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Image Denoising Using Lower Semi-Frames

Hemalatha M, P. Sam Johnson

arXiv:2609.27893v1eess.IVcs.CV

TL;DR

Blind image denoising must estimate unknown noise from the same observation that contains meaningful high-frequency image information. This paper uses a directional lower semi-frame and covariance-whitened noise estimation, achieving a mean PSNR gain of 7.45 dB and a mean SSIM gain of 0.367 across its experiments.

  • Problem

    Blind denoising remains challenging because noise must be estimated from the same observation that contains meaningful high-frequency image information.

    Source: §I · p. 1

  • Method

    The framework regularizes an infinite directional lower semi-frame with a resolvent and estimates noise via covariance-whitened chi-square statistics from four channels’ lower-tail energies.

    Source: §I · p. 1, §IX · p. 7

  • Results

    3.28% mean relative sigma error, 7.45 dB mean PSNR gain, and 0.367 mean SSIM gain were achieved across three images and four noise levels.

    Source: §IX · p. 7

  • Takeaways & Limitations

    At noise level 30/255, PSNR gains differed by image content: 11.27 dB for moon and 6.39 dB for astronaut.

    Source: §IX · p. 7

    After sigma estimation, structure/noise discrimination remains a limitation, with smaller SSIM gains for texture-rich content.

    Source: F. Component scope, limitations, and… · p. 6

Abstract

from arXiv · show

A blind image denoising framework based on an infinite directional lower semi-frame (DLSF) is proposed for additive white Gaussian noise. The model employs scale-dependent directional analysis with resolvent regularization of the unbounded semi-frame operator. Noise variance is estimated directly in the DLSF domain by modeling the joint covariance of four directional difference channels and applying covariance whitening to obtain a chi-square statistic. A lower-tail moment estimator provides blind noise estimation without median absolute deviation. The estimated noise level is incorporated into channel-wise Wiener-type shrinkage and canonical-dual synthesis, followed by a data-consistent iterative reconstruction with automatic stopping. Experiments on three standard grayscale images at noise levels 15--30 yield a mean relative noise-estimation error of 3.28\%, with average improvements of 7.45 dB in PSNR and 0.367 in SSIM. At 30/255 noise, the estimation error decreases to 1.73\%, with a mean PSNR gain of 8.31 dB. Results demonstrate effective noise suppression and structural preservation, with the strongest performance on smooth and edge-dominated images.

I. INTRODUCTION · II. INFINITE DIRECTIONAL LOWER SEMI-FRAME

The paper presents blind denoising in an infinite directional lower semi-frame (DLSF), using the same representation for noise estimation, coefficient processing, and reconstruction. The second section specifies directional sum/difference channels and an unbounded scale symbol that distinguishes the construction from conventional tight frames.

I. INTRODUCTION

  • Blind denoising is challenging because unknown noise must be estimated from the same observation that contains meaningful high-frequency image structure.The paper addresses this by deriving noise statistics from the directional lower semi-frame representation.Source: §I · p. 1
  • The proposed novelty is an infinite DLSF with scale weights lacking a finite global upper frame bound, stabilized through resolvent-based regularization.The directional filters are not claimed as new; the contribution is their formulation in the unbounded lower semi-frame setting.Source: §I · p. 1
  • Explicit cross-direction covariance modeling and whitening yield a chi-square statistic for estimating noise standard deviation without relying on median absolute deviation.The covariance treatment accounts for correlation among directional noise coefficients from filters applied to the same noisy image.Source: §I · p. 1
  • Channel-specific noise calibration, local Wiener-type shrinkage, canonical-dual synthesis, and data-consistent fixed-point iteration form the denoising and reconstruction pipeline.Automatic convergence control avoids prescribing a fixed number of reconstruction passes.Source: §I · p. 1
  • The study assesses the pipeline’s quantitative and visual behavior on standard images and multiple noise levels, not state-of-the-art superiority over nonlocal or learned denoisers.The paper says broader benchmarking and ablation studies would be required for such a superiority claim.Source: §I · pp. 1–2

II. INFINITE DIRECTIONAL LOWER SEMI-FRAME

  • Following Shen et al., in using 45°-separated orientations, the construction uses horizontal, vertical, main-diagonal, and anti-diagonal pairwise sum/difference responses.Unlike their Parseval tight-frame construction and masks, this system requires a canonical dual, resolvent regularization, and blind coefficient statistics for the lower semi-frame setting.Source: §II · p. 2
  • The antisymmetric directional channels are differences central to the blind noise estimator because their responses to white noise can be characterized exactly.The symmetric channels emphasize locally averaged directional content.Source: §II · p. 2
  • Because b_j ≥ 1 gives the symbol a positive lower bound while b_j → ∞ makes its essential supremum unbounded, the system is a lower semi-frame,.The unboundedness holds when infinitely many bands have nonzero measure.Source: §II · p. 2

A. Derivation of the canonical dual system

The section derives the canonical dual response from the inverse lower semi-frame operator and explains how the DLSF’s directional and scale structure grounds the later covariance model and synthesis.

A. Derivation of the canonical dual system

  • Because M(ω) > 0 almost everywhere, the inverse operator on its natural domain yields the canonical dual response through S^-1, not an ad hoc normalization.The response r,j/M is identified as the Fourier-domain consequence of canonical-dual synthesis.Source: A. Derivation of the canonical dual… · p. 2
  • The DLSF fixes scale, direction, symmetric/antisymmetric decomposition, and reconstruction symbol; its four-direction organization follows Shen et al.,, not an independent novelty.This structure ties the covariance model and canonical-dual synthesis to the same representation.Source: A. Derivation of the canonical dual… · p. 3
  • Replacing DLSF with an unspecified transform breaks that link, removing directional splits mixes edge orientations, and uniformly bounded weights erase the role of lower semi-frame regularization.The directional split supports interpretable noise statistics and shrinkage responses; the unbounded scale sequence distinguishes the method from a conventional frame model.Source: A. Derivation of the canonical dual… · p. 3

III. RESOLVENT REGULARIZATION AND DLSF NOISE

Resolvent regularization bounds the scale response of the unbounded DLSF operator and determines each channel’s noise gain. Covariance whitening of correlated directional channels provides the chi-square statistical basis for blind noise estimation.

III. RESOLVENT REGULARIZATION AND DLSF NOISE

  • Resolvent regularization keeps the scale multiplier bounded and calibrates each DLSF channel’s noise gain.Without it, high-scale responses would increasingly dominate as discretization is refined, invalidating the finite noise calibration.Source: §III · p. 3
  • Whitening the correlated four-channel directional coefficients yields a Mahalanobis energy with a central chi-square noise-only model.Treating the coefficients as independent would misrepresent their covariance and invalidate that chi-square calibration, potentially biasing the estimator.Source: §III · p. 3

IV. BLIND COVARIANCE-WHITENED CHI-SQUARE NOISE ESTIMATION

The section estimates the unknown noise variance blindly from lower-tail whitened DLSF energies, limiting contamination from image structure. Its model-based truncated-moment procedure uses multiple tail fractions and bandwise fusion rather than MAD.

IV. BLIND COVARIANCE-WHITENED CHI-SQUARE NOISE ESTIMATION

  • Lower-tail whitened DLSF energies estimate the noise variance needed by the denoiser while reducing inflation from structure-dominated coefficients.Without this estimate, the algorithm would require externally supplied σ; removing lower-tail selection could inflate variance and cause excessive shrinkage.Source: §IV · p. 3
  • For each band j, the observed energies Qj[m] are sorted, and the lowest fraction p is retained as set Lj,p for the bandwise estimate.Source: §IV · p. 3
  • The multi-p rule balances contamination from edges and textures against estimator variance when too few tail samples remain.Omitting the chi-square truncation correction would bias the sample mean downward relative to the full rather than truncated expectation.Source: §IV · p. 3

V. NOISE-CALIBRATED DLSF DENOISING

The denoising stage calibrates noise separately for each DLSF channel, applies local Wiener-type attenuation, and uses canonical-dual synthesis to reconstruct the modified channels.

V. NOISE-CALIBRATED DLSF DENOISING

  • Channel-specific noise standard deviations derived from effective transfer functions avoid mismatched denoising across channels with different gains.Using one global variance would mismatch denoising strength across scales and directions.Source: §V · p. 4
  • Local Wiener adaptation estimates coefficient statistics spatially, while nonzero minimum gains reduce suppression of small, potentially meaningful coefficients.The method estimates local means and variances for symmetric channels and a local signal variance for the generalized Wiener gain.Source: §V · p. 4
  • Starting from the blind estimate σ0, the continuation schedule decreases σk to a floor of cσ0, and automatic termination checks relative ℓ2 change and maximum pixel change.The continuation parameters satisfy 0 < c < 1 and 0 < ρ < 1.Source: §V · p. 4
  • Canonical-dual synthesis combines the modified directional channels and compensates for the nonuniform symbol M; using primal filters instead would cause scale-dependent amplitude distortion.The Wiener-type map attenuates coefficients, whereas the canonical dual handles their reconstruction in image space.Source: §V · p. 4

VI. CONVERGENCE-CONTROLLED DATA-CONSISTENT RECONSTRUCTION

The section embeds DLSF denoising in a relaxed, data-consistent iteration rather than assuming one pass is optimal. Reanchoring to the observation, limiting update size, and continuing noise shrinkage address accumulated smoothing and premature convergence.

VI. CONVERGENCE-CONTROLLED DATA-CONSISTENT RECONSTRUCTION

  • The iteration starts from x(0) = y and uses a data-consistency parameter 0 < τ < 1 to reanchor each reconstruction to the measured image.The reanchoring counters attenuation from repeated regularized analysis/synthesis, while β limits each denoising update.Source: §VI · p. 4
  • Noise continuation reduces shrinkage after large-noise components are suppressed, avoiding persistent oversmoothing or updates too small for meaningful convergence.Maintaining the estimated noise level can oversmooth late stages; reducing σk too quickly can create premature apparent convergence.Source: §VI · p. 4
  • Without data consistency or relaxation, repeated denoising can accumulate smoothing or produce larger oscillations, and required iteration counts vary by noise level and image.A fixed iteration count is difficult to justify because experiments show the count increases with noise level and differs across images.Source: §VI · p. 4

VII. FINITE PERIODIC REALIZATION

The section specifies how the infinite DLSF model is computed on finite images, while keeping its numerical FFT realization distinct from the underlying lower semi-frame theory.

VII. FINITE PERIODIC REALIZATION

  • For an N1 × N2 image, the implementation samples T2 on the discrete torus ZN1 × ZN2, retaining nonempty bands and replacing frequency integrals with arithmetic averages.This finite realization specifies numerical evaluation without making the model dependent on a single image size.Source: §VII · p. 4
  • Local statistics use overlapping periodic Hann-window patches; reported experiments use 32 × 32 patches with an 8-pixel stride.The finite implementation preserves the infinite model’s sequence of regularized analysis, covariance-aware estimation, calibrated processing, dual synthesis, and convergence-controlled iteration.Source: §VII · p. 4
  • The algorithm estimates noise by lower-tail chi-square moment matching, then iterates channel-calibrated DLSF Wiener denoising with canonical-dual synthesis and continuation-updated noise.Each iteration first applies a data-consistency step; it returns the reconstruction and blind noise estimate after relative and maximum successive-image changes meet stopping conditions with required patience.Source: §VII · p. 5

VIII. RESULTS AND DISCUSSION … C. Denoising performance across noise levels

Across three grayscale images and four noise levels, the blind estimator and denoiser achieve average improvements of 7.45 dB in PSNR and 0.367 in SSIM, with a mean relative noise-estimation error of 3.28%. Results show stronger denoising gains for smooth regions, while the comparisons provide within-method validation rather than evidence of superiority to external baselines.

A. Experimental protocol

  • The 12 tests use camera, moon, and grayscale astronaut with unclipped AWGN at 15/255, 20/255, 25/255, and 30/255; clipping occurs only for display and metrics.All three images are grayscale inputs, with astronaut converted from color.Source: A. Experimental protocol · p. 5
  • All 12 experiments use the same parameters without selection from clean references, and evaluation reports relative sigma error, PSNR, and SSIM.PSNR and SSIM are reported for both the noisy observation and final reconstruction.Source: A. Experimental protocol · p. 5

B. Blind noise estimation: quantitative behavior

  • 5.55% mean relative sigma error at noise level 15 falls to 1.73% at level 30, as noise increasingly dominates the lower-energy tail.At low noise, weak image structure can contaminate lower DLSF energies; at higher noise, the chi-square model is easier to identify relative to signal perturbation.Source: B. Blind noise estimation:… · p. 5
  • At noise level 30, sigma errors are 2.81% for camera, 0.31% for moon, and 2.07% for astronaut.Moon's smoother regions and sparser high-frequency structure are consistent with its smaller error; camera and astronaut contain more structure that can contaminate low-energy coefficients.Source: B. Blind noise estimation:… · p. 5
  • Covariance whitening removes cross-direction correlations before lower-tail selection matches low-energy coefficients to a calibrated chi-square law.The passage states that without either step, the observed estimator trend could not be interpreted through the central model.Source: B. Blind noise estimation:… · p. 5

C. Denoising performance across noise levels

  • 3.28% mean relative sigma error, 7.45 dB mean PSNR improvement, and 0.367 mean SSIM improvement were measured across all 12 experiments.The results are within-method validation, not evidence of superiority to external baselines.Source: C. Denoising performance across noise… · p. 5
  • Moon's PSNR gains are 9.17, 10.18, 10.73, and 11.27 dB at levels 15, 20, 25, and 30, versus astronaut's 4.81-to-6.39 dB range.Smooth regions permit more aggressive local Wiener attenuation, whereas fine textures and repeated edges increase estimated local variance and complicate detail preservation.Source: C. Denoising performance across noise… · p. 5

D. Visual comparison at a common noise level · E. Convergence behavior and interaction of algorithmic components

At the common 30/255 noise level, denoising gains vary with image structure, while convergence depends on reconstruction changes and the interaction of algorithmic components. The results highlight limits in preserving weak local detail and motivate joint data consistency, continuation, and stopping criteria.

D. Visual comparison at a common noise level

  • At 30/255, moon PSNR rises from 18.62 to 29.89 dB and SSIM from 0.0948 to 0.5724, the strongest quantitative improvement among the three cases.Large smooth dark regions and sparse, coherent crater boundaries suit local variance shrinkage, allowing substantial noise suppression while preserving important boundaries.Source: Table II · p. 6, D. Visual comparison at a common noise… · p. 6
  • At 30/255, camera noisy PSNR of 19.12 dB rises to 26.38 dB and SSIM rises from 0.2539 to 0.6187.Broad sky and background regions become cleaner while major boundaries remain identifiable, but grass and fine camera detail remain soft because weak directional structures can be attenuated with noise.Source: Table II · p. 6, D. Visual comparison at a common noise… · p. 6
  • At 30/255, astronaut PSNR rises from 19.25 to 25.64 dB and SSIM from 0.3119 to 0.5779, with a smaller improvement than moon.Fine hair, facial structure, suit texture, lettering, and equipment edges overlap spectrally with noise; accurate global σ estimation alone does not ensure preservation of weak local structure.Source: Table II · p. 6, D. Visual comparison at a common noise… · p. 6
  • Across the three images, blind noise estimation remains accurate despite differing visual difficulty, but denoising quality also depends on how signal structure is represented in local coefficient statistics.The results point to structure-aware coefficient gains as a more promising improvement for difficult cases than further tuning the global noise estimator.Source: D. Visual comparison at a common noise… · p. 6

E. Convergence behavior and interaction of algorithmic components

  • The average iteration count rises from 5.33 at noise level 15 to 7.33 at level 30; at level 30, camera and astronaut stop after seven iterations and moon after eight.Stopping depends on successive reconstruction changes shaped by data consistency, shrinkage, continuation, and image content, rather than simply stopping earlier for the easiest visual image.Source: E. Convergence behavior and… · p. 6
  • All final relative changes are below 10^-2 and maximum-pixel changes below 5 × 10^-2; at level 30, relative errors are 4.14 × 10^-3 for camera, 3.65 × 10^-3 for moon, and 4.72 × 10^-3 for astronaut.These values show termination is controlled by the reconstruction sequence rather than a predetermined iteration budget.Source: E. Convergence behavior and… · p. 6
  • Requiring both relative-change and maximum-pixel criteria for two consecutive iterations balances global change against localized pixel changes and reduces sensitivity to transient small updates.Removing the patience requirement would shorten some runs by one iteration but make stopping more sensitive to transient small updates.Source: E. Convergence behavior and… · p. 6
  • Joint continuation and data consistency with the two-error stopping rule avoids reported failures from repeated detail attenuation or overly rapid σ_k decay, though formal fixed-point convergence remains unproved.Without data consistency, repeated regularized passes could attenuate detail after most noise is removed; if σ_k decayed too rapidly, updates could become small before the reconstruction stabilized.Source: E. Convergence behavior and… · p. 6

F. Component scope, limitations, and research implications … CREDIT AUTHORSHIP CONTRIBUTION STATEMENT

The paper presents a blind DLSF denoising method with accurate noise estimation and strong average gains, while identifying structure-aware shrinkage, reduced resolvent attenuation, and broader comparisons as priorities for improvement.

F. Component scope, limitations, and research implications

  • At the most difficult noise level, the estimator’s mean error is below 2%, while the substantial visual gap between moon and astronaut points to structure/noise discrimination—not sigma estimation—as the bottleneck.Moon’s error is below 0.4% at noise levels 25 and 30, consistent with smaller SSIM gains for texture-rich content.Source: F. Component scope, limitations, and… · p. 6
  • Repeated resolvent-regularized analysis can cause cumulative attenuation because analysis/synthesis need not be identity when coefficient gains are one; data consistency and relaxation mitigate but do not theoretically eliminate it.Suggested alternatives include applying the resolvent once, using an unregularized covariance model, or designing a fixed-point operator with explicit identity behavior when shrinkage is disabled.Source: F. Component scope, limitations, and… · p. 7
  • The 12 controlled experiments assess internal trends but lack external denoiser baselines, so they do not support a comparative state-of-the-art claim.A broader study should compare classical transform shrinkage, BM3D, a nonlocal baseline, and learned blind denoisers, while reporting runtime, parameter sensitivity, and statistical significance across more images.Source: F. Component scope, limitations, and… · p. 7

IX. CONCLUSIONS

  • The framework combines an infinite directional lower semi-frame, covariance-whitened chi-square noise estimation, lower-tail moment estimation, and channel-wise Wiener shrinkage with canonical-dual, data-consistent reconstruction.An explicit resolvent handles unbounded scale weights; four antisymmetric channels yield a whitened chi-square Mahalanobis energy, and the blind estimator avoids MAD.Source: F. Component scope, limitations, and… · p. 6, §IX · p. 7
  • Across three images and four noise levels, the method achieves a mean relative sigma error of 3.28%, a mean PSNR gain of 7.45 dB, and a mean SSIM gain of 0.367.At 30/255 noise, moon gains 11.27 dB in PSNR, whereas astronaut gains 6.39 dB.Source: §IX · p. 7

F. Component scope, limitations, and research implications

  • Future work should use noncentral-chi-square structure confidence to guide Wiener gains, reduce cumulative resolvent attenuation, analyze convergence, and benchmark against established denoisers.The proposed structure confidence comes from treating whitened energy under image structure as noncentral chi-square; incorporating it into Wiener gains avoids post-hoc reinjection that also restores noise.Source: §IX · p. 7, F. Component scope, limitations, and… · p. 7

DATA AND IMPLEMENTATION AVAILABILITY

  • The numerical tables use a CSV containing all 12 experiments, the fixed-noise visual comparison uses saved outputs at 30/255, and a single-cell Python/Colab implementation can be distributed with the manuscript.The reproducibility materials are described as distributable with the manuscript.Source: DATA AND IMPLEMENTATION AVAILABILITY · p. 7

CREDIT AUTHORSHIP CONTRIBUTION STATEMENT

  • Hemalatha M is credited with conceptualization, methodology, software, formal analysis, investigation, and original-draft writing; P. Sam Johnson with validation, supervision, and review and editing.Source: CREDIT AUTHORSHIP CONTRIBUTION STATEMENT · p. 7
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