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REVIEW 3 major objections 5 minor 39 references

Accurate Shear Estimation with Fourth-Order Moments

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper shows that fourth-order galaxy shape moments, corrected analytically, reach shear bias below 0.003 and cut shape noise by about 35% when combined with the standard estimator.

desk verdict A solid extension of FPFS to fourth-order moments with credible bias tests; the headline shape-noise gain is real but quoted inconsistently and measured in-sample. read the letter →

arxiv 2411.13648 v1 pith:VLSXOPZ7 submitted 2024-11-20 astro-ph.CO

Andy Park , Xiangchong Li , Rachel Mandelbaum This is my paper · ORCID
classification astro-ph.CO
keywords weakgravitationallensingcosmicshearshapeletmomentsestimationanalyticcalibrationnoisebiasshapegalaxyblending
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Weak lensing shear is usually measured from the second-order moments of galaxy light, but this paper asks whether fourth-order moments contain additional, usable information about the shear. Using the FPFS shapelet framework and the AnaCal analytic calibration, the authors construct a fourth-order ellipticity from the $M_{42}$ moment and correct its detection, selection, and noise biases without external simulation-based calibration. In HSC- and LSST-like image simulations, the corrected estimator has multiplicative bias $|m|$ below $3\times10^{-3}$ at the $3\sigma$ level for both isolated and blended galaxies. When combined with the second-order estimator, it reduces per-galaxy shape noise by about 35% for isolated galaxies, roughly equivalent to the precision gain from a 70% larger survey area. If the claim holds, higher-order moments become a practical resource for tightening cosmic shear constraints while staying inside the LSST systematic error budget.

What carries the argument

The central object is the fourth-order spin-2 polar shapelet moment $M_{42}$ and its linear shear response, which couples $M_{42}$ to $M_{20}$, $M_{60}$, and $M_{64}$ (Eqs. 17–18). Dividing by $M_{00}+C^{(4)}$ turns this mode into a dimensionless fourth-order ellipticity, and the smooth selection weights of the FPFS/AnaCal framework give its detection and selection responses analytically. Noise bias is removed by the renoising procedure: an extra noise layer with the same statistics, rotated by 90 degrees, is added to the image so that the spin-2 anisotropies cancel after PSF deconvolution, making the estimator noise-bias-free to second order in shear without computing noisy high-order derivatives. Finally, the second- and fourth-order estimators are combined with the variance-minimizing weight $\mu$ derived from their covariance, so that independent information from each order is used where it is most constraining.

What would settle it

A decisive test would be to run the fourth-order estimator on simulated images whose noise has anisotropic pixel-to-pixel correlations and check whether the residual multiplicative and additive bias stays within $|m|<3\times10^{-3}$ after renoising; if the added rotated noise layer leaves a nonzero spin-2 correlation after PSF deconvolution, the cancellation assumed by Eq. (20) is incomplete and the analytic correction is biased for higher-order moments.

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Extended reading notes

Core claim

The central claim is that fourth-order polar shapelet moments carry shear information that is partly independent of the second-order moments, and that this information can be extracted at sub-percent accuracy through the analytic AnaCal correction scheme. The shear estimator built from the normalized fourth-order spin-2 moment $M_{42}/(M_{00}+C^{(4)})$ has multiplicative bias $|m| < 3\times10^{-3}$ with $99.7\%$ confidence after analytic correction for detection, selection, and noise bias, in isolated and blended galaxy simulations under HSC and LSST observing conditions. Combining this fourth-order estimator with the existing second-order estimator through a variance-minimizing weight $\mu$ reduces per-galaxy shape noise by roughly 35% for isolated galaxies, matching the precision gain of about a 70% larger survey area, while for blended galaxies the gain is only about 2% because the fourth-order moment amplifies the effects of blending and image noise. The paper therefore claims that the two estimators are complementary rather than redundant in high-SNR, isolated, or space-based regimes.

Load-bearing premise

The result rests on the renoising noise-bias correction: the added 90-degree-rotated noise layer must cancel the relevant spin-2 anisotropies after PSF deconvolution for fourth-order moments; if that cancellation is incomplete, the analytic corrections are biased and the $|m|<3\times10^{-3}$ claim fails.

Editorial extensions

If this is right

  • For isolated, high-SNR galaxy samples, combining second- and fourth-order estimators reduces shape noise by about 35%, equivalent to increasing survey area by roughly 70% at fixed statistical precision.
  • The fourth-order estimator alone satisfies the LSST ten-year multiplicative-bias requirement $|m|<3\times10^{-3}$ in HSC-like and LSST-like simulations for both isolated and blended galaxies.
  • Because the two estimators weight different spatial scales, their combination provides an internal cross-check that can expose PSF leakage and modeling errors that affect one order more than the other.
  • In blended ground-based images the fourth-order estimator contributes only about 2% to effective number density, concentrating the practical gain in high-SNR, isolated, or space-based observations.
  • The same analytic calibration machinery can be applied to further moment orders or to redshift-dependent shear without rerunning external image-calibration simulations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if the ~35% isolated-galaxy gain carries over to space-based surveys where blending is negligible, the fourth-order combination could raise effective number density by tens of percent, a testable prediction for Euclid- or Roman-like simulations.
  • Editorial inference: the optimal weighting parameters $C^{(2)}=7$ and $C^{(4)}=10$ were chosen on the same simulations used for validation, so an out-of-sample optimization would measure how much of the reported precision gain is tuning rather than information.
  • Editorial inference: the fourth-order moment's distinct sensitivity to small radii could make the ratio of second- to fourth-order shear estimates a practical diagnostic for PSF modeling error; the paper recommends but does not perform this test.
  • Editorial inference: a direct extension would inject correlated anisotropic noise into the renoising step and verify that the 90-degree rotation still cancels spin-2 correlations for fourth-order modes; this would distinguish the method's generic validity from the homogeneous-noise approximation used here.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper extends the FPFS/AnaCal shear estimator to use fourth-order polar shapelet moments (M42) to define a spin-2 ellipticity, with analytic corrections for detection/selection bias and noise bias via the renoising method of Li et al. (2024a). Using HSC-like and LSST-like image simulations of isolated and blended galaxies, the authors report multiplicative shear bias |m| below 3e-3 at the claimed 99.7% confidence and additive biases consistent with zero, while combining the second- and fourth-order estimators reduces shape noise by about 35% for isolated galaxies and by only about 2% in effective number density for blended galaxies.

Significance. If the claims hold, the paper demonstrates that fourth-order moments carry usable complementary shear information that can be calibrated analytically, improving precision by roughly 30-35% in isolated-galaxy samples without exceeding LSST multiplicative-bias requirements. The work is backed by extensive simulation tests (4000 isolated and 5000 blended subfields, ring tests, two survey setups) and public code, which are notable strengths. The main reservations are that the renoising noise-bias correction is applied to higher-order moments without a dedicated derivation or isolated validation, and that the headline shape-noise improvement is measured after in-sample parameter optimization, so the reported gain may be optimistic.

major comments (3)
  1. [Section 2.5, Eq. (20)] The renoising noise-bias correction is adopted by reference to Li et al. (2024a), but the paper does not derive or validate it for the fourth-order ellipticity e = M42/(M00+C(4)). The shear response of M42 couples to M20, M60, and M64 (Eqs. 17-18), and the effect of the 90-degree-rotated added noise layer on the covariance of these modes after PSF deconvolution is not demonstrated in the manuscript. Because the headline |m|<3e-3 result in Sections 4.1.4 and 4.2.2 is obtained with this correction, the end-to-end simulations cannot separate a residual renoising bias from other modeling choices. Please provide a dedicated test isolating the renoising correction for fourth-order moments, for example by comparing Eq. (20) with an independent noise-bias correction (Hessian-based AnaCal or a numerical method) on identical simulations, or an explicit analytic check of the spin-2 cancellation for the M42 response.
  2. [Section 4.1.1, Fig. 3 and Eqs. (22)-(24)] The values C(2), C(4), and the combination weight mu are selected by minimizing the measured variance on the same 100-subfield simulations that are then used to report the shape-noise reduction. This in-sample optimization makes the reported 30-35% improvement an optimistic estimate, and no cross-validation or bootstrap uncertainty on the gain is given. Please report the improvement with a split-sample or cross-validated procedure, or at minimum provide an uncertainty on the variance reduction.
  3. [Section 4.1.1 and Abstract/Conclusion] The paper quotes a "~30%" reduction in Section 4.1.1 and Section 4.1.2, while the abstract and conclusion quote "~35%". If these numbers refer to different sample definitions or different comparison baselines (e.g., with or without detection/selection cuts, or versus reGauss), that should be stated explicitly; as written, the headline improvement is internally inconsistent.
minor comments (5)
  1. [Section 1] The phrase "the fourth-order shear estimator, which is independent of the second-order" overstates the case; Eq. (23) explicitly includes the covariance rho between the two estimators, so they are correlated. Suggest replacing "independent" with "complementary" or "partially independent".
  2. [Section 2.4] The sentence beginning "In this work, use the same detection and selection" is missing the subject "we"; please correct the grammar.
  3. [Section 4.1.1] The claim that "A 30 per cent reduction in shape noise is equivalent to the increase in sample size that would be achieved by expanding the survey area by 70 per cent" appears arithmetically inconsistent with the usual scaling sigma proportional to 1/sqrt(area); please check whether the reduction refers to sigma or to the variance and correct the area-equivalence statement.
  4. [Section 4.2.1] The sentence "The n_eff ~ 15 arcmin^-2 for HSC setup is 35% smaller than the n_eff ~ 20 arcmin^-2" is inconsistent: 15 is 25% smaller than 20. Please correct the percentage or the quoted values.
  5. [Section 4.1.4] The statement that |m| is below 3e-3 at the 99.7% confidence interval would be clearer if the authors specified whether this is a one-sided upper limit or a two-sided interval, and how the confidence bound is derived from the displayed 1-sigma and 3-sigma error bars.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central bias claim is an externally tested simulation result, and the renoising correction is imported from prior work with independent validation.

full rationale

The paper's central accuracy claim (|m| < 3e-3) is not derived from the estimator's parameters by construction; it is measured on 4000 subfields of simulations with known input shear using equations (35)-(36), so it is an external falsifiable test. The fourth-order shear response (Eqs. 17-18) follows from the analytic shapelet coupling, and the detection/selection corrections are analytic derivatives (Sections 2.1-2.3), not fits to the bias. The renoising noise-bias correction (Section 2.5, Eq. 20) is imported from Li et al. (2024a) by self-citation, but that prior work contains an analytical proof and independent simulation tests; per the rules, this citation is real evidence rather than circular. The remaining in-sample element is the choice of C(4) and the combination weight mu on the same 100 subfields used to report the ~30-35% shape-noise reduction (Section 4.1.1, Eqs. 22-24). This is a statistical optimism/overfitting concern, not a circular reduction: the reported reduction is data-dependent and not forced by the fitting equations. The extension of the renoising correction to fourth-order moments is validated only end-to-end (Sections 4.1.4, 4.2.2), which is a limitation for isolating residual noise bias, but it is a correctness risk, not circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The free parameters C(2), C(4), and μ are tuned on the same 100-subfield simulations used to quote the shape-noise reduction; they are not determined by first principles. The axioms are standard shapelet mathematics plus domain assumptions about galaxy ensembles, noise fields, and the imported renoising correction. No new physical entities are introduced.

free parameters (3)
  • C(4) (fourth-order weighting parameter) = 10
    Chosen by minimizing per-galaxy shear uncertainty in Fig. 3 on isolated-galaxy simulations; it weights the fourth-order ellipticity denominator in Eq. (19).
  • C(2) (second-order weighting parameter) = 7
    Same variance-minimization procedure on the same simulations (Fig. 3), following Li and Mandelbaum (2023).
  • Combination weight μ = computed from measured variances and covariance per survey and shear component
    Eq. (22)-(24) set μ to minimize the combined variance using variances measured from the same 100-subfield simulations; the quoted 30-35 percent noise reduction uses those in-sample variances.
assumptions (5)
  • standard math Shapelet mode shear response equations (17)-(18) for fourth-order moments
    Borrowed from Massey and Refregier (2005); describes how sheared shapelet modes couple to M20, M60, M64.
  • domain assumption Vanishing ensemble mean of intrinsic spin-2 and spin-4 shapelet modes (average of M42 and M64 equal zero)
    Assumes no preferential orientation in the galaxy sample; used to infer shear from Eqs. (17)-(18).
  • domain assumption Renoising noise-bias correction is valid for fourth-order observables
    Section 2.5 imports the renoising method from Li et al. (2024a); the paper does not rederive it for higher-order moments.
  • domain assumption Homogeneous but correlated pixel noise model
    Section 4.1.1 assumes homogeneous noise fields, acknowledged as possibly inadequate for Stage IV surveys.
  • domain assumption Truncated sine selection weights approximate hard cuts
    Eq. (15) uses smooth step functions from Li and Mandelbaum (2023) to enable analytic shear derivatives.

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Cite this review

Pith. "Pith review of Accurate Shear Estimation with Fourth-Order Moments." pith.science (2026). https://pith.science/paper/VLSXOPZ7

@misc{pith2026241113648,
  author       = {Pith},
  title        = {Pith review of: Accurate Shear Estimation with Fourth-Order Moments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VLSXOPZ7}},
  note         = {Machine review of arXiv:2411.13648}
}
abstract

As imaging surveys progress in exploring the large-scale structure of the Universe through the use of weak gravitational lensing, achieving subpercent accuracy in estimating shape distortions caused by lensing, or shear, is imperative for precision cosmology. In this paper, we extend the \texttt{FPFS} shear estimator using fourth-order shapelet moments and combine it with the original second-order shear estimator to reduce galaxy shape noise. We calibrate this novel shear estimator analytically to a subpercent level accuracy using the \texttt{AnaCal} framework. This higher-order shear estimator is tested with realistic image simulations, and after analytical correction for the detection/selection bias and noise bias, the multiplicative shear bias $|m|$ is below $3\times10^{-3}$ ($99.7\%$ confidence interval) for both isolated and blended galaxies. Once combined with the second-order \texttt{FPFS} shear estimator, the shape noise is reduced by $\sim35\%$ for isolated galaxies in simulations with HSC and LSST observational conditions. However, for blended galaxies, the effective number density does not significantly improve with the combination of the two estimators. Based on these results, we recommend exploration of how this framework can further reduce the systematic uncertainties in shear due to PSF leakage and modelling error, and potentially provide improved precision in shear inference in high-resolution space-based images.

Figures

Figures reproduced from arXiv: 2411.13648 by the authors.

Figure 1
Figure 1. The real and imaginary components of the spin-2 second order and fourth-order shapelet basis. The fourth-order shapelets are sensitive to scales larger and smaller than that of the second moments, as referenced by the dashed lines. The color scale assigned to each basis function spans the interval [−𝐴, 𝐴], with 𝐴 representing the maximum absolute value of the corresponding basis function. and define the FPFS resolut… view at source ↗
Figure 2
Figure 2. The left panel shows a 128 pix × 128 pix (equivalent to 0.36 × 0.36 arcmin2 ) stamp image of the isolated galaxy image simulation with HSC seeing, where the dotted black lines show the boundaries of the 64 pix × 64 pix stamps. The right panel shows a random cut-out coadded image of 240 pixels × 240 pixels (equivalent to 0.8 × 0.8 arcmin2 ) of 𝑔𝑟 𝑖𝑧−bands of the LSST-like blended galaxy image simulation (Sheldon et a… view at source ↗
Figure 3
Figure 3. The 1𝜎 statistical uncertainty on a single component of the esti￾mated shear 𝛾ˆ1 for individual isolated galaxies (solid lines) as a function of the weighting parameter, 𝐶(𝑛) , in equation (19). For each second (blue) and fourth (orange) order estimator, the total uncertainty has contributions due to image noise (dotted lines) and intrinsic shape noise (dash-dotted lines). The vertical dashed lines show the values o… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The 1𝜎 statistical uncertainty on shear measurement 𝛾ˆ1 for indi￾vidual isolated galaxies as a function of the upper limit of FPFS magnitude (𝑚𝐹) for the galaxies included in the measurement. The uncertainty for each second (blue) and fourth (orange) order estimator in…
Figure 5
Figure 5. Figure 5: The 1𝜎 statistical uncertainty on shear measurement of individual galaxies from each estimator; second (blue) and fourth (orange) order, com￾bining the two FPFS estimators (green), and from reGauss (pink) method. The uncertainty values of reGauss are drawn in dotted ho…
Figure 6
Figure 6. Figure 6: The 1𝜎 statistical uncertainty per galaxy on shear measurement 𝛾ˆ1 for isolated galaxies as a function of the PSF size using different image noise levels and shear estimators. The uncertainty for each second (blue) and fourth (orange) order estimator includes contribut…
Figure 7
Figure 7. Figure 7: The multiplicative bias (upper panel) and additive bias (lower panel) of the AnaCal shear estimator on isolated galaxies using an HSC-like configuration. Blue (orange) lines are results using the second (fourth) order shear estimator. The error bars show the 1𝜎 and 3𝜎 …
Figure 8
Figure 8. Figure 8: The effective galaxy number density as a function of SNR cut for each AnaCal estimator and for each simulation setup (PSF seeing size and image noise level) using 100 subfields with blended galaxies. The statistical uncertainties that go into this effective number dens…
Figure 9
Figure 9. Figure 9: The multiplicative bias (upper panel) and additive bias (lower panel) of the AnaCal shear estimator on blended galaxies using an LSST￾like configuration. Blue (orange) lines show the second (fourth) order shear estimator. The error bars show the 1𝜎 and 3𝜎 uncertainties…

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Works this paper leans on

39 extracted references · 7 linked inside Pith

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTION or pop #1...

  2. [2]

    arXiv:1902.05569

    Akeson R., et al., 2019, @doi [arXiv e-prints] 10.48550/arXiv.1902.05569 , https://ui.adsabs.harvard.edu/abs/2019arXiv190205569A p. arXiv:1902.05569

    arXiv doi:10.48550/arxiv.1902.05569 2019

  3. [3]

    M., 2010, @doi [ ] 10.1111/j.1365-2966.2010.16883.x , https://ui.adsabs.harvard.edu/abs/2010MNRAS.406.2793B 406, 2793

    Bernstein G. M., 2010, @doi [ ] 10.1111/j.1365-2966.2010.16883.x , https://ui.adsabs.harvard.edu/abs/2010MNRAS.406.2793B 406, 2793

    arXiv 2010

  4. [4]

    M., Jarvis M., 2002, @doi [ ] 10.1086/338085 , https://ui.adsabs.harvard.edu/abs/2002AJ....123..583B 123, 583

    Bernstein G. M., Jarvis M., 2002, @doi [ ] 10.1086/338085 , https://ui.adsabs.harvard.edu/abs/2002AJ....123..583B 123, 583

    doi:10.1086/338085 2002

  5. [5]

    M., Armstrong R., Krawiec C., March M

    Bernstein G. M., Armstrong R., Krawiec C., March M. C., 2016, @doi [ ] 10.1093/mnras/stw879 , https://ui.adsabs.harvard.edu/abs/2016MNRAS.459.4467B 459, 4467

    doi:10.1093/mnras/stw879 2016

  6. [6]

    Bosch J., et al., 2018, @doi [ ] 10.1093/pasj/psx080 , https://ui.adsabs.harvard.edu/abs/2018PASJ...70S...5B 70, S5

    doi:10.1093/pasj/psx080 2018

  7. [7]

    Chang C., et al., 2013, @doi [ ] 10.1093/mnras/stt1156 , https://ui.adsabs.harvard.edu/abs/2013MNRAS.434.2121C 434, 2121

    doi:10.1093/mnras/stt1156 2013

  8. [8]

    R., et al., 2020, @doi [Nature] 10.1038/s41586-020-2649-2 , 585, 357

    Harris C. R., et al., 2020, @doi [Nature] 10.1038/s41586-020-2649-2 , 585, 357

    doi:10.1038/s41586-020-2649-2 2020

Show all 39 references
  1. [11]

    D., 2007, @doi [Computing in Science & Engineering] 10.1109/MCSE.2007.55 , 9, 90

    Hunter J. D., 2007, @doi [Computing in Science & Engineering] 10.1109/MCSE.2007.55 , 9, 90

    2007 doi

  2. [13]

    Ivezi \'c Z ., et al., 2019, @doi [ ] 10.3847/1538-4357/ab042c , https://ui.adsabs.harvard.edu/abs/2019ApJ...873..111I 873, 111

    2019 doi

  3. [14]

    Kaiser N., Squires G., Broadhurst T., 1995, @doi [ ] 10.1086/176071 , https://ui.adsabs.harvard.edu/abs/1995ApJ...449..460K 449, 460

    1995 doi

  4. [15]

    Kilbinger M., 2015, @doi [Reports on Progress in Physics] 10.1088/0034-4885/78/8/086901 , https://ui.adsabs.harvard.edu/abs/2015RPPh...78h6901K 78, 086901

    2015 doi

  5. [16]

    arXiv:0912.0201

    LSST Science Collaboration et al., 2009, @doi [arXiv e-prints] 10.48550/arXiv.0912.0201 , https://ui.adsabs.harvard.edu/abs/2009arXiv0912.0201L p. arXiv:0912.0201

    2009 doi arXiv

  6. [17]

    arXiv:1110.3193

    Laureijs R., et al., 2011, @doi [arXiv e-prints] 10.48550/arXiv.1110.3193 , https://ui.adsabs.harvard.edu/abs/2011arXiv1110.3193L p. arXiv:1110.3193

    2011 doi arXiv

  7. [18]

    Li X., Mandelbaum R., 2023, @doi [ ] 10.1093/mnras/stad890 , https://ui.adsabs.harvard.edu/abs/2023MNRAS.521.4904L 521, 4904

    2023 doi

  8. [19]

    Li X., Katayama N., Oguri M., More S., 2018, @doi [ ] 10.1093/mnras/sty2548 , https://ui.adsabs.harvard.edu/abs/2018MNRAS.481.4445L 481, 4445

    2018 doi

  9. [20]

    Li X., et al., 2022a, @doi [ ] 10.1093/pasj/psac006 , https://ui.adsabs.harvard.edu/abs/2022PASJ...74..421L 74, 421

    doi

  10. [21]

    Li X., Li Y., Massey R., 2022b, @doi [ ] 10.1093/mnras/stac342 , https://ui.adsabs.harvard.edu/abs/2022MNRAS.511.4850L 511, 4850

    doi

  11. [22]

    arXiv:2408.06337

    Li X., Mandelbaum R., The LSST Dark Energy Science Collaboration 2024a, @doi [arXiv e-prints] 10.48550/arXiv.2408.06337 , https://ui.adsabs.harvard.edu/abs/2024arXiv240806337L p. arXiv:2408.06337

    doi arXiv

  12. [23]

    Li X., Mandelbaum R., Jarvis M., Li Y., Park A., Zhang T., 2024b, @doi [ ] 10.1093/mnras/stad3895 , https://ui.adsabs.harvard.edu/abs/2024MNRAS.52710388L 527, 10388

    doi

  13. [24]

    Liaudat T. I., Starck J.-L., Kilbinger M., 2023, @doi [Frontiers in Astronomy and Space Sciences] 10.3389/fspas.2023.1158213 , https://ui.adsabs.harvard.edu/abs/2023FrASS..1058213L 10, 1158213

    2023

  14. [25]

    MacCrann N., et al., 2022, @doi [ ] 10.1093/mnras/stab2870 , https://ui.adsabs.harvard.edu/abs/2022MNRAS.509.3371M 509, 3371

    2022 doi

  15. [26]

    Mandelbaum R., 2018, @doi [ ] 10.1146/annurev-astro-081817-051928 , https://ui.adsabs.harvard.edu/abs/2018ARA&A..56..393M 56, 393

    2018 doi

  16. [27]

    Mandelbaum R., Lackner C., Leauthaud A., Rowe B., 2019, COSMOS real galaxy dataset, @doi 10.5281/zenodo.3242143 , https://doi.org/10.5281/zenodo.3242143

    2019 doi

  17. [28]

    H., Bosch J., Kannawadi A., Murphy M

    Mandelbaum R., Jarvis M., Lupton R. H., Bosch J., Kannawadi A., Murphy M. D., Zhang T., LSST Dark Energy Science Collaboration 2023, @doi [The Open Journal of Astrophysics] 10.21105/astro.2209.09253 , https://ui.adsabs.harvard.edu/abs/2023OJAp....6E...5M 6, 5

    2023 arXiv

  18. [29]

    Massey R., Refregier A., 2005, @doi [ ] 10.1111/j.1365-2966.2005.09453.x , http://adsabs.harvard.edu/abs/2005MNRAS.363..197M 363, 197

    2005

  19. [30]

    Massey R., et al., 2007, @doi [ ] 10.1111/j.1365-2966.2006.11315.x , https://ui.adsabs.harvard.edu/abs/2007MNRAS.376...13M 376, 13

    2007

  20. [31]

    Moffat A. F. J., 1969, , https://ui.adsabs.harvard.edu/abs/1969A&A.....3..455M 3, 455

    1969

  21. [32]

    Pujol A., Kilbinger M., Sureau F., Bobin J., 2019, @doi [ ] 10.1051/0004-6361/201833740 , https://ui.adsabs.harvard.edu/abs/2019A&A...621A...2P 621, A2

    2019 doi

  22. [33]

    Refregier A., 2003, @doi [ ] 10.1046/j.1365-8711.2003.05901.x , http://adsabs.harvard.edu/abs/2003MNRAS.338...35R 338, 35

    2003

  23. [34]

    Refregier A., Kacprzak T., Amara A., Bridle S., Rowe B., 2012, @doi [ ] 10.1111/j.1365-2966.2012.21483.x , https://ui.adsabs.harvard.edu/abs/2012MNRAS.425.1951R 425, 1951

    2012

  24. [35]

    Rowe B., Jarvis M., Mandelbaum R., 2014, GalSim: Modular galaxy image simulation toolkit , Astrophysics Source Code Library, record ascl:1402.009 ( @eprint ascl 1402.009 )

    2014

  25. [36]

    S., Huff E

    Sheldon E. S., Huff E. M., 2017, @doi [ ] 10.3847/1538-4357/aa704b , https://ui.adsabs.harvard.edu/abs/2017ApJ...841...24S 841, 24

    2017 doi

  26. [37]

    S., Becker M

    Sheldon E. S., Becker M. R., MacCrann N., Jarvis M., 2020, @doi [ ] 10.3847/1538-4357/abb595 , https://ui.adsabs.harvard.edu/abs/2020ApJ...902..138S 902, 138

    2020 doi

  27. [38]

    S., Becker M

    Sheldon E. S., Becker M. R., Jarvis M., Armstrong R., LSST Dark Energy Science Collaboration 2023, @doi [The Open Journal of Astrophysics] 10.21105/astro.2303.03947 , https://ui.adsabs.harvard.edu/abs/2023OJAp....6E..17S 6, 17

    2023 arXiv

  28. [39]

    arXiv:1809.01669

    The LSST Dark Energy Science Collaboration et al., 2018, @doi [arXiv e-prints] 10.48550/arXiv.1809.01669 , https://ui.adsabs.harvard.edu/abs/2018arXiv180901669T p. arXiv:1809.01669

    2018 doi arXiv

  29. [40]

    Virtanen P., et al., 2020, @doi [Nature Methods] 10.1038/s41592-019-0686-2 , https://rdcu.be/b08Wh 17, 261

    2020 doi

  30. [41]

    Zhang T., et al., 2023, @doi [ ] 10.1093/mnras/stad1801 , https://ui.adsabs.harvard.edu/abs/2023MNRAS.525.2441Z 525, 2441

    2023 doi

  31. [42]

    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.stat...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.