Pith. sign in

REVIEW 2 major objections 5 minor 44 references

Reconstructing compound objects by quantum imaging with higher-order correlation functions

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Parametric locality, visible as band structure in the Fisher information matrix, enables a sliding-window estimator that reconstructs super-resolved quantum images at linear cost and predicts the optimal photon correlation width.

desk verdict A solid, useful paper whose main algorithmic claim is real but whose linear-complexity statement is oversold in the super-resolution regime. read the letter →

arxiv 1908.07461 v2 pith:U4V63XU6 submitted 2019-08-20 quant-ph

A. B. Mikhalychev , B. Bessire , I. L. Karuseichyk , A. A. Sakovich , M. Unternahrer , D. A. Lyakhov , D. L. Michels , A. Stefanov
show 1 more author
D. Mogilevtsev
This is my paper · ORCID
classification quant-ph
keywords quantumimagingFisherinformationmatrixslidingwindowmethodparametriclocalitysuper-resolutionhigher-ordercorrelationfunctionspseudo-thermallightentangledtwinphotons
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

Parametric locality is the property that each measurement outcome depends on only a few neighbouring parameters of the object. The paper claims that for such problems the Fisher information matrix is banded and its inverse is approximately banded, so a parameter can be estimated from a small window of data around it. On that basis it constructs an iterative sliding-window method in which each step fits only a subset of parameters, making the total reconstruction effort linear in the number of parameters instead of nonlinear. Applied to quantum near-field imaging from second- and third-order correlation functions, the method reconstructs grey transmission objects beyond the Rayleigh limit using both pseudo-thermal and entangled twin-photon sources. The same Fisher-information analysis predicts an optimal photon correlation width, close to the smallest object detail to be resolved, and the paper reports experimental confirmation of that prediction.

What carries the argument

The central object is the Fisher information matrix of a parametrically l-local measurement, where each outcome probability has nonzero derivatives only with respect to parameters within distance l, so the matrix is l-banded. The argument uses the matrix fact that inverses of banded matrices are approximately banded, which turns the error bound for one parameter into a function of nearby parameters only. The sliding window method is the estimator built from that fact: after a coarse first pass with pixels large enough for diagonal dominance, it refines the grid and, in each step, fits only a core window of unknown pixels surrounded by a border whose width exceeds the number of major Fisher information bands; values outside the window are held fixed or set to zero. The border width, not the total number of pixels, sets the cost of every local fit, and shifting the core window across the object produces the claimed linear total complexity.

What would settle it

Calculate the number of significant Fisher information bands, n0 = Δl/d, for a one-dimensional near-field image on grids with d = Δl/2, Δl/4, and Δl/8. If the required border size grows with n0 rather than remaining bounded, the per-window cost grows with the number of pixels and the linear-complexity claim is refuted; the paper reports no benchmark of this scaling.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the hard nonlinear estimation problem of quantum imaging becomes tractable when the measurement is parametrically local: when each detected correlation event depends on a small cluster of nearby pixels, the Fisher information matrix is narrowly banded, and the standard lower bound on estimator variance for a given pixel depends only on data in a window around it. The authors build the sliding window method on this: first a coarse reconstruction with large pixels whose Fisher information matrix is diagonally dominant, then refinement on a finer pixel grid where each optimization step treats a core of unknown pixels, a border of known or ignored pixels, and a border size set by the number of major Fisher information bands. They demonstrate the method on simulated and experimental data, reconstructing grey transmission objects from measured second- and third-order correlation functions for pseudo-thermal light and for position-momentum entangled twin photons, and report resolution beyond the Rayleigh limit. The paper also establishes two further results from the same information analysis: an optimal correlation width of the imaging field, approximately the size of the smallest object detail, and an improvement of resolution from the estimation bias that arises when parameters sit at the boundary of their allowed range.

Load-bearing premise

The load-bearing premise is that the measurement is parametrically local at the finest pixel grid used: the width of the Fisher information bands stays small relative to the total number of parameters as the grid is refined below the Rayleigh limit, so a fixed-size window border can keep the local fits accurate.

Editorial extensions

If this is right

  • For objects with many pixels, each reconstruction step becomes a small least-squares fit over a window, so the total computational cost scales linearly with the number of parameters instead of with a high power of it.
  • In the super-resolution regime, the best photon-source correlation width is not the smallest possible but roughly the smallest object feature size, giving a design rule for choosing source speckle size or twin-photon correlation width.
  • Higher-order correlation measurements with both pseudo-thermal and entangled twin-photon sources can reconstruct grey transmission objects beyond the Rayleigh limit.
  • When estimated parameters lie at the boundary of their allowed range, the resulting estimate bias can lower the total reconstruction error, so binary objects can be resolved better than the unbiased error bound suggests.
  • The Fisher-information analysis can be performed per imaging setup without knowing the object, so the window structure and border size can be fixed in advance from the point-spread function and source correlations.

Reading between the lines

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

  • The same windowing principle should carry over to any nonlinear estimation problem whose Fisher information matrix is banded, not only optical imaging; the argument never uses the specific imaging model except to establish locality.
  • A direct stress test is to benchmark how the required border size grows as the pixel grid is refined below the Rayleigh limit; the linear-complexity claim assumes this growth stays bounded while the number of parameters increases.
  • The reported resolution gain from boundary bias suggests that encoding object constraints, such as known binary or grey-value bounds, is not just a prior but a measurable information resource worth quantifying separately from the correlation statistics.
  • One could test the optimal-correlation-width rule by measuring reconstruction infidelity versus source correlation width across several feature sizes and checking whether the minimum tracks the feature size, as the paper reports for one object.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The manuscript proposes an iterative sliding-window method (SWM) for nonlinear parameter estimation in problems with parametrically local measurements, i.e., measurements where each observed probability depends on a limited subset of the unknown parameters. The authors connect this locality to the banded structure of the Fisher information matrix (FIM), use the FIM to choose the window and border sizes, and apply the method to near-field quantum imaging with higher-order correlation functions. They derive the relevant detection coefficients for pseudo-thermal and SPDC sources, demonstrate reconstructions on simulated and experimental data, predict an optimal source correlation width from the trace of the inverse FIM, and experimentally confirm the prediction. They additionally discuss how estimation bias arising from parameter constraints can improve resolution.

Significance. If the linear-complexity claim is sustained, the SWM is a potentially valuable tool for quantum imaging and tomography with large parameter spaces. The paper has several concrete strengths: the FIM-based locality analysis is clearly presented, the pseudo-code is implementable, and the experimental demonstrations with a 32x32 SPAD array for both pseudo-thermal and SPDC sources are credible. The prediction of an optimal correlation width from the Fisher-information model and its confirmation with experimental reconstruction infidelity (Fig. 4a,b) is a nontrivial, falsifiable result. The biased-estimation discussion correctly draws on the constrained-estimation literature. However, the central scaling claim is not rigorously established for the super-resolution regime, and the window-truncation approximation lacks an error bound; these issues need to be addressed before the claimed complexity advantage can be accepted.

major comments (2)
  1. [Abstract; Results (Theoretical background); Methods (Sliding windows method)] The central complexity claim, stated in the abstract as 'this iterative scheme is linear on the total number of parameters,' is not established for the super-resolution regime in which the method is applied. In Supplementary Note 3, n0 = Delta_l / d is the width of the point-spread function in object-pixel units. For a fixed physical object, refining the pixel grid below the Rayleigh limit makes the total pixel count M grow while n0 grows proportionally to M. The Methods pseudo-code sizes the window border from the 'number and relative value of major bands of the FIM,' so each local minimization at a given shift involves O(n0) unknown and known pixels, and the number of shifts is only O(M/n0). The total cost is therefore O(M) only under the unstated assumption that n0 remains O(1) as the grid is refined. Since the experiments use a fixed 32x32 detector array, they do not exercise the scaling regime in which the problem is claimed to become hard. Please state the assumptions under which the claimed linearity holds, or qualify the claim in the abstract and introduction.
  2. [Methods (pseudo-code, first approximation and refinement)] The truncation approximation used in both algorithms is stated as 'the theoretical probabilities are computed assuming all pixels but those inside the window to be zero' (first approximation) and 'pixels outside the full window are set to zero' (refinement), but no error bound is given for this approximation. The size of the omitted-pixel contribution is controlled by the same PSF tails that define n0; when d is reduced toward the super-resolution regime, n0 grows and the number of omitted pixels with non-negligible weight grows as well, so the bias introduced by the truncation is not controlled. A quantitative error bound, or at least a numerical convergence check as a function of the window and border sizes, is needed to justify the accuracy of the SWM reconstructions in the claimed regime.
minor comments (5)
  1. [Supplementary Note 3, paragraph after Eq. (22)] The sentence stating that the coefficients are 'effectively zero for |j-m| << n0 or |k-n| << n0' appears to have the inequality reversed: if n0 is the PSF width in object pixels, the coefficients should be negligible for separations much larger than n0, not much smaller. Please correct this.
  2. [Supplementary Note 4] The text refers to 'the Appendix C' for the spatial correlation function, but the manuscript contains no Appendix C; the relevant description appears in Supplementary Note 3. Please update the cross-reference.
  3. [Results (Experiment); Fig. 3] The claim of resolution beyond the Rayleigh limit is supported visually by the red bars in Fig. 3, but a quantitative definition of achieved resolution (for example, estimated feature widths with uncertainties derived from the FIM or from repeated reconstructions) would make the claim more precise and easier to evaluate.
  4. [Data availability] The statement 'The code itself is available upon request' is weaker than the rest of the reproducibility effort; depositing the code in a permanent repository would allow readers to reproduce the reconstructions and the linear-complexity experiments.
  5. [Results (Theoretical background)] The definition of strict parametric l-locality is given for a one-dimensional ordering of parameters, while the imaging demonstrations are two-dimensional. Please state how the locality condition and the window construction generalize to two dimensions (for example, using a Chebyshev or Euclidean distance between pixel indices).

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the central claims are model-based predictions tested against known USAF benchmarks; self-citations are non-load-bearing.

full rationale

The paper's central derivation is self-contained. The SWM is constructed from the banded structure of the FIM, with locality derived from the PSF and source correlation functions (Supplementary Note 3, Eqs. (16)-(22)); the linear-complexity statement is an algorithmic property of the sliding-window iteration, not a quantity fitted from the data. The optimal-correlation-width prediction is computed from the trace of the inverse FIM (Fig. 4(a), Supplementary Note 4) and then tested against reconstruction infidelity for known USAF targets (Fig. 4(b)), so the predicted quantity is not used as an input to the reconstruction in a way that forces the claimed agreement. The only self-citations are to the detector characterization [36,37] and to data-pattern tomography as an example of parametric locality [15]; neither is load-bearing for the paper's central inference claims. The scaling concern about the window border growing with n0 = Delta_l/d is a correctness risk about the complexity claim in the deep super-resolution limit, not a circularity.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central claim relies on standard mathematical results (banded inversion bounds, CRB), two physical models for the light sources, the locality of the near-field PSF, and two algorithm-specific approximations (window truncation and rule-of-thumb border size). The free parameters are the reconstruction hyperparameters and the source correlation width, which is scanned in the prediction rather than fitted to the reconstruction. No new physical entities are postulated.

free parameters (3)
  • Window border/core sizes = Rule of thumb: border larger than the number of major FIM bands; no precise numeric value given
    These algorithm hyperparameters determine the accuracy and cost of each local minimization; they are chosen by heuristic, not derived.
  • Source correlation width wc = Varied; optimum at wc = 1.5 pix in Fig. 4(a), measured via Gaussian fit to G(2)
    wc is the physical correlation width of the pseudo-thermal or SPDC source and is a control parameter in the FIM optimization; it is calibrated from data but not fitted to the reconstruction target.
  • Initial pixel size d_initial = Chosen so the inverse FIM is diagonally dominant
    Determines the starting resolution of the iterative refinement; chosen by inspection of the FIM for the model setup.
assumptions (7)
  • standard math Inverses of banded matrices can be approximated by banded matrices (Refs [21,22])
    Used to conclude that an estimator of parameter θj depends mainly on probabilities in its vicinity, the basis of the SWM.
  • standard math Fisher information matrix Eq. (1) and Cramér-Rao bound Eq. (9), plus biased CRB Eq. (28) from Eldar [34,35]
    Used for error bounds and for designing and optimizing the measurement.
  • domain assumption Pseudo-thermal source has Gaussian statistics and nth-order correlations factor into pairwise products (Eq. 16, Supp Note 3)
    Needed to compute D coefficients and FIM for the thermal experiment.
  • domain assumption SPDC joint-position amplitude is approximately a Gaussian function with width wc (Supp Note 3)
    Needed to compute D coefficients for the SPDC experiment; an approximation to the real crystal correlations.
  • domain assumption Near-field PSF is a jinc function with phase factor approximately 1, and coefficients D vanish for |j-m| >> n0 = Δl/d (Supp Note 3)
    This is the parametric locality that justifies the sliding window.
  • ad hoc to paper In the first approximation, pixels outside the full window are set to zero; in refinement, pixels outside the core but inside the full window are treated as known constants (Methods pseudocode)
    This is the core approximation of the SWM; its error is controlled only by the assumption that the window border covers all significant FIM couplings.
  • ad hoc to paper Border size is chosen larger than the number of major FIM bands, with no rigorous guarantee of sufficient accuracy
    The method relies on this rule of thumb for the core claim of linear complexity and reconstruction accuracy.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Reconstructing compound objects by quantum imaging with higher-order correlation functions." pith.science (2026). https://pith.science/paper/U4V63XU6

@misc{pith2026190807461,
  author       = {Pith},
  title        = {Pith review of: Reconstructing compound objects by quantum imaging with higher-order correlation functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U4V63XU6}},
  note         = {Machine review of arXiv:1908.07461}
}
read the original abstract

Quantum imaging has a potential of enhancing precision of the object reconstruction by using quantum correlations of the imaging field. This is especially important for imaging requiring low-intensity fields up to the level of few-photons. However, quantum imaging generally leads to nonlinear estimation problems. The complexity of these problems rapidly increases with the number of parameters describing the object. We suggest a way to drastically reduce the complexity for a wide class of problems. The key point of our approach is connecting the features of the Fisher information with the parametric locality of the problem, and building the efficient iterative inference scheme reconstructing only a subset of the whole set of parameters in each step. This iterative scheme is linear on the total number of parameters. This scheme is applied to quantum near-field imaging, the inference procedure is developed resulting in super-resolving reconstruction of grey compound transmission objects. The functionality of the method is demonstrated with experimental data obtained by measurements of higher-order correlation functions for imaging with entangled twin-photons and pseudo-thermal light sources. By analyzing the informational content of the measurement, it becomes possible to predict the existence of optimal photon correlations providing for the best image resolution in the super-resolution regime. This prediction is experimentally confirmed. It is also shown how an estimation bias stemming from image features may drastically improve the resolution.

Figures

Figures reproduced from arXiv: 1908.07461 by the authors.

Figure 1
Figure 1. FIG. 1: Scheme of the measurement setup. A state of light [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The sliding window method. Examples of the Fisher [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4: Inverse Fisher information matrix trace and recon [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (8 more)
Figure 3
Figure 3. Figure 3: FIG. 3: Experimental data and reconstructed pixel transmis [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]
Figure 5
Figure 5. Figure 5: FIG. 5: An example of two adjacent reconstruction steps for [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Pseudo-thermal light imaging setup. A monochro [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: SPDC setup. A monochromatic laser is weakly fo [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Pseudo-thermal light imaging setup. A monochro [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: SPDC setup. A monochromatic laser is weakly fo [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Dependence of the inverse Fisher matrix trace (a) [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Mean value of the biased estimator [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

  1. [1]

    E. J. Candes, J. K. Romberg, and T. Tao, IEEE Trans. Inform. Theory 52, 489509 (2006)

    work page 2006

  2. [2]

    D. L. Donoho, IEEE Trans. Inform. Theory 52, 12891306 (2006)

    work page 2006

  3. [3]

    Gross, Yi-Kai Liu, S

    D. Gross, Yi-Kai Liu, S. T. Flammia, S. Becker, and J. Eisert, Phys. Rev. Lett. 105, 150401 (2010)

    work page 2010

  4. [4]

    C. A. Riofrio, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt and J. Eisert, Nat. Comm. 8, 15305 (2017)

    work page 2017

  5. [5]

    Cramer, M

    M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin and Yi-Kai Liu, Nat. Comm. 1, (2010)

    work page 2010

  6. [6]

    T´ oth, W

    G. T´ oth, W. Wieczorek, D. Gross, R. Krischek, C. Schwemmer, and H. Weinfurter, Phys. Rev. Lett. 105, 250403 (2010)

    work page 2010

  7. [7]

    Moroder, P

    T. Moroder, P. Hyllus, G. T´ oth, C. Schwemmer, A. Niggebaum, S. Gaile, O. G¨ uhne, and H. Weinfurter, New J. Phys. 14, 105001 (2012)

    work page 2012

  8. [8]

    Schwemmer, G

    C. Schwemmer, G. T´ oth, A. Niggebaum, T. Moroder, D. Gross, O. G¨ uhne, and H. Weinfurter, Phys. Rev. Lett. 113, 040503 (2014)

    work page 2014

Show all 44 references
  1. [9]

    Thiebaut, J

    E. Thiebaut, J. Young, JOSA A 34 904 (2017)

    2017

  2. [10]

    J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press Inc, San Diego, 1970)

    1970

  3. [11]

    H. R. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Korber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Ghne, W. Dur & R. Blatt, Nature 438, 643 (2005)

    2005

  4. [12]

    Bloch, J

    I. Bloch, J. Dalibard, & W. Zwerger, Rev. Mod. Phys. 80, 885 (2008)

    2008

  5. [13]

    Tetienne, N

    J.-Ph. Tetienne, N. Dontschuk, D. A. Broadway, A. Stacey, D. A. Simpson, and L. C. L. Hollenberg, Sci. Adv. 3, e1602429 (2017)

    2017

  6. [14]

    Shih, IEEE Journal of Selected Topics in Quantum Electronics, IEEE 13, 1016 (2007)

    See, for example, Y.H. Shih, IEEE Journal of Selected Topics in Quantum Electronics, IEEE 13, 1016 (2007)

    2007

  7. [15]

    Rehacek, D

    J. Rehacek, D. Mogilevtsev, and Z. Hradil, Phys. Rev. Lett. 105, 010402 (2010)

    2010

  8. [16]

    Tsang, R

    M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X6, 031033 (2016). 8

    2016

  9. [17]

    Nair and M

    R. Nair and M. Tsang, Phys. Rev. Lett. 117, 190801 (2016)

    2016

  10. [18]

    Motka, B

    L. Motka, B. Stoklasa, M. DAngelo, P. Facchi, A. Garuc- cio, Z. Hradil, S. Pascazio, F.V. Pepe, Y.S. Teo, J. Re- hacek, L.L. Sanchez-Soto, The European Physical Jour- nal Plus, 131, 130 (2016)

    2016

  11. [19]

    Rehacek, Z

    J. Rehacek, Z. Hradil, B. Stoklasa, M. Paur, J. Grover, A. Krzic, L.L. Sanchez-Soto, Phys. Rev. A 96 062107 (2017)

    2017

  12. [20]

    Seveso, M

    L. Seveso, M. A. C. Rossi, and M. G. A. Paris, Phys. Rev. A 95, 012111 (2017)

    2017

  13. [21]

    Bickel and M

    P. Bickel and M. Lindner, Theory Probab. Appl. 56, 1 (2012)

    2012

  14. [22]

    Demko, W

    S. Demko, W. F. Moss, and P. W. Smith, Math. Comp., 43 491 (1984)

    1984

  15. [23]

    Born and E

    M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, UK, 1999)

    1999

  16. [24]

    Zhang, W

    P. Zhang, W. Gong, Xia Shen, D. Huang, and S. Han, Opt. Lett. 34, 1222 (2009)

    2009

  17. [25]

    Chen, I.N

    X.-H. Chen, I.N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M.V.Chekhova, and L.-A. Wu, Opt. Lett. 53, 1166 (2010)

    2010

  18. [26]

    Y. Zhou, J. Simon, J. Liu, and Y. Shih, Phys. Rev. A 81, 043831 (2010)

    2010

  19. [27]

    Gatto Monticone, K

    D. Gatto Monticone, K. Katamadze, P. Traina, E. Moreva, J. Forneris, I. Ruo-Berchera, P. Olivero, I.P. De- giovanni, G. Brida, and M. Genovese, Phys. Rev. Lett. 113, 143602 (2014)

    2014

  20. [28]

    Israel, R

    Y. Israel, R. Tenne, D. Oron and Y. Silberberg, Nat. Comm. 8, 14786 (2017)

    2017

  21. [29]

    Tenne, U

    R. Tenne, U. Rossman, B. Rephael, Y. Israel, A. Krupinski-Ptaszek, R. Lapkiewicz, Y. Silberberg and D. Oron, Nat. Phot. 13, 116 (2019)

    2019

  22. [30]

    Classen, F

    A. Classen, F. Waldmann, S. Giebel, R. Schneider, D. Bhatti, Th. Mehringer, and J. von Zanthier, Phys. Rev. Lett. 117, 253601 (2016)

    2016

  23. [31]

    Classen, J

    A. Classen, J. von Zanthier, M. O. Scully, and G. S. Agarwal, Optica 4, 580 (2017)

    2017

  24. [32]

    Martienssen and E

    W. Martienssen and E. Spiller, Am. J. Phys. 32(12), 919 (1964)

    1964

  25. [33]

    Statistical properties of laser speckle patterns

    J. W. Goodman, “Statistical properties of laser speckle patterns”, in Laser speckle and related phenomena , vol. 9 of Toppics in Applied Physics, J. C. Dainty, ed., (Springer, Berlin, 1975)

    1975

  26. [34]

    Y. C. Eldar, Foundations and Trends in Signal Process- ing, 1 305 (2008)

    2008

  27. [35]

    Y. C. Eldar, IEEE Trans. Signal Processing, 52, 1916 (2004)

    2004

  28. [36]

    Gasparini, M

    L. Gasparini, M. Zarghami, H. Xu, L. Parmesan, M. M. Garcia, M. Unternhrer, B. Bessire, A. Stefanov, D. Stoppa, and M. Perenzoni, “A 32 ×32-pixels time- resolved singlephoton image sensor with 44.64- µm pitch and 19.48% fill-factor with on-chip row/frame skipping features reach...

    2018

  29. [37]

    Unternahrer, B

    M. Unternahrer, B. Bessire, L. Gasparini, M. Perenzoni, and A. Stefanov, Optica 5(9), 1150–1154 (2018)

    2018

  30. [38]

    S. P. Walborn, C. H. Monken, S. Padua, and P. H. Ribeiro, Physics Reports 495, 87–139 (2010)

    2010

  31. [39]

    Liu, T.Z

    X.Q. Liu, T.Z. Huang and Y.D. Fu, Appl. Math. Lett., 19 590 (2006)

    2006

  32. [40]

    Politi, M

    T. Politi, M. Popolizio, JIPAM 9, 31 (2008)

    2008

  33. [41]

    Zhuohong Huang, Jianzhou Liu, Applied Mathematics E-Notes, 10, 11-18 (2010)

    2010

  34. [42]

    WoIkowicz, G

    H. WoIkowicz, G. P. H. Styan, Linear Algebra and its Applications 29, 471 (1980)

    1980

  35. [43]

    Emanueli, and A

    S. Emanueli, and A. Arie, Appl. Opt. 42, 6661 (2003)

    2003

  36. [44]

    Schneeloch, and J

    J. Schneeloch, and J. C. Howell, J. Opt. 18, 053501 (2016) Supplementary note 1. Bounds for inverses of banded and approximately banded matrices Here we give a number of known results about inverses of banded and approximately banded matrices useful for our discussion. First o...

    2016

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.