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A Unified Theory of Image Scaling: Point Spread Functions, Discrete Kernels, and Scale-Spaces

Assaf Shocher

arXiv:2610.22472v1eess.IVcs.CV

TL;DR

Common resizing rules often lack optical grounding and fail basic consistency properties, while Gaussian scale-space does not exactly describe finite-aperture PSFs. This paper derives a camera-specific scaling family from the PSF, then reports optical validation and practical relevance to stability in super-resolution and classification. Its scope depends on assumptions including alias-free sampling and a calibrated, shift-invariant PSF.

  • Problem

    Common resizing rules generally lack optical grounding and fail composition, commutativity, and round trips, while Gaussian scale-space kernels cannot be exact finite-aperture PSFs.

    Source: §1 · p. 1

  • Method

    The paper derives a camera-specific scaling family from the PSF, treating pixels as secondary sources that reconstruct a band-limited scene before rescaling and recapture.

    Source: §1 · p. 2

  • Results

    The authors report complete derivation and optical simulation, tests on real images and noisy signals, and relevance to super-resolution and classification stability.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The paper argues that image transformations should inherit image-formation physics and advocates camera consistency as a computer-vision design criterion.

    Source: §9 · p. 8

    Optical prediction assumes alias-free sampling and one calibrated shift-invariant PSF, and regularization trades exact identities for noise control.

    Source: §8 · p. 8

Abstract

from arXiv · show

We derive the optically aligned system of image-scaling operators $\mathbf{K}_s$ that reproduces optical zoom (exact zoom-out, canonical zoom-in), by identifying a Huygens-like consistency principle in which each pixel acts as a secondary source. The system is determined in closed form by the camera's point spread function (PSF), itself the operator $\mathbf{K}_0$. The operators compose, $\mathbf{K}_{s'} \circ \mathbf{K}_s = \mathbf{K}_{ss'}$ (with band projection for down-up), and account for both upscaling and downscaling, which form a Moore-Penrose pseudoinverse pair, $\mathbf{K}_{1/s} = \mathbf{K}_s^\dagger$. We show that image scaling, as routinely practiced, is neither optically faithful nor internally consistent; for example, common anti-aliasing rules generally correspond to no camera, and two resizes by a factor of $1/2$ generally do not equal one resize by a factor of $1/4$. Gaussian scale-space kernels cannot be exact finite-aperture PSFs. Gaussian scale-space is an idealized special case of what emerges here: a generalized scale-space for sampled images, derived entirely from camera optics, bidirectional, and not limited to Gaussian kernels. The system goes beyond scaling and supports composable and invertible subpixel geometric transformations. We empirically validate its optical predictions with real captures and noisy optics simulations, and demonstrate its practical impact on the stability of tasks such as super-resolution and classification.

1. Introduction

The paper argues that common resizing rules are often neither optically faithful nor compositionally consistent, while Gaussian scale-space cannot represent a finite-aperture PSF exactly. It derives a camera-specific scaling family from the PSF, with pixels acting as secondary sources, and tests the construction’s optical and application-level relevance.

1. Introduction

  • Common resizing choices generally fail composition, commutativity, and round trips because they are prescribed without requiring them to describe real optics.Examples include bilinear, bicubic, Lanczos, and learned resampling.Source: §1 · p. 1
  • Gaussian scale-space derives a semigroup of Gaussian kernels from axioms, but finite-aperture PSFs are necessarily non-Gaussian.The paper presents its construction as joining idealized scale-space with photograph-oriented but cross-scale-inconsistent resizing.Source: §1 · p. 1
  • The paper situates its approach among generalized sampling, PSF matching, and super-resolution kernels, extending prior relations to cross-scale kernels, fractional scaling, and upscaling.Its cited continuous-convolution view connects discrete super-resolution kernels to interpolation [35].Source: §1 · pp. 1–2
  • Given a camera, the paper derives a consistent scaling family from its PSF across scales and grid phases, with exact coarsening, canonical refinement, and capture as an endpoint.The construction treats pixels as secondary sources whose signals reconstruct a band-limited scene before rescaling and recapture.Source: Figure 1 · p. 1, §1 · p. 2
  • The authors report complete derivation and optical simulation, tests on real images and noisy signals, and relevance to super-resolution and classification stability.Figure 1 depicts the secondary-source interpretation and the reverse route, which restores resolution but not discarded frequencies.Source: Figure 1 · p. 1, §1 · p. 2

2. Preliminaries

The preliminaries connect camera image formation and conventional resizing to the paper’s optical scaling construction. They also introduce scale-space and generalized inverses as mathematical context for the later bidirectional operators.

2.1. The PSF and image formation

  • A camera convolves continuous scene radiance with its PSF and samples the result; the PSF combines lens blur and pixel integration, and its Fourier transform is the OTF.The construction uses the camera-visible frequency set and extends beyond separable one-dimensional cases.Source: §2.1 · p. 2

2.2. Discrete downscaling kernels

  • For integer reduction, conventional downscaling convolves samples with a discrete kernel and then subsamples; the kernel is related to the camera PSF.Figure 2 depicts projected output locations, neighboring samples, signed-offset weights, and their weighted sum.Source: §2.2 · p. 2, Figure 2 · p. 2

2.3. Scaling sampled images

  • A sampled-image resize projects output pixels onto input coordinates, gathers neighbors, weights them by an interpolant, and sums; downscaling stretches and normalizes the interpolant for anti-aliasing.The default output length is the ceiling of scale times input length.Source: §2.3 · p. 2

2.4. Image scaling is a continuous convolution

  • The four discrete resizing phases can be expressed continuously by lifting samples to a bed of nails, convolving with a continuous kernel, and resampling.For integer reduction, this continuous process collapses to a discrete kernel.Source: Figure 3 · p. 2, §2.4 · p. 3

2.5. Scale-space

  • Scale-space studies progressively smoothed image families with a semigroup, causality, and self-similarity; symmetric isotropic convolution semigroups yield stable kernels, with finite variance selecting the Gaussian.The cited lattice counterpart is the discrete Gaussian.Source: §2.5 · p. 3

2.6. Generalized inverses

  • The Moore–Penrose pseudoinverse maps attainable outputs to minimum-norm preimages; for convolution, its multiplier inverts nonzero frequencies and is zero at kernel zeros.Convolutional generalized inverses may differ from the pseudoinverse by a null-space term.Source: §2.6 · p. 3

3. Consistent Scaling Operator

The paper defines scaling through a Huygens-like process: pixels act as sources that reconstruct a band-limited scene for rescaling and recapture. Under stated optical assumptions, the camera PSF determines the scaling operator, while a canonical kernel choice and practical regularization address realization.

3. Consistent Scaling Operator

  • Pixels act as secondary sources whose superposition reconstructs the band-limited scene, which is then recaptured by the camera as if sampling had not intervened.The paper presents this as analogous to Huygens’ wave principle.Source: §3 · p. 3

3.1. The consistency equation and solution set

  • Assuming alias-free sampling and that scaling reproduces the matching optical-zoom capture, the operator follows by reconstructing through the camera pseudoinverse, stretching, and recapturing.The reconstruction is the canonical camera-visible continuous scene, and the relation recovers prior discrete downscaling at one grid phase for alias-free integer factors [33, Sec. 2].Source: §3.1 · p. 3
  • A consistent kernel must realize the operator on every translated grid; suppressing noncentral sampling replicas yields a solution with a free multiplier at camera-PSF zeros, and the canonical choice sets it to zero.If the ratio is unbounded near OTF zeros, the exact multiplier is defined on its natural domain rather than necessarily as an ordinary convolution kernel.Source: §3.1 · pp. 3–4
  • No nonnegative integrable pure-stretch family can reproduce optical coarsening at both reciprocal integer scales, even near DC; the obstruction is PSF nonnegativity.The result is not caused by finite kernel support.Source: §3.1 · p. 4

3.2. Realization

  • Realization can use a regularized normal equation, but finite regularization controls noise amplified at weak camera frequencies while making structural identities approximate.Exact bounded identities instead use a thresholded camera symbol; nesting-dependent claims also require the modified visible set to remain dilation-nested.Source: §3.2 · p. 4
  • Different consistent kernels can yield the same output, while adding frequencies can produce different outputs; Table 1 summarizes the uniqueness cases under closed-band and dilation-nested visibility conditions.Figure 5 illustrates kernels that agree at pixel centers and replica-reading nonmembers that add fine-Nyquist aliases.Source: Table 1 · p. 4, Figure 5 · p. 4

4. Properties of the Operator Family

The operator family obeys a conditional scale law: compatible steps compose, while mixed down-up chains retain a band projection. Its canonical kernels and completions are constrained by uniqueness, endpoint behavior, and the visible frequency band.

4.1. Composition

  • The composition tests find that baselines fail while the proposed operators close to numerical precision.The figure compares cycle, unequal-step composition, and order-swapping tests.Source: §4.1 · p. 5, Figure 6 · p. 5
  • For dilation-nested bands, compatible chains compose to the product scale; mixed-direction chains retain a projector, and middle cameras cancel unless an earlier step discarded needed frequencies.The family also commutes for two coarsenings or two refinements; the mixed-direction case follows the projector law.Source: §4.1 · p. 5
  • Sampled composition on every translated grid forces the coarse-limit support to be a band, and the bounded ratio converse also composes.The converse excludes finite-support interpolants; the limit need not be physical, and actions outside the band remain free.Source: §4.1 · p. 5

4.2. Kernels, actions, and completions

  • The operator is unique, and its limits connect image formation, identity on band-limited signals, and camera pseudoinversion.As scale tends to zero, resizing becomes PSF convolution followed by sampling; at scale one, many kernels can act as identity on band-limited signals.Source: §4.2 · pp. 5–6
  • The canonical refinement is the matching downscaler’s least-squares pseudoinverse; a down-up cycle gives the closest recoverable-subspace approximation.Other completions can add undetermined frequencies, whereas the canonical completion adds none.Source: §4.2 · p. 5
  • For nested bands, one inter-image zoom kernel determines every canonical scale through a product construction.This product also appears as a refinement-mask infinite product and describes a physical camera when the recovered PSF is nonnegative.Source: §4.2 · p. 6

5. Generalized Scale-Spaces

The camera-defined family generalizes scale-space beyond Gaussian smoothing, retaining composition and bidirectionality without requiring self-similar kernels. Gaussian behavior is an idealized special case, while refinement involves deconvolution rather than smoothing.

Generalized scale-space

  • Under isotropy, regularity, and finite variance, classical scale-space is Gaussian under its stated axioms.Those axioms include linearity, shift invariance, the semigroup law, scale invariance, and causality.Source: §5 · p. 6
  • Camera kernels retain a conditional scale-product law but need not be self-similar or preserve local extrema; mixed chains retain the intermediate band.The generalized family is defined through linearity, coordinate-adjusted shift invariance and composition, plus band and bidirectionality conditions.Source: §5 · p. 6
  • For an idealized Gaussian PSF, coarsening kernels are Gaussian, but refinement requires negative variance time and becomes band-limited deconvolution.The operators cease to form a group when coarsening loses information; the no-cutoff idealization is exceptional.Source: §5 · p. 6

6. Beyond scale: camera-coordinate actions

The camera-conditioned construction extends from scaling to linear geometric actions on captures. Composition and inversion hold in camera coordinates when the action bijectively preserves the camera-visible scene space.

Camera-coordinate actions

  • A linear geometric action on continuous scenes induces a camera-conditioned reconstruct-transform-resample operator on captures.Translations preserve every visible set; rotations do so when the camera’s visible band is rotation-invariant, while general warps use spatially varying kernels.Source: §6 · p. 6
  • When an action bijectively preserves the camera-visible scene space, the intermediate round trip cancels and composition and inversion hold in camera coordinates.The figure tests inversion after a 15° rotation and a (0.37, 0.21) px shift.Source: §6 · p. 6, Figure 9 · p. 7

7. Conventions Audit and Validation

The audit finds that common resizing conventions generally lack optical fidelity or internal consistency, whereas camera-derived scaling predicts optical captures more closely. Application tests report benefits in multistep stability and down-up feature matching, with the fixed-PSF model remaining an approximation for real lenses.

7.1. Inconsistency of common conventions

  • Pure anti-aliasing stretch is not an optical scale law for nonnegative integrable PSFs, and common finite-support interpolants generally fail to invert under up-then-down resizing.Gaussian AA also uses the wrong variance increment; finite-pupil cameras cannot produce exact Gaussian or isotropic stable scale-spaces.Source: §7.1 · p. 7

7.2. Optical alignment and robustness

  • In optical simulations, predicted far captures have relative RMSE 0.15 and 0.20, versus cubic AA’s 0.53 and 0.91 and same-band Fourier’s 0.59 and 1.05.Matched AA ties at 2d but reaches 0.24 at 3d; the unregistered power-spectrum ratio follows the predicted kernel until noise dominates.Source: §7.2 · p. 7
  • RealSR evaluates kernel transfer from ×2 to ×3/×4 captures, but per-pair fits reach only 32–33 dB amid registration, exposure, and noise; a fixed pixel-domain PSF across zoom is approximate.Center MTF50 varies about 1% with focal length under diffraction and 7–11% wide open.Source: §7.2 · pp. 7–8

7.3. Effects on applications

  • At ten training steps, band-limited bicubic gains 2.58–5.02 dB, while each arm’s best result gains 0.08–0.17 dB.The simplified ZSSR test changes only the training downscaler; longer chains test robustness, not optimal super-resolution.Source: §7.3 · p. 8
  • Across one to five resize steps, library outputs and top-1 predictions shift, while the proposed families change negligibly; this measures path stability, not accuracy.The test uses 50 crops resized from 2047 to 255.Source: §7.3 · p. 8, Figure 10 · p. 8
  • At 32× reduction, the down-up pair gains 17.1, 9.1, and 5.9 points over linear, cubic, and Lanczos-3 AA/interpolation pairs, respectively.Direct low-resolution scores nearly tie; the pair realizes ideal retained-band projection rather than a new interpolator.Source: §7.3 · p. 8

8. Limitations

The evaluation has practical boundaries: real scale pairs inherit registration resampling, while finite tests truncate kernels, extend boundaries, and exclude margins. Optical prediction assumes alias-free sampling and a calibrated shift-invariant PSF, and omits several image-processing stages.

8. Limitations

  • Finite tests truncate kernels, extend boundaries, and exclude margins; real scale pairs also inherit registration resampling.Optical prediction assumes alias-free sampling, which a finite aperture does not guarantee, and uses one calibrated shift-invariant PSF.Source: §8 · p. 8
  • The HPatches down–up test compares homography success after reduction and restoration, with success defined as median error below five original pixels.Finite testing also entails margin exclusion and kernel truncation.Source: Figure 11 · p. 8, §8 · p. 8
  • The model omits demosaicing, tone curves, and compression, and kernel-transfer gains depend on calibration.Lenses can vary with field and focus.Source: §8 · p. 8

9. Conclusion

The conclusion advocates camera consistency as a design criterion and frames scale transformations as a family determined by one camera’s optics. The discussion develops that family across notation, grid alignment, reduction, consistency, and regularized refinement.

A. Notation (Sections 2–7)

  • The camera PSF h = K0 defines the OTF and visible-frequency set, while Ks denotes scale-s operators and K∞ the canonical un-capture.The notation also distinguishes the centered grid, camera-visible bands, and regularization level.Source: A. Notation (Sections 2–7) · p. 11

B. Extended discussion: connections across fields

  • The paper’s scale-space follows one possibly non-Gaussian camera PSF, whose visible bands determine kernels and the canonical reverse action under stated sampling and visibility conditions.The conclusion advocates camera consistency and proposes that blind super-resolution estimate a PSF-represented all-scale family.Source: §9 · p. 8, B. Extended discussion: connections… · p. 12

C. The projected grid (Section 2.3)

  • A resize requires both scale, which sets output spacing, and integer size, which sets sample count; conflating them can shift the grid and break size round trips.For N0 = 6, s = 0.47, and N = 3, shortcut grids shift the center by +0.191 and −0.372 pixels.Source: C. The projected grid (Section 2.3) · p. 12, Figure 12 · p. 12

D. Integer-factor reduction (Eq. 7)

  • Integer-factor reduction places samples at the centers of a-pixel blocks, with fractional center offset entering the taps and integer offset entering padding.For rounded output sizes, the general projected-grid offset replaces the usual block-center offset.Source: D. Integer-factor reduction (Eq. 7) · p. 13

E. Solving the consistency equation (Section 3.1)

  • The consistency equation determines the scale kernel up to a camera-null convolution; requiring all translated grid phases is stronger than requiring only their average.Free multiplier values lie outside source replicas, not on recoverable new scene frequencies.Source: E. Solving the consistency equation… · p. 14

F. Wiener realization (Section 3.2)

  • Tikhonov regularization uniquely damps weak camera frequencies, while finite regularization makes structural identities approximate.Exact thresholded identities instead use the same modified camera in both numerator and denominator.Source: F. Wiener realization (Section 3.2) · p. 14

G. Noisy refinement (Section 3.2)

  • At noise 4/255, exact, regularized, and bicubic refinement yield −4.03, 36.72, and 32.16 dB, respectively.Noiseless canonical refinement matches the input-determined target within 4 × 10^-14 per channel; regularization controls weak-frequency noise amplification at the cost of exact inverse and composition identities.Source: G. Noisy refinement (Section 3.2) · p. 15

H. Identifiability from sampled data (Corollary 4.8)

This section examines when sampled data identify kernels, the constraints composition imposes across sampling phases, and the distinction between mathematical consistency and optical admissibility. It also covers experimental checks of resize paths and geometric transforms, controls and calibration for HPatches and RealSR, and lens measurements assessing the practical scope of the fixed-PSF assumption.

H. Identifiability from sampled data (Corollary 4.8)

  • A sampled slice is identifiable from its complete bi-infinite taps only under the stated conditions: irrational-scale offsets are dense, while rational-scale offsets require a band limit for injective recovery.For reduced rational scale p/q′, the offsets form a translate of p−1Z and determine a square-integrable slice with centered Fourier cutoff at most pπ; finite taps cannot determine an arbitrary continuous slice. A capture pair identifies a kernel only when its multirate convolution map is injective within the chosen class.Source: H. Identifiability from sampled data… · p. 15

I. Proof of Theorem 4.1 (Composition)

  • The scale operators compose on the full lattice, and the reverse-scale operator is the Moore–Penrose pseudoinverse; cropped finite matrices need not preserve that pairing.The round-trip products project onto the active source and output bands. The lattice identities require compatible boundaries for finite implementations.Source: §I · p. 16

J. Proof of Proposition 4.6 (Uniqueness)

  • At rational scales, distinct consistent kernels can have identical taps when their difference lies in replica-free spectral gaps; at irrational scales, dense offsets make shared taps determine continuous kernels uniquely.The rational-scale construction applies when source replicas leave an open spectral gap.Source: J. Proof of Proposition 4.6 (Uniqueness) · p. 17

K. Proof of Proposition 4.7 (Endpoints and Midpoint)

  • At scale 1 the multiplier is the identity on attainable captures, while one-slice telescoping recovers the slice under dilation nesting; a spectral hole can obstruct recovery.The endpoint statements concern pointwise-multiplier limits, not operator-norm convergence or stability of an unbounded deconvolver.Source: K. Proof of Proposition 4.7 (Endpoints… · p. 17

L.2. All-phase sampling (Theorem 4.5)

  • Under the all-phase sampled-composition hypotheses, the coarse limit determines the multiplier ratio and a dilation-nested band B = (−β, β), with β ≤ π; conversely, the stated bounded normalized multipliers yield composition.Replica nonoverlap limits the band to β ≤ π; the converse assumes H is bounded, Hermitian, continuous at zero with H(0) = 1, and each Fs is bounded.Source: §L.1 · p. 17, §L.2 · p. 18

L.3. What the hypotheses do (Theorem 4.5)

  • Composition alone does not ensure optical admissibility: ideal pure widening has a signed sinc PSF, and no continuous all-phase pure-stretch law can match two successive optical coarsenings for a normalized nonnegative L1 PSF.Optical pupil realizability and nonnegativity must be imposed separately; the obstruction requires no finite-support or analyticity assumption on the kernel.Source: §L.3 · p. 19, §L.5 · p. 19

M. Experimental details: geometric transformation (Figure 7); O. Experimental details: HPatches (Section 7.3)

N. Experimental details: library audit (Figure 10)

  • Each consistent Gaussian, annular-pupil, and astigmatic-pupil control showed 0.0% flips, versus Pillow bilinear’s 16.7% and 16.0% chain-flip rates.Matched-RMS white noise often caused as many or more flips; camera controls had only numerical path errors and no flips.Source: N. Experimental details: library audit… · p. 21

O. Experimental details: HPatches (Section 7.3)

  • Across 1,044 HPatches comparisons, float32 consistent-pair outputs differed from direct ideal Fourier low-pass projection by at most 2.76 × 10−7 per pixel.The reported gain reflects retained-band preservation, not improvement over ideal sinc or a distinct kernel-dependent down–up advantage; bitwise SIFT/RANSAC equality was not asserted.Source: O. Experimental details: HPatches… · p. 21

P. Experimental details: RealSR (Table 2)

  • RealSR’s ×2-fitted finite-support kernel was used for calibrated, not blind, predictions at ×3 and ×4, with the lower disjoint block selecting the ridge weight.The fitted reference is limited to the chosen finite-support model and is not a guaranteed held-out ceiling; scores use PSNR = −20 log10(RMSE) on [0, 1].Source: P. Experimental details: RealSR (Table… · pp. 21–22

Q. Experimental details: real lenses (Section 7.2)

  • Stopped-down center MTF50 varied by 0.9% across tested focal lengths for each lens, supporting approximate center-sharpness stability but not equal OTFs.Wide-open center variation was 7–11%, with greater variation in the corners.Source: Q. Experimental details: real lenses… · p. 22

R. Experimental details: the computed camera (Figure 8)

The camera was derived from a pupil rather than fitted to a resizing kernel, and its scale predictions were compared with alternative resizers. The consistent family tracked the camera more closely at the off-anchor scale and composed with much smaller error than the anti-aliased alternatives.

R. Experimental details: the computed camera (Figure 8)

  • A pupil-derived camera with defocus and spherical aberration was tested on a shaped-noise scene, using camera-ratio operators and cubic-AA, matched-AA, and Fourier comparators.The pupil was P(u) = exp[2πi(0.18u^2 + 0.10u^4)] for |u| ≤ 1, with full-pitch pixel integration.Source: R. Experimental details: the computed… · p. 22
  • At 1/2 and 1/3, consistent-family relative RMSE was 0.152 and 0.200, versus 0.525 and 0.906 for cubic AA, 0.152 and 0.235 for matched AA, and 0.592 and 1.045 for Fourier.At 1/3, consistent RMSE was 15% below matched AA in this single seeded scene/noise realization.Source: R. Experimental details: the computed… · p. 23
  • The reported 2σ/ std(I1/2) value of 0.20 is not an exact prediction-noise floor, which depends on operator row norms.The noise standard deviation was 0.005 on [0, 1], and shot noise was omitted.Source: R. Experimental details: the computed… · pp. 22–23
  • Composition relative errors were 3.8 × 10^-5 for consistent, 0.154 for cubic AA, and 0.244 for matched AA.The Fourier arm's finite-grid residual was 3.10 × 10^-3; on a compatible periodic grid, direct evaluation closed within 9.5 × 10^-14 maximum absolute error.Source: R. Experimental details: the computed… · p. 23
  • The measured spectral ratio followed |bK0(ν)/bK0(ν/2)|^2 until noise dominated near 0.7 of coarse Nyquist, unlike cubic AA and Fourier below cutoff.The comparison used Welch estimates; the ratio was compared with the camera-derived prediction.Source: R. Experimental details: the computed… · p. 23

S. Clean one-shot upscaling (Table 3)

The experiments compare camera-derived refinement with common interpolation and test whether a consistent scale law transfers better than stretching from a shared anchor. Camera-derived refinement leads the reported clean-upscaling comparisons, while transfer gains are not uniform across noisy cases.

S. Clean one-shot upscaling (Table 3)

  • Clean one-shot scores use unquantized observations, and finite sampled degradations still include aliasing, truncation, and boundary effects; the full-reflection Fourier implementation is slower than local interpolation.The results therefore do not establish bicubic-level runtime.Source: S. Clean one-shot upscaling (Table 3) · p. 23
  • Camera-derived refinement applies the camera transfer-function ratio on source-visible frequencies, with Gaussian and bicubic-AA anchors evaluated on centered output grids.For the bicubic-AA anchor, the refinement multiplier telescopes to 1/U(ω).Source: S. Clean one-shot upscaling (Table 3) · p. 23
  • For quantized Set5 Gaussian ×2, exact refinement scored 10.70 dB clipped (3.63 raw), while fixed regularization reached 32.59 dB clipped (32.55 raw).The exact refinement was ill-conditioned because it amplified quantization error; the regularization heuristic used no target pixels and was not necessarily optimal.Source: S. Clean one-shot upscaling (Table 3) · pp. 23–24
  • Camera-derived refinement beats bicubic and Lanczos-3 in all 30 Set5 image–degradation–scale combinations and leads dataset means in all six BSD100, Set5, and Set14 conditions.The comparison uses fixed interpolation baselines, not other kernel-aware super-resolution or optimal natural-image reconstruction.Source: S. Clean one-shot upscaling (Table 3) · pp. 23–24, Table 4 · p. 24, Table 5 · p. 24

T. Shared-solver kernel transfer

  • The shared-solver transfer test gives both laws the same ×2 anchor and compares their ×3 and ×4 predictions using the same forward operator, grid, boundaries, solver, and regularization.It isolates the scale law and is neither blind super-resolution nor another RealSR experiment; the Gaussian realization is approximate because unprojected Gaussian sampling does not meet alias-free identities.Source: T. Shared-solver kernel transfer · pp. 24–25
  • With a common anchor and solver, transfer favors the consistent law in these means, but not uniformly under noise: two images favor stretch for noisy pupil ×3, whose interval includes zero.The test supplies the same exact ×2 relative kernel to both scale laws.Source: T. Shared-solver kernel transfer · pp. 24–25

U. Sensitivity to estimated-kernel error

The sensitivity tests examine how errors in an estimated camera kernel affect scaling results. Beyond alias-free sampling, algebraic composition can persist even as optical prediction accuracy fails.

U. Sensitivity to estimated-kernel error

  • The sensitivity test uses 20 random seeds and sets the ×2-anchor error to 1%, 2%, or 5%, keeping the grid, solver, regularization, and paired noise draws fixed.No target or score selects the perturbations.Source: U. Sensitivity to estimated-kernel error · p. 25
  • Some 2% pupil ×3 anchor errors reverse the ranking, while small smooth errors preserve most gains; error magnitude alone does not determine the result.Perturbed anchors need not be nonnegative spatial PSFs, and no particular kernel estimator is modeled.Source: U. Sensitivity to estimated-kernel error · p. 26
  • A coherent 10% Gaussian-width bias produces 6.22% weighted anchor error and noisy ×3/×4 gains of −0.88/−0.88 dB.The bias nearly cancels AA stretching’s Gaussian under-blur, but is not typical random kernel-estimation noise.Source: U. Sensitivity to estimated-kernel error · p. 26

V. Optical prediction beyond alias-free sampling (Section 8)

  • The optical test compares predicted captures with independently rendered captures and compares two half-steps against one quarter-step.It varies optical cutoff and evaluates clean and input-noisy scenes; the one-dimensional signals are not an image benchmark.Source: §V · p. 26
  • At cutoff 3π, projected scaling has 20.86% quarter-scale relative error versus 0.61% for the full-band heuristic, whose path discrepancy is 6.16%.At cutoff 0.9π, both methods predict the optical target and compose to numerical precision; no source-sample operator can recover distinct captures from aliased scenes with identical source samples.Source: §V · p. 26

W. Learned super-resolution (Section 7.3, Figure 9)

The learned super-resolution experiments test whether a camera-derived scale law improves robustness across repeated learned resizing. Additional figures illustrate properties and boundaries of the camera-based scaling operators.

W. Learned super-resolution (Section 7.3, Figure 9)

  • The controlled ZSSR-style test changes only the training downsampler; network, budget, initialization, crop schedule, and bicubic upsampling are shared.Each image is reduced at factors 2, 3, or 4, and a freshly trained network reverses each of 1–10 repeated steps without external training data or back-projection.Source: W. Learned super-resolution (Section… · p. 26
  • Consistent degradation does not ensure improvement with more steps because learning, boundary, sampling, and finite-support errors remain.On Set5, Gaussian ×4 at ten steps falls below bicubic interpolation; the cause of that decline is not uniquely identified.Source: W. Learned super-resolution (Section… · p. 29
  • Across BSD100, Set5, and Set14, the camera law favors matched AA at late steps, but not one-shot bicubic at every step.The main curves cover 100 BSD100 images; Figures 15 and 16 cover five Set5 and 14 Set14 images.Source: W. Learned super-resolution (Section… · p. 29, Figure 15 · p. 29, Figure 16 · p. 29

X. Additional figures (Sections 2–7)

  • Consistent operators close the tested round trip to numerical precision, unlike same-kernel baselines; a 3% simulated half-scale kernel error yields about 1% camera error.The camera estimate uses a telescoping product, and route disagreement diagnoses inconsistency rather than providing a universal error bound.Source: Figure 19 · p. 31, Figure 25 · p. 32

W. Learned super-resolution (Section 7.3, Figure 9)

  • On Set5 ordinary bicubic observations, the camera law overtakes AA from steps 3, 4, and 5 at ×2, ×3, and ×4, with ten-step gains of 3.39, 3.26, and 2.35 dB.AA still has the higher best mean at every scale; the result supports greater many-step robustness, not universal superiority or exact sampled consistency under aliasing.Source: Figure 17 · p. 30, W. Learned super-resolution (Section… · p. 31

X. Additional figures (Sections 2–7)

  • The camera endpoint depends on the anchor scale, while ratios can diverge near source zeros; the shown 3% denominator cutoff makes the figure a regularized illustration, not an exact identity.The figure depicts telescoping kernels for bicubic, tent, and box anchors.Source: §X · p. 32
  • Alias-free sampling does not guarantee a nonnegative physical PSF: truncated defocus, motion, and coma profiles can be nonphysical, while slit-aperture sinc2 has interior zeros and ringing.The coma-like profile is asymmetric beyond symmetric conventions.Source: Figure 21 · p. 32
  • At scale 2, exact kernels are fixed on the source band but retain target-band freedom between source and target bands.The displayed target-band freedom vanishes for s ≤ 1, though the full null space need not.Source: Figure 23 · p. 32
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