Solving equations after dense scan to improve the resolutions of microscopes

Yaohua Xie

REVIEW 4 major objections 4 minor 6 references

Solving equations after dense scan to improve the resolutions of microscopes

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A square linear system built from dense scans of only the region of interest recovers the sharp high-resolution image exactly, making peripheral scanning unnecessary.

desk verdict A sincere but naive methods note: the linear algebra is standard, the ROI-only scan idea is mildly useful, but the exact-recovery and unlimited-resolution claims die on first contact with noise. read the letter →

arxiv 1908.01284 v2 pith:TWSJAXO4 submitted 2019-08-04 eess.IV

Yaohua Xie This is my paper · ORCID

classification eess.IV

keywords super-resolutionmicroscopydeconvolutiondensescanlinearequationsystemregionofinterestilluminationspotSTEDimagereconstruction

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

Using a known illumination spot and a dense scan that visits only the region of interest (ROI), the paper builds one brightness equation per scan position and claims that the exact solution of the resulting square linear system is the sharp, high-resolution image. This removes the need, present in an earlier dense-scan deconvolution technique, to scan a peripheral border around the ROI before filtering. Two simulated experiments recover simulated ground-truth images to average pixel differences of about $8.03\times10^{-12}$ at a $3\,\mathrm{nm}$ step and $1.11\times10^{-6}$ at a $0.1\,\mathrm{nm}$ step, with the spot larger than the ROI in the second case. The paper concludes that the approach can reach arbitrarily high resolution in principle because only the ROI's own pixels must be scanned no matter how fine the step becomes.

What carries the argument

The load-bearing object is the square linear system assembled from Eq. (1). Each dense-scan measurement at spot center $(i,j)$ is written as the sum over the pixel-wise product of the known illumination spot $I$ and the unknown image $E$, so the matrix $A$ has one row per scan position and one column per unknown image pixel, and the system takes the form $Ax=b$. The zero-periphery preprocessing guarantees that every equation references only pixels inside the ROI, which is what makes the system square and, in the idealized model, exactly invertible. This mechanism replaces the filtering-based deconvolution of the earlier dense-scan method and is what lets the paper drop the peripheral scan.

What would settle it

Compute the condition number of the matrix $A$ for the $101\times101$ spot and $60\times60$ ROI used in the second experiment; if the condition number times the measurement noise exceeds one, the solved image will not track the true image, refuting exact recovery outside the noise-free simulation. A direct experiment would add Poisson noise to each simulated $S(i,j)$ and watch the recovery error grow with the noise level.

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Extended reading notes

Core claim

The central claim is that deconvolution after dense scan can be replaced by solving an equation system, and that the solution of that system is exactly the sharp and high-resolution image. The measurements are modeled by Eq. (1), $S(i,j)=\sum_u\sum_v I(u,v)E(i+u,j+v)$, where $I$ is the known illumination spot centered at $(i,j)$ and $E$ is the unknown image. Because preprocessing makes the optical response outside the ROI zero, the sums involve only the $R\cdot C$ unknown pixels, giving $R\cdot C$ equations in $R\cdot C$ unknowns. On the paper's idealized model this is an exact convolution inversion rather than an approximate filtering operation, so peripheral scanning is unnecessary. The paper further claims that, in principle, the achievable resolution is unlimited because the dense step can be made arbitrarily small while the number of scanned positions remains the ROI's pixel count.

Load-bearing premise

The approach stands or falls on the assumption that every measured brightness sum is exactly the noiseless discrete convolution of the true image with a known, shift-invariant illumination spot, and that the sample outside the ROI has a known (usually zero) optical response.

Editorial extensions

If this is right

For an ROI with $R\cdot C$ pixels, exactly $R\cdot C$ scan positions are needed regardless of spot diameter, so the savings over scanning a surrounding border grow as the demanded resolution (and hence the spot-to-step ratio) grows.
The recovered image's resolution is set by the scan step, not by the spot size; the second experiment recovers a $0.1\,\mathrm{nm/pixel}$ image from a spot about one hundred times larger.
The peripheral area only needs to be made optically silent or known, not scanned, shifting the experimental burden from acquisition time to sample preparation.
The same equation-solving pipeline applies to any scanning microscope with a known, shift-invariant illumination spot, so it can be layered on existing super-resolution scanning schemes.

Reading between the lines

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

The exactness of the recovered image is an artifact of the noiseless discrete convolution model; under real detector noise the conditioning of $A$ will determine whether the solve is stable, so an error bound involving the condition number would turn the claim into a practical guarantee.
A testable extension is to apply SEDS to real confocal or STED image stacks with a measured point-spread function and a photobleached or absorbing boundary, comparing the solve against Wiener-filter deconvolution on the same ROI.
If the near-zero peripheral condition is only approximate, boundary leakage enters the right-hand side $b$; quantifying this sensitivity would give a required silence level for the preprocessing stage.
The 'unlimited resolution in principle' claim depends on an arbitrarily fine scan with no noise; in practice, scan precision and photon noise set a floor, so the practical claim is more naturally stated as resolution limited by scan mechanics rather than by spot size.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

A quick read of arXiv:1908.01284. It's a short proposal to replace the filtering step in DDS with a direct linear solve, scanning only the ROI while assuming the periphery is optically zero. The forward model in Eq. (1) is correctly written for a noiseless, shift-invariant, known spot. The observation that you only need to scan the ROI if the boundary is exactly known is the one practical kernel here, and it might save scans in a niche setting if the boundary condition can actually be enforced on a real microscope. That is the paper's only real contribution.

The rest is standard textbook deconvolution, and the paper is honest enough to cite Lay et al. for the linear algebra. The two simulations are self-consistency checks: the data are generated from the same convolution equation the solver inverts, so agreement at the 1e-6 level just confirms that the code solves its own forward model. No independent validation.

The soft spots are load-bearing. There is no noise model, no conditioning analysis, and no regularization. For a fixed spot, reducing the scan step makes adjacent rows of the system nearly identical, so the condition number of A blows up; any measurement noise is violently amplified. The claim that 'unlimited high resolution in principle' is therefore false for any real signal. The boundary assumption—that peripheral areas can be made optically zero—is asserted but not demonstrated, and it is essential to the square system. The efficiency comparison also ignores the cost of solving an R*C by R*C system, which for the 260x260 example would dominate the saved scan time.

Is it a serious thinker? Yes, in the limited sense that the model is set up clearly and the extreme-case caveat shows some care. But the missing analysis is not a minor omission; it kills the central promise. For a microscopy journal I would desk reject. The idea could perhaps be repackaged as a short note if the author adds realistic noise, computes condition numbers, and tests on a real dataset with an empirically measured PSF. As it stands, it's not there.

In short: not worth referee time in its current form. If you see it in a general venue, I'd skip it.

Referee Report

4 major / 4 minor

Summary. The paper proposes a super-resolution microscopy approach called SEDS (Solving Equations after Dense Scan). After preprocessing the sample so that areas outside the region of interest (ROI) are optically zero, the ROI is scanned densely with a known illumination spot. Each measured brightness value S(i,j) is modeled as a discrete convolution of the unknown high-resolution image E with the spot I, leading to a square linear system of R*C equations in R*C unknowns. The claimed contribution is that solving this system exactly recovers the sharp high-resolution image of the ROI without scanning the periphery, which is said to be more efficient than the existing DDS method and, in principle, capable of unlimited resolution. Two simulations, one with a 3x3-pixel spot and one with a 101x101-pixel spot, show that the recovered image matches the expected image to within tiny numerical differences.

Significance. If the method worked as claimed, it would offer a modest practical improvement over deconvolution-after-dense-scan by avoiding the need to scan peripheral areas. The manuscript clearly states the forward model and the linear-system formulation, and it explicitly acknowledges a degenerate case. However, the current evidence is only self-consistency: the simulated measurements are generated from the very same convolution equation that is then inverted. No noise, no conditioning analysis, no experimental data, and no validation against a different forward model are provided. The central claim of 'unlimited high resolution in principle' is not supported and is, in fact, contradicted by the ill-conditioning that inevitably arises as the scan step shrinks. As a result, the significance for real microscopy is currently unestablished.

major comments (4)

[Equation (1) and following paragraph] The assertion that the solution of the equation system is 'exactly the sharp and high resolution image' rests entirely on the assumption that Eq. (1) is an exact, noise-free representation of the measurement and that the R*C coefficient matrix is nonsingular. The paper only counts equations and unknowns to argue solvability; it provides no proof of nonsingularity or conditioning for realistic spots. Indeed, the admitted degenerate case of a constant spot shows that nonsingularity is not guaranteed, and smooth realistic spots (e.g., Gaussian) will produce near-singular matrices as the scan step becomes much smaller than the spot, making 'exact' recovery impossible in practice with any finite precision arithmetic.
[Simulation experiments (Fig. 4 and Fig. 5)] The two validation experiments are circular. The 'expected image' is used in a simulated imaging procedure that applies the same convolution relationship as Eq. (1), and then the proposed solver recovers the image by solving that same equation. The reported differences of 8.03e-12 and 1.11e-06 are merely numerical solve residuals, not evidence that the method would recover the true structure from a real microscope measurement. To substantiate the effectiveness claim, the authors need to test with noisy measurements, with an unknown or perturbed spot function, or with a more realistic optical forward model that differs from the inversion model.
[Final paragraph before the Summary] The claim that 'the proposed approach can actually achieve unlimited high resolution in principle' is unsupported and is mathematically misleading. For a fixed physical spot, reducing the scanning step increases the linear dependence among rows of the coefficient matrix A in A*x=b, so the condition number grows and the unregularized solution becomes increasingly sensitive to measurement noise. In any real measurement with noise, the reconstruction error will grow without bound as the step tends to zero. A conditioning analysis or at least a numerical condition-number study is required before such a claim can be made.
[Preprocessing stage (Section 2)] The boundary assumption that the optical property of peripheral areas is made zero (or known) is asserted without any experimental support or physical justification. In fluorescence and reflection microscopy, rendering an extended region completely non-emitting or non-reflecting is highly nontrivial. If the peripheral values are not exactly zero, the zero-padding used in Eq. (2) is an uncontrolled source of model error that can dominate the recovery. The paper should either demonstrate this preprocessing experimentally or analyze the sensitivity of the solution to violations of the zero-boundary assumption.

minor comments (4)

[Equation (2)] The bounds of the summation indices u and v are not explicitly defined; the spot size and the indexing conventions (e.g., negative indices) should be stated clearly.
[Abstract and Introduction] The abbreviation DDS is introduced but the related work citation 'Xie, Y. (2019). Improving the resolution of microscope by deconvolution after dense scan. arXiv.' is incomplete; a full arXiv identifier or journal reference should be provided.
[Figs. 3-5] The figures are referenced but not presented in the text; the reader cannot see the claimed visual similarity between the expected and recovered images. The 'averaged difference of pixels' metric should be defined, and ideally supplemented with standard image-quality metrics such as PSNR or SSIM.
[Efficiency comparison] The efficiency gain is quantified only by the number of scanned footprints. The computational cost of solving a dense R*C by R*C linear system, which is O((RC)^3) without special structure, should be discussed and compared with the cost of DDS filtering, especially for large images.

Circularity Check

2 steps flagged · score 6.0 of 10

The 'exact recovery' claim and the simulated validations are self-consistency checks: the equation system defines the measurements in terms of the unknown image, and the simulations generate data from that same convolution relation before solving it back.

self definitional [Introduction ('The solution of the equation system is exactly...') and Eq. (1)]
"The solution of the equation system is exactly the sharp and high resolution image. ... Such a relationship can be represented by the following equation system: S(i,j)=∑ ∑ I(u,v)∙E(i+u,j+v) (1)"

In Eq. (1), E is explicitly the expected image's pixels and S is defined as the sum over the product of the known spot I with E. Solving the square system Ax=b is therefore inverting the very relation that defines the measurements. The claim that the solution is 'exactly' the sharp image is true by construction whenever the noiseless convolution model is assumed; it is not an empirical finding. The simulations only confirm that a linear solve recovers the E that was inserted into Eq. (1).
other [Simulation experiments (around Fig. 5 and the reported averaged pixel differences)]
"The expected image is the simulation of the physical sample, and its size equals the total number of scanned footprint. It is used in a simulated imaging procedure, and also used for verifying the accuracy of recovery."

The 'simulated imaging procedure' is the same convolution model (Eq. (1)) that the proposed method later inverts; no other imaging model is defined in the paper. Generating measurements from an assumed model and then solving that model is a self-consistency test of the numeric solver, not an independent test of the physical claim. The small averaged differences (8.03e-12, 1.11e-06) therefore do not validate the method against real microscope data; they only show that the noiseless linear algebra is consistent with itself.

full rationale

The central derivation in SEDS is a linear deconvolution: Eq. (1) is a discrete convolution of the unknown image E with an illumination spot I, and the proposed method solves that equation system. The 'exactness' statement is definitional under the paper's assumptions, and the two simulations are generated from the same convolution relation that is then inverted, so the agreement between the recovered and expected images is a numerical self-consistency check rather than independent validation. This is the main circularity. The paper does not fit parameters to a subset of data, so there is no 'fitted input called prediction' pattern. The self-citation to Xie (2019) for DDS is used only as a baseline/preprocessing convention, not as a load-bearing uniqueness theorem, so it does not add to the score. No noise model, conditioning analysis, or real experimental test is provided; those are correctness risks beyond circularity. Given the central claim's support rests on inverting the same equation that defines both the measured quantities and the simulations, a moderate-to-high circularity score is warranted.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data; the only inputs are the simulated spot and scan geometry. The assumptions are all idealizations of a real microscope: zero known periphery, exact known shift-invariant spot, noiseless measurements, and an invertible convolution matrix. These are precisely the conditions that make the inverse problem trivial.

assumptions (4)

domain assumption Peripheral areas around the ROI can be made optically zero (or perfectly known).
The equation system in Eq. (1) only includes ROI pixels E(i,j); contributions from outside are set to known values. The paper states this can be done 'in various ways' but gives no physical demonstration.
domain assumption The illumination spot I is known exactly and is shift-invariant across the scan.
Eq. (1) uses the same I(u,v) at every scan position, and the simulations use a fixed spot. Any drift or calibration error breaks the model.
domain assumption The measured sum S(i,j) is noiseless and equals the sum of products in Eq. (1).
No noise model is introduced; the two experiments simulate data with exactly this equation, so the recovery error is only numerical rounding.
ad hoc to paper The R*C by R*C matrix A is nonsingular for realistic spots.
The paper proves no invertibility condition and dismisses the constant-spot counterexample as rare. This is needed for the claim that solving yields a unique exact image.

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Cite this review

Pith. "Pith review of Solving equations after dense scan to improve the resolutions of microscopes." pith.science (2026). https://pith.science/paper/TWSJAXO4

@misc{pith2026190801284,
  author       = {Pith},
  title        = {Pith review of: Solving equations after dense scan to improve the resolutions of microscopes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TWSJAXO4}},
  note         = {Machine review of arXiv:1908.01284}
}

read the original abstract

Super-resolution techniques overcome the diffraction-limit and get very high resolutions. A category of these techniques, e.g., STED achieves this by creating an illumination spot smaller than the Airy Disk. As a result, points are distinguishable even if they are as small as the spot. In order to further observe structures smaller than the spot itself, a technique called DDS scans the sample more densely, and recovers the expected image by deconvolution. In that technique, the deconvolution is achieved by filtering which requires some peripheral areas to be scanned together with the region of interest. In this study, an approach is proposed which has the same preprocessing stage as DDS. But it requires to scan only the region of interest. After that, an equation system is got from the scanned data. Finally, the expected image is recovered by solving the equation system. Experiments are performed on simulated data, and the results demonstrate the effectiveness of the proposed approach. The experiments also suggest that the proposed approach is more efficient than the existing one especially when the expected resolution is high.

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

Gonzalez, R. C. and R. E. Woods (2014). Digital Image Processing (3rd Edition), PEARSON

work page 2014
[2]

Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy

Hell, S. W. and J. Wichmann (1994). "Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy." Optics Letters 19(11): 780-782

work page 1994
[3]

Lay, D. C., S. R. Lay and J. J. McDonald (2015). Linear Algebra and Its Applications (5th Edition), PEARSON

work page 2015
[4]

Extracting super-resolution details directly from a diffraction -blurred image or part of its frequency spectrum

Sheffield, E. Y . (2019). "Extracting super-resolution details directly from a diffraction -blurred image or part of its frequency spectrum." PeerJ Preprints 7: e27591v27594

work page 2019
[5]

Visualizing and discovering cellular structures with super -resolution microscopy

Sigal, Y . M., R. B. Zhou and X. W. Zhua ng (2018). "Visualizing and discovering cellular structures with super -resolution microscopy." Science 361(6405): 880-887

work page 2018
[6]

Improving the resolution of microscope by deconvolution after dense scan

Xie, Y . (2019). "Improving the resolution of microscope by deconvolution after dense scan." arXiv

work page 2019

Pith tools

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

[1] [1]

Gonzalez, R. C. and R. E. Woods (2014). Digital Image Processing (3rd Edition), PEARSON

work page 2014

[2] [2]

Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy

Hell, S. W. and J. Wichmann (1994). "Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy." Optics Letters 19(11): 780-782

work page 1994

[3] [3]

Lay, D. C., S. R. Lay and J. J. McDonald (2015). Linear Algebra and Its Applications (5th Edition), PEARSON

work page 2015

[4] [4]

Extracting super-resolution details directly from a diffraction -blurred image or part of its frequency spectrum

Sheffield, E. Y . (2019). "Extracting super-resolution details directly from a diffraction -blurred image or part of its frequency spectrum." PeerJ Preprints 7: e27591v27594

work page 2019

[5] [5]

Visualizing and discovering cellular structures with super -resolution microscopy

Sigal, Y . M., R. B. Zhou and X. W. Zhua ng (2018). "Visualizing and discovering cellular structures with super -resolution microscopy." Science 361(6405): 880-887

work page 2018

[6] [6]

Improving the resolution of microscope by deconvolution after dense scan

Xie, Y . (2019). "Improving the resolution of microscope by deconvolution after dense scan." arXiv

work page 2019