3D Power Spectrum Research Articles

Extracting the 21 cm EoR signal using MWA drift scan data

ABSTRACT The detection of redshifted hyperfine line of neutral hydrogen (H i) is the most promising probe of the epoch of reionization (EoR). We report an analysis of 55 h of Murchison Widefield Array (MWA) phase II drift scan EoR data. The data correspond to a central frequency $\nu _0 = 154.24 \, \rm MHz$ (z ≃ 8.2 for the redshifted H i hyperfine line) and bandwidth $B = 10.24 \, \rm MHz$. As one expects greater system stability in a drift scan, we test the system stability by comparing the extracted power spectra from data with noise simulations and show that the power spectra for the cleanest data behave as thermal noise. We compute the H i power spectrum as a function of time in one and two dimensions. The best upper limit on the 1D power spectrum are: $\Delta ^2(k) \simeq (1000~\rm mK)^2$ at k ≃ 0.2h Mpc−1 and at k ≃ 1h Mpc−1. The cleanest modes, which might be the most suited for obtaining the optimal signal to noise, correspond to k ≳ 1h Mpc−1. We also study the time-dependence of the foreground-dominated modes in a drift scan and compare with the expected behaviour.

Projected two- and three-point statistics: forecasts and mitigation of non-linear RSDs

ABSTRACT The combination of two- and three-point clustering statistics of galaxies and the underlying matter distribution has the potential to break degeneracies between cosmological parameters and nuisance parameters and can lead to significantly tighter constraints on parameters describing the composition of the Universe and the dynamics of inflation. Here, we investigate the relation between biases in the estimated parameters and inaccurate modelling of non-linear redshift-space distortions for the power spectrum and bispectrum of projected galaxy density fields and lensing convergence. Non-linear redshift-space distortions are one of the leading systematic uncertainties in galaxy clustering. Projections along the line of sight suppress radial modes and are thus allowing a trade-off between biases due to non-linear redshift-space distortions and statistical uncertainties. We investigate this bias-error trade-off for a CMASS-like survey with a varying number of redshift bins. Improved modelling of the non-linear redshift-space distortions allows the recovery of more radial information when controlling for biases. Not modelling non-linear redshift-space distortions inflates error bars for almost all parameters by $20{{\ \rm per\ cent}}$. The information loss for the amplitude of local non-Gaussianities is smaller, since it is best constrained from large scales. In addition, we show empirically that one can recover more than 99 per cent of the 3D power spectrum information if the depth of the tomographic bins is reduced to 10 h−1 Mpc.

H i intensity mapping with MeerKAT: 1/f noise analysis

ABSTRACT The nature of the time correlated noise component (the 1/f noise) of single dish radio telescopes is critical to the detectability of the H i signal in intensity mapping experiments. In this paper, we present the 1/f noise properties of the MeerKAT receiver system using South Celestial Pole tracking data. We estimate both the temporal power spectrum density and the 2D power spectrum density for each of the antennas and polarizations. We apply singular value decomposition to the data set and show that, by removing the strongest components, the 1/f noise can be drastically reduced, indicating that it is highly correlated in frequency. With two-mode subtraction, the knee frequency over a $20\,$ MHz averaging is about $3\times 10^{-3}\, {\rm Hz}$, indicating that the system induced 1/f-type variations are well under the thermal noise fluctuations over a few hundred seconds time-scales. We also show that such cleaning on the time ordered data has very little impact on the 21-cm signal itself. The 2D power spectrum shows that the 1/f-type variations are restricted to a small region in the time–frequency space, either with long-wavelength correlations in frequency or in time. This gives a wide range of cosmological scales where the H i signal can be measured without further need to calibrate the gain time fluctuations. Finally, we demonstrate that a simple power spectrum parameterization is sufficient to describe the data and provide fitting parameters for both the 1D and 2D power spectrum.

Cooperative Multi-Point Vehicular Positioning Using Millimeter-Wave Surface Reflection

Multi-point vehicular positioning is an essential operation for autonomous vehicles. However, the state-of-the-art positioning technologies, relying on reflected signals from a target (i.e., RADAR and LIDAR), cannot work without line-of-sight (LoS). Besides, it takes significant time for environment scanning and object recognition with potential detection inaccuracy, especially in complex urban situations. Some recent fatal accidents involving autonomous vehicles further expose such limitations. In this article, we aim at overcoming these limitations by proposing a novel relative positioning approach, called Cooperative Multi-point Positioning (COMPOP). The COMPOP establishes cooperation between a target vehicle (TV) and a sensing vehicle (SV) if a LoS path exists, where a TV explicitly lets an SV to know the TV’s existence by transmitting positioning waveforms. This cooperation makes it possible to remove the time-consuming scanning and target recognizing processes, facilitating real-time positioning. One prerequisite for the cooperation is a clock synchronization between a pair of TV and SV. To this end, we use a phase-differential-of-arrival (PDoA) based approach to remove the TV-SV clock difference from the received signal. With clock difference correction, the TV’s position can be obtained via peak detection over a 3D power spectrum constructed by a Fourier transform (FT) based algorithm. The COMPOP also incorporates nearby vehicles, without knowing their locations, into the above cooperation for the case without a LoS path. Specifically, several strong non-LoS (NLoS) links from the TV to the SV can be generated via mirror-like reflections over the neighboring vehicles’ metal surfaces. Following the same procedures in the LoS case, virtual TVs mirrored by nearby vehicles can be detected. By exploiting the geometric relation between the virtual and actual TVs, COMPOP can be achieved by intelligently combining the virtual TVs to position the actual TV. The effectiveness of the COMPOP is verified by several simulations concerning practical channel parameters.

Three-Dimensional Atomic Structure of Grain Boundaries Resolved by Atomic-Resolution Electron Tomography

Summary Grain boundaries (GBs) determine the properties of polycrystals, and tailoring the GB structure offers a promising method for the discovery and engineering of new materials. However, GB structures are far from well understood because of their structural complexity and limitations of conventional projection imaging methods. Here, we decipher three-dimensional atomic structure and crystallography of GBs in nanometals using atomic-resolution electron tomography. Unlike conventional descriptions, whereby they are either straight or curved planar planes with one-dimensional translational symmetry, we show that the high-angle GBs completely lose translational symmetry due to undulated curvature related to configurations of structural units. Moreover, we directly visualize kinks and jogs at the single-atom scale in dislocation-type GBs and investigate their mobilities. Our findings bring new insights to the conventional wisdom of GBs and show the importance of developing methods to include the non-planar nature of GBs to statistically evaluate the behavior of GBs in modeling studies.

Measuring interstellar delays of PSR J0613−0200 over 7 yr, using the Large European Array for Pulsars

ABSTRACT Using data from the Large European Array for Pulsars, and the Effelsberg telescope, we study the scintillation parameters of the millisecond pulsar PSR J0613−0200 over a 7 yr timespan. The ‘secondary spectrum’ – the 2D power spectrum of scintillation – presents the scattered power as a function of time delay, and contains the relative velocities of the pulsar, observer, and scattering material. We detect a persistent parabolic scintillation arc, suggesting scattering is dominated by a thin, anisotropic region. The scattering is poorly described by a simple exponential tail, with excess power at high delays; we measure significant, detectable scattered power at times out to ${\sim}5 \, \mu {\rm s}$, and measure the bulk scattering delay to be between 50 to 200 ns with particularly strong scattering throughout 2013. These delays are too small to detect a change of the pulse profile shape, yet they would change the times of arrival as measured through pulsar timing. The arc curvature varies annually, and is well fitted by a one-dimensional scattering screen ${\sim}40{{\ \rm per\ cent}}$ of the way towards the pulsar, with a changing orientation during the increased scattering in 2013. Effects of uncorrected scattering will introduce time delays correlated over time in individual pulsars, and may need to be considered in gravitational wave analyses. Pulsar timing programmes would benefit from simultaneously recording in a way that scintillation can be resolved, in order to monitor the variable time delays caused by multipath propagation.

Statistical Detection of IGM Structures during Cosmic Reionization Using Absorption of the Redshifted 21 cm line by H i against Compact Background Radio Sources

Abstract Detecting neutral hydrogen structures in the intergalactic medium (IGM) during cosmic reionization via absorption (21 cm forest) against a background radiation is considered independent and complementary to the three-dimensional tomography and power spectrum techniques. The direct detection of this absorption requires very bright (≳10–100 mJy) background sources at high redshifts (z ≳ 8), which are evidently rare; very long times of integration; or instruments of very high sensitivity. This motivates a statistical one-dimensional (1D) power spectrum approach along narrow sightlines but with fainter background objects (∼1–10 mJy), which are likely to be more abundant and significant contributors at high redshifts. The 1D power spectrum reduces cosmic variance and improves sensitivity especially on small spatial scales. Using standard radiative transfer and fiducial models for the instrument, the background sources, and the evolution of IGM structures during cosmic reionization, the potential of the 1D power spectrum along selected narrow directions is investigated against uncertainties from thermal noise and the chromatic synthesized point spread function (PSF) response. Minimum requirements on the number of high-redshift background sources, the telescope sensitivity, and the PSF quality are estimated for a range of instrumental, background source, and reionization model parameters. The 1D power spectrum is intrinsically stronger at higher redshifts. A ∼1000 hr observing campaign targeting ∼100 narrow sightlines to radio-faint, high-redshift background objects with modern radio telescopes, especially the Square Kilometre Array, can detect the 1D power spectrum on a range of spatial scales and redshifts, and potentially discriminate between models of cosmic reionization.

Automated Detection of Respiratory Movements for Image Quality Assurance

In this study, we aimed to develop a rapid quantification technique that can circumvent radiodiagnostic errors that occur during respiratory movements in chest imaging. We also intended to use this technique to improve data interpretation, diagnosis, and image quality assurance. Chest X-ray imaging was performed in a total of 45 patients between 2006 and 2011 at a general hospital. During imaging, the respiratory movements of the patients were visually evaluated by radiologists and radiological technicians. The radiographic images were retrospectively analyzed. A statistical analysis was conducted using equal-variance Student’s t-test. A significantly distinct pattern in the frequency domain was observed in the power spectra of radiographs of patients without and with respiratory movements. Similarly, 3D power spectrum images showed a decreased power intensity and a wider base region in images of patients with different disease states that reflected different patterns. These results suggest that the proposed method can easily detect respiratory movements; hence, it may be used for image quality assurance and improving data interpretation and diagnosis.

The impact of AGN feedback on the 1D power spectra from the Ly α forest using the Horizon-AGN suite of simulations

ABSTRACT The Lyman-α forest is a powerful probe for cosmology, but it is also strongly impacted by galaxy evolution and baryonic processes such as active galactic nucleus (AGN) feedback, which can redistribute mass and energy on large scales. We constrain the signatures of AGN feedback on the 1D power spectrum of the Lyman-α forest using a series of eight hydro-cosmological simulations performed with the adaptive mesh refinement code ramses. This series starts from the Horizon-AGN simulation and varies the subgrid parameters for AGN feeding, feedback, and stochasticity. These simulations cover the whole plausible range of feedback and feeding parameters according to the resulting galaxy properties. AGNs globally suppress the Lyman-α power at all scales. On large scales, the energy injection and ionization dominate over the supply of gas mass from AGN-driven galactic winds, thus suppressing power. On small scales, faster cooling of denser gas mitigates the suppression. This effect increases with decreasing redshift. We provide lower and upper limits of this signature at nine redshifts between z = 4.25 and 2.0, making it possible to account for it at post-processing stage in future work given that running simulations without AGN feedback can save considerable amounts of computing resources. Ignoring AGN feedback in cosmological inference analyses leads to strong biases with 2 per cent shift on σ8 and 1 per cent shift on ns, which represents twice the standards deviation of the current constraints on ns.

Contribution of Galactic free–free emission to the foreground for EoR signal in SKA experiments

ABSTRACT The overwhelming foreground contamination hinders the accurate detection of the 21-cm signal of neutral hydrogen during the Epoch of Reionization (EoR). Among various foreground components, the Galactic free–free emission is less studied, so that its impact on the EoR observations remains unclear. In this work, we employ the observed H α intensity map with the correction of dust absorption and scattering, the simfast21 software, and the latest SKA1-Low layout configuration to simulate the SKA ‘observed’ images of Galactic free–free emission and the EoR signal. By calculating the 1D power spectra from the simulated image cubes, we find that the Galactic free–free emission is about 103.5–102.0, 103.0–101.3, and 102.5–101.0 times more luminous than the EoR signal on scales of 0.1 Mpc−1 < k < 2 Mpc−1 in the 116–124, 146–154, and 186–194 MHz frequency bands. We further analyse the 2D power spectra inside the properly defined EoR window and find that the leaked Galactic free–free emission can still cause non-negligible contamination, as the ratios of its power (amplitude squared) to the EoR signal power can reach about 200, 60, and 15 per cent on scales of k ∼ 1.2 Mpc−1 in three frequency bands, respectively. Therefore, we conclude that the Galactic free–free emission, as a severe contaminating foreground component, needs to be carefully treated in the forthcoming deep EoR observations.

Predictions for measuring the 21-cm multifrequency angular power spectrum using SKA-Low

ABSTRACT The light-cone effect causes the mean as well as the statistical properties of the redshifted 21-cm signal ${T_{\rm b}}(\hat{\boldsymbol {n}}, \nu)$ to change with frequency ν (or cosmic time). Consequently, the statistical homogeneity (ergodicity) of the signal along the line-of-sight (LoS) direction is broken. This is a severe problem particularly during the Epoch of Reionization (EoR) when the mean neutral hydrogen fraction ($\bar{x}_{\rm {H\,{\small I}}}$) changes rapidly as the Universe evolves. This will also pose complications for large bandwidth observations. These effects imply that the 3D power spectrum P(k) fails to quantify the entire second-order statistics of the signal as it assumes the signal to be ergodic and periodic along the LoS. As a proper alternative to P(k), we use the multifrequency angular power spectrum (MAPS) ${\mathcal {C}}_{\ell }(\nu _1,\nu _2)$, which does not assume the signal to be ergodic and periodic along the LoS. Here, we study the prospects for measuring the EoR 21-cm MAPS using future observations with the upcoming SKA-Low. Ignoring any contribution from the foregrounds, we find that the EoR 21-cm MAPS can be measured at a confidence level ≥5σ at angular scales ℓ ∼ 1300 for total observation time tobs ≥ 128 h across ∼44 MHz observational bandwidth. We also quantitatively address the effects of foregrounds on MAPS detectability forecast by avoiding signal contained within the foreground wedge in $({\boldsymbol {k}}_\perp , k_\parallel)$ plane. These results are very relevant for the upcoming large bandwidth EoR experiments as previous predictions were all restricted to individually analysing the signal over small frequency (or equivalent redshift) intervals.

The Schrödinger-Poisson method for Large-Scale Structure

We study the Schrödinger-Poisson (SP) method in the context of cosmological large-scale structure formation in an expanding background. In the limit 0ℏ → , the SP technique can be viewed as an effective method to sample the phase space distribution of cold dark matter that remains valid on non-linear scales. We present results for the 2D and 3D matter correlation function and power spectrum at length scales corresponding to the baryon acoustic oscillation (BAO) peak. We discuss systematic effects of the SP method applied to cold dark matter and explore how they depend on the simulation parameters. In particular, we identify a combination of simulation parameters that controls the scale-independent loss of power observed at low redshifts, and discuss the scale relevant to this effect.

Compressive Joint Angular and Frequency Spectrum Sensing Based on MUSIC Spectrum Reconstruction

This paper aims to introduce a new tool for wideband spectrum sensing in cognitive radio applications based on the multiple signal classification (MUSIC) two-dimensional (2D)-pseudospectrum, which is able to locate the active sources [licenced users (LUs)] or other secondary users in both the angular and frequency domains. The proposed tool consists of two-stages compression and reconstruction procedures. The spatial-domain compression is introduced by activating only some of the antennas in the underlying uniform linear array (ULA) leading to a non-ULA of active antennas. The time-domain compression is introduced by allowing the receivers connected to every active antenna to sample the analog signals below the Nyquist rate. This compression aims to alleviate the burden on the analog-to-digital converter especially when a broad frequency band has to be sensed. The simulation study shows that the constructed MUSIC 2D-pseudospectrum can be used to locate the direction-of-arrival (DoA) and the frequency band of the LUs with a much better resolution than the one that can be afforded by the already proposed 2D power spectrum. Note that the number of sources is beyond that of active antennas and the sampling rate in each receiver is only 17 / 42 times the Nyquist rate. Moreover, the evaluation on the root mean square error in terms of DoA estimation indicates that the proposed MUSIC 2D-pseudospectrum construction approach outperforms the already proposed 2D power spectrum reconstruction method. In fact, the performance of the proposed approach based on the sub-Nyquist-rate samples is only slightly below the approach that is based on the pseudospectrum constructed from Nyquist-rate samples. Furthermore, the evaluation on the detection performance of the spectrum sensing tool based on the MUSIC 2D-pseudospectrum underlines the promising benefits of the proposed approach.

Matter power spectrum: from Ly α forest to CMB scales

ABSTRACT We present a new compilation of inferences of the linear 3D matter power spectrum at redshift $z\, {=}\, 0$ from a variety of probes spanning several orders of magnitude in physical scale and in cosmic history. We develop a new lower noise method for performing this inference from the latest Ly α forest 1D power spectrum data. We also include cosmic microwave background (CMB) temperature and polarization power spectra and lensing reconstruction data, the cosmic shear two-point correlation function, and the clustering of luminous red galaxies. We provide a Dockerized Jupyter notebook housing the fairly complex dependences for producing the plot of these data, with the hope that groups in the future can help add to it. Overall, we find qualitative agreement between the independent measurements considered here and the standard ΛCDM cosmological model fit to the Planck data.

Structural framework of the Wagad uplift and adjoining regions, Kutch rift basin, India, from aeromagnetic data

The Kutch sedimentary basin formed during the Late Triassic breakup of Gondwanaland is characterised by horst and graben structures consisting of several east–west trending uplifts surrounded by low-lying plains. The eastern part of the basin has a diverse landscape comprising the Wagad uplift, Banni plain, Island Belt uplift and the Rann of Kutch. This area is bounded by major faults like the South Wagad Fault (SWF), Gedi fault and the Island Belt Fault. The lineaments/faults present in the region at different depth levels and the propagation of these features through the different sedimentary layers are studied using the semi-detailed aeromagnetic data collected over the basin. The aeromagnetic anomaly map depicts several major E–W, NE–SW and NW–SE oriented lineaments/faults, which probably represent structural trends associated with different stages of evolution of this rift basin. Power spectral analysis of the differential reduced to pole magnetic data indicates the presence of four magnetic interfaces. The slopes identified from the 1D power spectra were used for designing matched bandpass filters for isolating and enhancing the magnetic signatures present within those interfaces. Different edge detection techniques were used to delineate the magnetic contacts/faults/lineaments present in those interfaces. In addition, we have computed the radially averaged power spectrum of 121 subset grids each with a dimension of 20 km × 20 km from which three magnetic interfaces were delineated and compared with the stratigraphic sequence of the Wagad uplift and adjoining regions. A major NE–SW fault is delineated from this analysis and suggests that this fault has depth persistence as it dislocates the different magnetic interfaces. Integration with stratigraphic data suggests that this fault was formed prior to the deposition of Miocene Kharinadi formation. We have interpreted that this fault, forming the eastern limit of the Banni basin, might have formed during the passage of the Indian plate over the Reunion hotspot. Based on the results of the aeromagnetic data analysis and other published data, we propose a generalised evolutionary model for the study region.

The one-dimensional power spectrum from the SDSS DR14 Lyα forests

We present a measurement of the 1D Lyα forest flux power spectrum, using the complete Baryon Oscillation Spectroscopic Survey (BOSS) and first extended-BOSS (eBOSS) quasars at zqso > 2.1, corresponding to the fourteenth data release (DR14) of the Sloan Digital Sky Survey (SDSS). Our results cover thirteen bins in redshift from zLyα = 2.2 to 4.6, and scales up to k = 0.02 (km/s)−1. From a parent sample of 180,413 visually inspected spectra, we selected the 43,751 quasars with the best quality; this data set improves the previous result from the ninth data release (DR9), both in statistical precision (achieving a reduction by a factor of two) and in redshift coverage. We also present a thorough investigation of identified sources of systematic uncertainties that affect the measurement. The resulting 1D power spectrum of this work is in excellent agreement with the one from the BOSS DR9 data.

Impact of inhomogeneous reionization on the Lyman-α forest

The \lya forest at high redshifts is a powerful probe of reionization. Modeling and observing this imprint comes with significant technical challenges: inhomogeneous reionization must be taken into account while simultaneously being able to resolve the web-like small-scale structure prior to reionization. In this work we quantify the impact of inhomogeneous reionization on the \lya forest at lower redshifts ($2 < z < 4$), where upcoming surveys such as DESI will enable precision measurements of the flux power spectrum. We use both small box simulations capable of handling the small-scale structure of the \lya forest and semi-numerical large box simulations capable of representing the effects of inhomogeneous reionization. We find that inhomogeneous reionization could produce a measurable effect on the \lya forest power spectrum. The deviation in the 3D power spectrum at $z_{\rm obs} = 4$ and $k = 0.14 \ \rm{Mpc}^{-1}$ ranges from $19 - 36\%$, with a larger effect for later reionization. The corrections decrease to $2.0 - 4.1\%$ by $z_{\rm obs} = 2$. The impact on the 1D power spectrum is smaller, and ranges from $3.3 - 6.5\%$ at $z_{\rm obs}=4$ to $0.35 - 0.75\%$ at $z_{\rm obs}=2$, values which are comparable to the statistical uncertainties in current and upcoming surveys. Furthermore, we study how can this systematic be constrained with the help of the quadrupole of the 21 cm power spectrum.

Cosmological Hydrodynamic Simulations with Suppressed Variance in the Lyα Forest Power Spectrum

Abstract We test a method to reduce unwanted sample variance when predicting Lyα forest power spectra from cosmological hydrodynamical simulations. Sample variance arises due to sparse sampling of modes on large scales and propagates to small scales through nonlinear gravitational evolution. To tackle this, we generate initial conditions in which the density perturbation amplitudes are fixed to the ensemble average power spectrum—and are generated in pairs with exactly opposite phases. We run 50 such simulations (25 pairs) and compare their performance against 50 standard simulations by measuring the Lyα 1D and 3D power spectra at redshifts z = 2, 3, and 4. Both ensembles use periodic boxes of 40 h − 1 Mpc containing 5123 particles each of dark matter and gas. As a typical example of improvement, for wavenumbers k = 0.25 h Mpc − 1 at z = 3, we find estimates of the 1D and 3D power spectra converge 34 and 12 times faster in a paired–fixed ensemble compared with a standard ensemble. We conclude that, by reducing the computational time required to achieve fixed accuracy on predicted power spectra, the method frees up resources for exploration of varying thermal and cosmological parameters—ultimately allowing the improved precision and accuracy of statistical inference.

Image Evolution Using 2D Power Spectra

Procedurally generated images and textures have been widely explored in evolutionary art. One active research direction in the field is the discovery of suitable heuristics for measuring perceived characteristics of evolved images. This is important in order to help influence the nature of evolved images and thereby evolve more meaningful and pleasing art. In this regard, particular challenges exist for quantifying aspects of style and shape. In an attempt to bridge the divide between computer vision and cognitive perception, we propose the use of measures related to image spatial frequencies. Based on existing research that uses power spectral density of spatial frequencies as an effective metric for image classification and retrieval, we posit that Fourier decomposition can be effective for guiding image evolution. We refine fitness measures based on Fourier analysis and spatial frequency and apply them within a genetic programming environment for image synthesis. We implement fitness strategies using 2D Fourier power spectra and phase, with the goal of evolving images that share spectral properties of supplied target images. Adaptations and extensions of the fitness strategies are considered for their utility in art systems. Experiments were conducted using a variety of greyscale and colour target images, spatial fitness criteria, and procedural texture languages. Results were promising, in that some target images were trivially evolved, while others were more challenging to characterize. We also observed that some evolved images which we found discordant and “uncomfortable” show a previously identified spectral phenomenon. Future research should further investigate this result, as it could extend the use of 2D power spectra in fitness evaluations to promote new aesthetic properties.

The Effect of Baseline Layouts on the Epoch of Reionization Foreground Wedge: A Semianalytical Approach

Abstract The 2D power spectrum is a cornerstone of the modern toolkit for analysis of the low-frequency radio interferometric observations of the 21 cm signal arising from the early universe. Its familiar form disentangles a great deal of systematic information concerning both the sky and telescope and is displayed as a foreground-dominated “brick” and “wedge” on large line-of-sight scales and a complementary “window” on smaller scales. This paper builds on many previous works in the literature that seek to elucidate the varied instrumental and foreground factors that contribute to these familiar structures in the 2D power spectrum. In particular, we consider the effects of uv sampling on the emergence of the wedge. Our results verify the expectation that arbitrarily dense instrument layouts in principle restore the missing information that leads to mode mixing and can therefore mitigate the wedge feature. We derive rule-of-thumb estimates for the required baseline density for complete wedge mitigation, showing that these will be unachievable in practice. We also discuss the optimal shape of the layout, showing that logarithmic regularity in the radial separation of baselines is favorable. While complete suppression of foreground leakage into the wedge is practically unachievable, we find that designing layouts that promote radial density and regularity is able to reduce the amplitude of foreground power by one to three orders of magnitude.