Phase-Resolved Mapmaking

• Phase-resolved mapmaking is a reconstruction approach that retains phase variables (e.g., orbital, Fourier, pulsar) to recover both static and dynamic features in observational data.
• It enhances signal extraction by explicitly modeling instrumental responses and time-dependent systematics, leading to improved localization and reduced degeneracies in inverse problems.
• Applications range from lunar interferometry and gravitational-wave mapping to exoplanet cartography and CMB polarisation, each using customized basis functions and regularization methods.

Phase-resolved mapmaking is a class of reconstruction methods in which the map is inferred while explicitly retaining or exploiting a phase variable: orbital phase, Fourier phase, scan angle, pulsar phase, or mechanical vibration phase. In lunar-orbit radio interferometry it denotes inversion with a time-dependent lunar mask and, when variability is expected, separate orbital-phase maps (Huang et al., 2018). In gravitational-wave astronomy it denotes phase-coherent reconstruction of the full complex sky field, preserving amplitude, phase, and polarisation rather than collapsing the data to power (Curyło et al., 21 Apr 2026, Romano et al., 2015). In exoplanet cartography, CMB analyses, gamma-ray phase studies, and holographic vibrometry it denotes inversion conditioned on orbital, oscillation, source, or stroboscopic phase (Rauscher et al., 2018, Guan, 12 Nov 2025, Lange et al., 6 Nov 2025, Verrier et al., 2012).

1. Conceptual scope and terminology

Two meanings recur across the literature. In one, phase is a conditioning variable: data are partitioned or folded by a known periodic phase, and separate maps are reconstructed for phase bins or harmonic phase modes. In the other, phase is part of the object being mapped: the unknown field is complex, and the reconstruction preserves amplitude and phase in every pixel or mode. The two meanings are mathematically close, but they answer different questions. Phase-conditioned maps ask how the sky or source varies with phase; phase-coherent maps ask what complex field produced the data.

Setting	Phase variable	Reconstructed object
Lunar interferometer	Orbital phase and time-dependent occultation	Static sky or phase-resolved sky maps
PTA / ground-based GW mapping	Fourier phase and detector-response phase	Complex polarisation maps or grad/curl modes
Exoplanet, CMB, Fermi-LAT, vibrometry	Orbital phase, trial oscillation phase, pulsar/binary phase, mechanical phase	Brightness maps, phase-frequency maps, per-bin counts/TS maps, vibration amplitude/phase maps

A further distinction concerns where the phase enters the inference. In wide-field interferometry and GW mapping, the phase is embedded directly in the measurement operator. In exoplanet, CMB phase-folding, Fermi-LAT, and vibrometry, the data are first reorganized by phase and then inverted. Scan-modulated CMB polarisation mapmaking occupies an intermediate position: the relevant phase is the instrument rotation or HWP angle, and the mapmaker separates sky and systematics by their spin-dependent phase modulation (Wallis et al., 2015, Collaboration et al., 2013).

2. Common inverse-problem structure

Despite the variety of applications, phase-resolved mapmaking is typically cast as a linear inverse problem with an operator that encodes geometry, modulation, beam response, masking, and noise weighting. In interferometric 21 cm mapmaking this appears as

$\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{n},$

with the dirty-map estimator

$$\hat{\mathbf{x}}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{y},$$

and point-spread matrix

$$\mathbf{P}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{A}.$$

The matrix $\mathbf{P}$ carries the full complex instrument response, aggregated across baselines, times, and frequencies, and the resulting covariance propagates directly into quadratic power-spectrum estimation (Dillon et al., 2014).

The same algebra appears in full-sky lunar interferometric imaging. There the visibility equation explicitly includes non-coplanar baselines, the primary beam, and the time-dependent lunar occultation mask. After discretization on the sphere, the mapmaker solves

$v=P\,x+n,$

or, with regularization,

$(P^\top N^{-1}P+R)\,x=P^\top N^{-1}v.$

The operator rows encode the gain $A(\Omega,\nu)$, lunar mask $M(\Omega,t)$, and wide-field Fourier kernel with the $w$-term. Pixel and harmonic discretizations are both used; because short dipoles have smooth wide beams, the spherical-harmonic operator is localized at low $\ell$ and therefore sparse (Huang et al., 2018).

Phase-coherent PTA mapping uses the same generalized least-squares logic. For a single frequency bin,

$$\hat{\mathbf{x}}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{y},$$0

with maximum-likelihood estimate

$$\hat{\mathbf{x}}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{y},$$1

The paper further distinguishes a dirty map, a Fisher matrix, a clean map, and a per-pixel radiometer estimator. Rank deficiency is intrinsic because $$\hat{\mathbf{x}}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{y},$$2, so truncated SVD is used to construct a pseudoinverse, retaining a fixed fraction of well-constrained modes in the tests (Curyło et al., 21 Apr 2026).

This common structure suggests that the essential design choice is not the existence of a linear estimator but the construction of the phase-aware operator, the selection of basis functions, and the regularization of weakly constrained modes.

3. Phase-conditioned and phase-folded reconstructions

When the relevant physics is periodic or quasi-periodic, phase-resolved mapmaking often reorganizes the time stream by phase before inversion. In the lunar-orbit interferometer this appears as phase-resolved sky reconstruction when variability is expected. Data are partitioned into phase bins $$\hat{\mathbf{x}}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{y},$$3, and a separate map $$\hat{\mathbf{x}}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{y},$$4 is assigned to each bin through a block-diagonal system. A coupling prior,

$$\hat{\mathbf{x}}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{y},$$5

can enforce smoothness in angle and coherence across phase. The static-sky alternative simply stacks all visibilities with their instantaneous masks into one operator and was shown to remain unbiased even though the blockage fraction varies from near zero at the ecliptic poles to $$\hat{\mathbf{x}}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{y},$$6 near the ecliptic plane (Huang et al., 2018).

In exoplanet mapping, the phase variable is orbital phase and secondary-eclipse phase. The forward model is linear,

$$\hat{\mathbf{x}}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{y},$$7

but eclipse lightcurves do not preserve the orthogonality of spherical harmonics in the time domain. The solution in "A More Informative Map: Inverting Thermal Orbital Phase and Eclipse Lightcurves of Exoplanets" is to compute basis lightcurves from spherical harmonics up to $$\hat{\mathbf{x}}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{y},$$8, form a mean-centered matrix $$\hat{\mathbf{x}}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{y},$$9, diagonalize its covariance, and construct orthogonal eigencurves $$\mathbf{P}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{A}.$$0. For the Spitzer $$\mathbf{P}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{A}.$$1 observations of HD 189733b, the sampling yielded $$\mathbf{P}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{A}.$$2 and eight non-zero eigenvalues; model selection favored the first two eigencurves, with preferred fit $$\mathbf{P}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{A}.$$3, $$\mathbf{P}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{A}.$$4, and $$\mathbf{P}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{A}.$$5. The retrieved equatorial hotspot lay $$\mathbf{P}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{A}.$$6 east of the substellar point, and the maximum-to-minimum dayside flux ratio was $$\mathbf{P}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{A}.$$7 (Rauscher et al., 2018).

Phase-folded CMB mapmaking pushes the same idea to much higher modulation frequencies. For a trial frequency $$\mathbf{P}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{A}.$$8, the timestream is binned by the signal phase rather than by clock time. With phase-bin operators $$\mathbf{P}=\mathbf{D}\mathbf{A}^\dagger \mathbf{N}^{-1}\mathbf{A}.$$9,

$\mathbf{P}$0

and the discrete Fourier transform across phase bins produces frequency-mode maps,

$\mathbf{P}$1

The $\mathbf{P}$2 mode contains the static sky, while $\mathbf{P}$3 modes contain instrumental noise plus the targeted time-varying signal. The paper emphasizes that these $\mathbf{P}$4 modes are cosmic-variance-free and that phase-folding extends sensitivity from the $\mathbf{P}$5 Hz regime of time-division maps up to the detector Nyquist frequency, typically $\mathbf{P}$6 Hz (Guan, 12 Nov 2025).

Fermi-LAT phase-resolved analysis adopts a more classical phase-binning architecture. FermiPhased automates per-bin event selection, counts cubes, livetime cubes, and likelihood fits, and supports standard equal-phase bins, adaptive fixed-counts bins, and joint phase-resolved analysis across multiple time ranges. The adaptive mode chooses variable-width phase windows so each bin contains approximately a target number of photons, giving more uniform S/N and more homogeneous map quality, while the joint mode combines likelihood contributions across bins and epochs with shared parameters where appropriate (Lange et al., 6 Nov 2025).

4. Phase-coherent and complex-field mapmaking

In gravitational-wave mapping, phase-resolved mapmaking usually means phase coherence rather than phase binning. The mapped quantity is the complex sky field itself. In the PTA framework MIMOSIS, the GW perturbation is decomposed into $\mathbf{P}$7 and $\mathbf{P}$8 polarisations, and the fundamental sky field is $\mathbf{P}$9. The reconstruction returns four complex-valued maps per frequency bin: $v=P\,x+n,$0, $v=P\,x+n,$1, $v=P\,x+n,$2, and $v=P\,x+n,$3. Retaining phase and polarisation enables coherent localization, preserves polarisation content, and allows coherent combination across frequency bins for continuous-wave sources. The approach is explicitly contrasted with cross-correlation power mapping, which integrates out phase and polarisation (Curyło et al., 21 Apr 2026).

The same conceptual distinction was developed earlier for ground-based laser interferometers. There the complex sky can be represented either in the $v=P\,x+n,$4 basis or in tensor gradient/curl harmonics. A key result is that, unlike PTAs, a network of ground-based interferometers is sensitive to both gradient and curl components. Earth’s rotation and orbit synthesize many spatially separated, differently oriented virtual detectors, so the response functions acquire the phase delays needed to recover both components. In the static single-interferometer, origin-centered limit, by contrast, only the $v=P\,x+n,$5 gradient response is nonzero and the curl response vanishes (Romano et al., 2015).

Phase-coherent mapmaking also changes the role of deconvolution and uncertainty quantification. In the PTA setting, clean maps invert the full Fisher matrix and suppress polarisation leakage, while radiometer maps ignore inter-pixel covariance and remain optimal point estimates for isolated pixels. The two products serve different purposes. In the MPTA-like best-covered direction, the total-power radiometer map reached SNR $v=P\,x+n,$6 but spread across the sky, while the clean map reached SNR $v=P\,x+n,$7 and produced a compact single-pixel hotspot at the true location. In that same regime the radiometer recovery fraction for $v=P\,x+n,$8 at the true pixel was $v=P\,x+n,$9. In poorly covered directions, clean-map SNR dropped to $(P^\top N^{-1}P+R)\,x=P^\top N^{-1}v.$0, detection regions broadened to up to $(P^\top N^{-1}P+R)\,x=P^\top N^{-1}v.$1 pixels, and amplitude recovery became marginal when the clean-map excess over noise was $(P^\top N^{-1}P+R)\,x=P^\top N^{-1}v.$2 (Curyło et al., 21 Apr 2026).

A common misconception is that any sky map that preserves angular information is already phase-resolved. The GW literature makes the opposite point: power maps are not phase-coherent maps. Once amplitude, phase, and polarisation are discarded, one no longer has a minimally processed representation from which stochastic-background characterization, anisotropy searches, and individual-source identification can all be derived within a single framework (Curyło et al., 21 Apr 2026, Romano et al., 2015).

5. Scan-phase, modulation phase, and instrumental systematics

A separate branch of phase-resolved mapmaking uses controlled instrumental phase modulation to disentangle sky signal from calibration drifts or leakage. Planck HFI is a prominent example. The satellite scans in stable pointing periods known as rings, and the mapmaking pipeline constructs HPRs, solves for ring offsets, and calibrates the low-frequency channels against the kinematic dipole. In destriping notation,

$(P^\top N^{-1}P+R)\,x=P^\top N^{-1}v.$3

while the HFI-specific model writes

$(P^\top N^{-1}P+R)\,x=P^\top N^{-1}v.$4

Calibration and mapmaking are tightly coupled because the orbital dipole is time-variable and low-frequency noise is modeled as ring baselines. The 2013 release used solar-dipole absolute calibration and an iterative correction, bogopix, for apparent ring-by-ring gain variations; reported calibration uncertainties ranged from a few $(P^\top N^{-1}P+R)\,x=P^\top N^{-1}v.$5 to several per cents from 100 to 857 GHz (Collaboration et al., 2013).

For CMB polarisation experiments, phase resolution can be used to remove temperature-to-polarisation leakage directly in the mapmaker. Wallis et al. exploit the fact that different leakages transform with different spin under instrument rotation: differential gain is spin-0, differential pointing is spin-1, and differential ellipticity is spin-2. With a HWP, the differenced TOD can be written as

$(P^\top N^{-1}P+R)\,x=P^\top N^{-1}v.$6

Two algorithms are then proposed: a Fourier method for extensive crossing-angle coverage and a Legendre-plus-Fourier method for limited coverage. In EPIC-like simulations with a HWP, simultaneous removal of differential gain, pointing, and ellipticity recovered unbiased $(P^\top N^{-1}P+R)\,x=P^\top N^{-1}v.$7-mode spectra with a noise power penalty of $(P^\top N^{-1}P+R)\,x=P^\top N^{-1}v.$8. In LSPE-like simulations, modeling the leakage as a linear function of crossing angle reduced the $(P^\top N^{-1}P+R)\,x=P^\top N^{-1}v.$9-mode bias by $A(\Omega,\nu)$0 orders of magnitude, to below $A(\Omega,\nu)$1 of the statistical error bars, again with a noise power penalty of $A(\Omega,\nu)$2 (Wallis et al., 2015).

Phase-resolved heterodyne holographic vibrometry uses yet another modulation architecture. A strobe local oscillator is frequency-shifted to heterodyne the first optical sideband of a vibrating object and amplitude-modulated to freeze successive mechanical phase states. The reconstructed holograms are demodulated twice: first across camera frames to extract a frozen-state phasor, then across the sequence of stroboscopically sampled states to recover the spatially resolved vibration phase $A(\Omega,\nu)$3 and amplitude $A(\Omega,\nu)$4. The method is explicitly designed to alleviate phase retrieval and yields full-field maps of sinusoidal out-of-plane motion (Verrier et al., 2012).

6. Conditioning, degeneracies, and practical limits

Phase resolution improves identifiability only when the measurement operator has sufficient diversity. The lunar-orbit interferometer provides a clear example. If all baselines lie in a plane, the sky suffers a mirror symmetry between $A(\Omega,\nu)$5 and $A(\Omega,\nu)$6, and the lunar mask alone does not remove the degeneracy. Orbital-plane precession naturally generates nonzero $A(\Omega,\nu)$7 components and spreads the baselines in 3D; in simulations, even a modest fraction of out-of-plane baselines, $A(\Omega,\nu)$8, visibly suppressed the mirror image, and a fully 3D baseline cloud removed it entirely. The same paper shows that inhomogeneous $A(\Omega,\nu)$9 coverage distorts PSF cores and increases sidelobes, although major structures remain recoverable even with missing short baselines (Huang et al., 2018).

Regularization is therefore not an implementation detail but a structural feature of phase-resolved mapmaking. In lunar imaging, SVD-based pseudoinversion retained the top modes accounting for $M(\Omega,t)$0 of cumulative singular-value power in the demonstration problem. In PTA mapping, truncated SVD retained 30% of eigenmodes in the tests, because the Fisher matrix is rank-deficient. The benefit is stability and leakage suppression; the cost is bias in poorly measured extended structures. The PTA analysis states this directly: clean maps decorrelate pixels and polarisations, but radiometer amplitudes are robust only where clean-map SNR significantly exceeds noise (Huang et al., 2018, Curyło et al., 21 Apr 2026).

Another recurrent misconception is that an orthogonal basis on the sphere automatically yields an orthogonal basis in the data domain. The exoplanet literature shows that this is true for phase-only thermal mapping with full-orbit sampling, where spherical-harmonic lightcurves are orthogonal sinusoids, but false for eclipse mapping, where the occultation geometry breaks temporal orthogonality. PCA eigencurves are introduced precisely to restore orthogonality in the sampled lightcurve space and to rank the modes the data can support (Rauscher et al., 2018).

The gains of phase resolution can also be highly specific to the observable. In phase-folded CMB searches, the statement that $M(\Omega,t)$1 modes are cosmic-variance-free relies on a static sky and a targeted periodic signal; it is not a universal property of all phase-resolved maps. In GW mapping, ground-based interferometers recover curl modes because detector translations and time-varying antenna patterns provide the required phase structure, whereas PTAs remain insensitive to half of the gravitational-wave sky in the curl sector. In gamma-ray phase studies, equal-phase bins may produce low-count maps and noisy TS surfaces, which is why adaptive fixed-count bins are introduced (Guan, 12 Nov 2025, Romano et al., 2015, Lange et al., 6 Nov 2025).

Taken together, these results define phase-resolved mapmaking not as a single algorithm but as a methodological family. Its unifying principle is the explicit use of phase information that standard averaging, cross-correlation, or power-only mapping would discard. The practical consequences are improved localization, access to otherwise degenerate modes, direct treatment of time-variable occultation or modulation, and a more transparent propagation of systematics and covariance through the mapmaker. The corresponding costs are familiar: larger parameter spaces, stronger dependence on coverage and conditioning, and a central role for regularization, weighting, and null tests (Huang et al., 2018, Curyło et al., 21 Apr 2026, Wallis et al., 2015).