevertheless, we achieve high
accuracy even for very large band-limits. For our optimal-dimensionality
sampling scheme, we develop a fast and accurate algorithm to compute the
spherical harmonic transform (and inverse), with computational complexity
comparable with existing schemes in practice. We conduct numerical experiments
to study in detail the stability, accuracy and computational complexity of the
proposed transforms. We also highlight the advantages of the proposed sampling
scheme and associated transforms in the context of potential applications.
... The mask has both large and small scale features, and so algorithms must account for the full range of scales when calculating the noise covariance. Estimators constructed using matched filters, for example [5], typically assume that the noise is isotropic, which is not the case in practice. The resolution of the JCAP10(2013)001 images also presents computational challenges. ...
... where D l mm is the Wigner D-matrix (for a definition see, for example, [5]). Using the identity D l m0 ( n) = 4π/(2 + 1)Y * m ( n), we obtain eq. ...
We present algorithms for searching for azimuthally symmetric features in CMB
data. Our algorithms are fully optimal for masked all-sky data with
inhomogeneous noise, computationally fast, simple to implement, and make no
approximations. We show how to implement the optimal analysis in both Bayesian
and frequentist cases. In the Bayesian case, our algorithm for evaluating the
posterior likelihood is so fast that we can do a brute-force search over
parameter space, rather than using a Monte Carlo Markov chain. Our motivating
example is searching for bubble collisions, a pre-inflationary signal which can
be generated if multiple tunneling events occur in an eternally inflating
spacetime, but our algorithms are general and should be useful in other
contexts.
... The problem to design optimal filters for signals defined on the sphere has been well studied. [9][10][11] At their core, these investigations assume the signal and/or noise process to be isotropic and present the formulation of optimal filters in either spatial (pixel) domain or spectral domain, which is enabled through the spherical harmonic transform. 12,13 Assuming the process to be isotropic, the classical approach to remove the effect of noise is using the the mean-square error linear optimal filter (Wiener filter) formulated in the spectral domain. ...
... 12,13 Assuming the process to be isotropic, the classical approach to remove the effect of noise is using the the mean-square error linear optimal filter (Wiener filter) formulated in the spectral domain. 10,14 Since such an optimal filter makes profit based on the knowledge of the energy distribution of the noise in the spectral domain and does not take into account the anisotropic properties of the process, it is not suitable for filtering out anisotropic noise. 9 The extension of the mean-square error optimal filter for the anisotropic process results in a spatially-varying anisotropic filter (also referred to as an anisotropic non-symmetric filter 9 ) for which a simple spectral domain formulation cannot be obtained. ...
... For isotropic processes, there exist simple spectral domain formulations for optimal filters which minimizes the mean-square error between the estimate and the signal 11 or equivalently the variance of the unbiased estimate. 10 In our present formulation the signal and noise processes can be anisotropic, and this puts us in the realm of designing spatially-varying optimal filters for the estimation of a signal from its noise contaminated version. We design such an optimal filter in the next section. ...
In this paper, we present an optimal filter for the enhancement or estimation
of signals on the 2-sphere corrupted by noise, when both the signal and noise
are realizations of anisotropic processes on the 2-sphere. The estimation of
such a signal in the spatial or spectral domain separately can be shown to be
inadequate. Therefore, we develop an optimal filter in the joint
spatio-spectral domain by using a framework recently presented in the
literature --- the spatially localized spherical harmonic transform ---
enabling such processing. Filtering of a signal in the spatio-spectral domain
facilitates taking into account anisotropic properties of both the signal and
noise processes. The proposed spatio-spectral filtering is optimal under the
mean-square error criterion. The capability of the proposed filtering framework
is demonstrated with by an example to estimate a signal corrupted by an
anisotropic noise process.
... The optimal matched filters are constructed in harmonic space: thus the assumption of an azimuthally-symmetric beam profile and isotropic noise simplifies their construction and application considerably, while remaining highly accurate for the relatively low band-limit considered. Once the source profile and stochastic background are defined, the filters are constructed on the sphere as outlined by Ref. [36]. ...
... The construction of optimal filters is implemented in the S2FIL code [36] (which in turn relies on the codes S2 [39] and HEALPix [40]), while the COMB code [36] has been used to simulate bubble collision signatures embedded in a CMB background. 3 The candidate object detection algorithm described here is implemented in a modified version of S2FIL that will soon be made publicly available. ...
... The construction of optimal filters is implemented in the S2FIL code [36] (which in turn relies on the codes S2 [39] and HEALPix [40]), while the COMB code [36] has been used to simulate bubble collision signatures embedded in a CMB background. 3 The candidate object detection algorithm described here is implemented in a modified version of S2FIL that will soon be made publicly available. ...
A number of theoretically well-motivated additions to the standard
cosmological model predict weak signatures in the form of spatially localized
sources embedded in the cosmic microwave background (CMB) fluctuations. We
present a hierarchical Bayesian statistical formalism and a complete data
analysis pipeline for testing such scenarios. We derive an accurate
approximation to the full posterior probability distribution over the
parameters defining any theory that predicts sources embedded in the CMB, and
perform an extensive set of tests in order to establish its validity. The
approximation is implemented using a modular algorithm, designed to avoid a
posteriori selection effects, which combines a candidate-detection stage with a
full Bayesian model-selection and parameter-estimation analysis. We apply this
pipeline to theories that predict cosmic textures and bubble collisions,
extending previous analyses by using: (1) adaptive-resolution techniques,
allowing us to probe features of arbitrary size, and (2) optimal filters, which
provide the best possible sensitivity for detecting candidate signatures. We
conclude that the WMAP 7-year data do not favor the addition of either cosmic
textures or bubble collisions to the standard cosmological model, and place
robust constraints on the predicted number of such sources. The expected
numbers of bubble collisions and cosmic textures on the CMB sky within our
detection thresholds are constrained to be fewer than 4.0 and 5.2 at 95%
confidence, respectively.
... It is important to fill that gap because spherical data arise in numerous fields, including measurements of atmospheric pressure, cosmic microwave background, or surface reflectance. Consequently, methods for filtering, spatio-spectral and spatioscale analysis of spherical data using spherical harmonics have attracted the attention of the research community [7][15] [18]. However, existing research has not provided a detailed understanding of the importance of phase of spherical harmonics, either on its own or in relation to the harmonic magnitude. ...
This paper examines filtering on a sphere, by first examining the roles of
spherical harmonic magnitude and phase. We show that phase is more important
than magnitude in determining the structure of a spherical function. We examine
the properties of linear phase shifts in the spherical harmonic domain, which
suggest a mechanism for constructing finite-impulse-response (FIR) filters. We
show that those filters have desirable properties, such as being associative,
mapping spherical functions to spherical functions, allowing directional
filtering, and being defined by relatively simple equations. We provide
examples of the filters for both spherical and manifold data.
