BMND: Direct Poisson Denoising by N-Dimensional Block Matching and Collaborative Filtering

Christof Duhme; Lars Schiefelbein; Florian Büther; Xiaoyi Jiang

cs.CVSep 28, 2026

BMND: Direct Poisson Denoising by N-Dimensional Block Matching and Collaborative Filtering

Authors: Christof Duhme, Lars Schiefelbein, Florian Büther, Xiaoyi Jiang

Organizations: University of Münster

Abstract

Poisson denoising of scientific data requires methods that account for signal-dependent noise while accommodating different data dimensionalities and preserving quantitative intensity information. We present BMND, a dimension-independent extension of block matching and collaborative filtering for Gaussian and Poisson observations. Building on the two-stage structure of BM3D and BM4D, BMND processes Poisson data directly, without a variance-stabilizing transform, by combining noise-aware patch matching with propagation of signal-dependent noise variances through collaborative filtering and aggregation. A dimension-independent reference-patch traversal scheme supports arrays with an arbitrary number of axes. An optional aggregation-aware mass conservation preserves the observed total intensity after weighted overlap-add. We evaluate the framework on one-dimensional physiological signals, two-dimensional images, and three-dimensional volumes, using controlled noise experiments and measured fluorescence microscopy acquisitions. The experiments demonstrate improved reconstruction quality from noise-aware matching and Wiener filtering, while low-count phantom experiments show reduced denoising-induced intensity loss through mass conservation. The framework provides a unified, non-learning-based approach to denoising across arbitrary data dimensions and is released as an open-source library.

Figures & tables

Problem and existing limitations	bmnd approach
Different data modalities and noise models. Parameters, dimensions, and noise assumptions are tightly coupled in existing implementations, which limits their reuse across imaging modalities and acquisition settings.	A common formulation with configurable profiles for geometry, noise, matching, shrinkage, and aggregation.
Signal-dependent Poisson noise. Standard squared Euclidean block matching and fixed-variance shrinkage are designed for data-independent, stationary Gaussian noise and do not account for the signal dependence of Poisson noise.	Direct scaled-Poisson processing with a plug-in variance map and Poisson-aware matching metrics.
Correlated or heteroscedastic noise. A single scalar variance per coefficient is inadequate for nonstationary or correlated noise.	A common covariance interface that propagates noise through transforms, thresholds, Wiener gains, and aggregation weights.
Arbitrary-dimensional data. bm3d and bm4d are formulated for specific data dimensionalities and therefore require dimension-dependent implementations of patch extraction, grouping, transforms, and traversal.	Coupled reference-patch schedules and group transforms for arrays with an arbitrary number of data axes.
Low-count measurements. High-count asymptotics for discrepancy moments become inaccurate for small pooled counts.	Exact finite-count conditional moments for calibrating acceptance rules and matching scales.
Noise-aware block matching. Raw ssd ( ssd ) includes the expected contribution of noise and does not distinguish structural mismatch from noise fluctuations.	Poisson deviance, Pearson and Anscombe discrepancies, and noise-bias-corrected ssd under Gaussian or Poisson models.

Table 1 : Top-down overview of the problems addressed by bmnd , the limitations of existing approaches, and the corresponding methodological solutions.

Figure 2 : Reference-patch count and coverage uniformity. Counts are normalized by those of the official schedule and averaged across images. Horizontal error bars show their range across image shapes. Uniformity is measured by the median cv of reference-patch coverage across images where lower values indicate more uniform coverage. Dotted lines mark the official schedule. G and B denote generated and balanced schedules, with tuples specifying (shift density, schedule density).

Figure 3 : Denoising quality versus reference-patch count under Gaussian noise. psnr changes are relative to the official schedule and averaged equally across BSD68, Kodak24, Set12, and scikit-image. Counts are normalized by those of the official schedule and averaged across images. Horizontal error bars show their range across image shapes. Panels correspond to

\sigma_{255}\in\{15,25,50\}

. G and B denote generated and balanced schedules, with tuples specifying (shift density, schedule density).

Figure 4 : Source-balanced marginal effects of the finite-count structural allowance on psnr . Effects are measured relative to

\beta_{s}=0

for the structure factor (top) and

\kappa_{s}=1

for the intensity exponent (bottom), averaging uniformly over the levels of the other factor. Positive values indicate improved reconstruction quality. Error bars denote 95% nonparametric source-unit bootstrap intervals.

Figure 5 : Shepp–Logan phantom with three fixed regions of interest (left) and its clean sinogram (right). The regions sample the gray background within the object, a dark ellipse with zero phantom activity, and the upper circular structure.

Figure 6 : Mass-conservation effects when denoising Poisson-corrupted Shepp–Logan images directly. Panels show bias in the gray region (a), dark ellipse (b), upper circle (c), object support (d), and whole image (e), measured against the clean phantom. Bias is expressed as a percentage of reference mass, except in (b), which shows the signed mean-intensity difference. Panel (f) shows the psnr change relative to denoising without mass conservation. Curves show means over 100 Poisson realizations, with shaded 95% bootstrap confidence intervals.

Figure 7 : Mass-conservation effects when denoising Poisson-corrupted Shepp–Logan sinograms. Panels (a)–(d) show bias in the reconstructed gray region, dark ellipse, upper circle, and object support, measured against the reconstruction of the clean sinogram. Panel (e) shows whole-sinogram bias relative to the clean sinogram. Bias is expressed as a percentage of reference mass, except in (b), which shows the signed mean-intensity difference. Panels (f) and (g) show psnr changes relative to denoising without mass conservation in sinogram and reconstructed-image space, respectively. Curves show means over 100 Poisson realizations, with shaded 95% bootstrap confidence intervals.

Figure 8 : Marginal effects of the matching strategy on psnr relative to SSD with fixed calibration. PD denotes Poisson deviance; A-SSD denotes Anscombe-transformed SSD. Finite-count denotes reference-based finite-count calibration; standardized denotes candidate standardization. Error bars denote 95% source-unit bootstrap intervals.

Figure 9 : Marginal psnr effects of the thresholding rule relative to hard thresholding (top) and the threshold multiplier relative to

\lambda_{\mathrm{HT}}=3

(bottom). Error bars denote 95% source-unit bootstrap intervals. Note the different horizontal scales.

Figure 10 : Marginal effects of the Wiener gain formulation on psnr relative to classic gains. Error bars denote 95% source-unit bootstrap intervals.

Figure 11 : Marginal psnr effects of aggregation weights relative to classic weighting in the first filtering stage (top) and Wiener stage (bottom). Error bars denote 95% source-unit bootstrap intervals. Note the different horizontal scales.

Figure 12 : Source-balanced marginal psnr effects by noise condition, with dataset groups in columns and algorithmic factors in rows. Effects are relative to the indicated reference settings, averaging over the other factors. Logarithmic horizontal axes show peak expected counts or acquisition averages for fmd . In the matching row, colors identify distances, and line styles and markers identify calibrations. Vertical scales are shared within rows.

Figure 13 : Denoising examples from the five ablation dataset groups. Columns show the reference, noisy observation, Anscombe bmnd , and direct Poisson bmnd with the globally top-ranked ablation configuration. Simulated observations use $\lambda_{\mathrm{pk}}=5$ ; the fmd example uses a single acquisition with a 50-acquisition average as reference. Labels report psnr / ssim .

Method	Metric	$1$	$2$	$4$	$8$	$16$
Reported by Zhang et al. [ 51 ]
Raw	PSNR (dB)	$27.22$	$30.08$	$32.86$	$36.03$	$39.70$
	SSIM	$0.5442$	$0.6800$	$0.7981$	$0.8892$	$0.9487$
VST + BM3D	PSNR (dB)	$32.71$	$34.09$	$36.05$	$38.01$	$40.61$
	SSIM	$0.7922$	$0.8430$	$0.8970$	$0.9336$	$0.9598$
PURE-LET	PSNR (dB)	$31.95$	$33.49$	$35.29$	$37.25$	$39.59$

Table 2 : FMD denoising performance by averaging level. Published results are from Zhang et al. [ 51 ] . For the full test set, bold marks the best non-deep-learning score and underlining the best overall. Within each modality subset, bold marks the best score.

Figure 14 : Single-acquisition FMD denoising examples from held-out FOV 19. Rows show confocal, two-photon, and widefield microscopy. Columns show the reference, noisy input, Anscombe bmnd , and direct Poisson bmnd with modality-specific profiles. Yellow boxes indicate enlarged regions. Labels report full-image psnr / ssim .

Figure 15 : Mean snr improvement across confirmation subjects at each input snr level. Shaded areas denote 95% subject-bootstrap confidence intervals.

Figure 16 : Paired subject-level comparison of bmnd with each baseline, averaged across input snr levels. Points above the diagonal indicate greater snr improvement with bmnd . Colors identify the corresponding baseline.

QRS RMSE $\downarrow$
Method	0 dB	6 dB	12 dB	18 dB
BMND	$\mathbf{0.1105}$ $[0.0965,0.1254]$	$\mathbf{0.0638}$ $[0.0557,0.0724]$	$\mathbf{0.0388}$ $[0.0341,0.0437]$	$\mathbf{0.0238}$ $[0.0210,0.0267]$
Wavelet	$0.1901$ $[0.1699,0.2113]$	$0.1188$ $[0.1055,0.1322]$	$0.0726$ $[0.0644,0.0808]$	$0.0430$ $[0.0383,0.0478]$
NLM	$\underline{0.1395}$ $[0.1184,0.1627]$	$0.0768$ $[0.0682,0.0858]$	$0.0532$ $[0.0475,0.0592]$	$0.0350$ $[0.0308,0.0393]$
SSCF	$0.2234$ $[0.1505,0.3079]$	$\underline{0.0700}$ $[0.0615,0.0791]$	$\underline{0.0415}$ $[0.0366,0.0465]$	$\underline{0.0247}$ $[0.0220,0.0276]$
Savitzky–Golay	$0.1583$ $[0.1384,0.1787]$	$0.0860$ $[0.0761,0.0962]$	$0.0533$ $[0.0477,0.0588]$	$0.0401$ $[0.0353,0.0453]$

Table 3 : QRS morphology preservation across input snr levels, measured by waveform rmse , relative amplitude error, and signed relative amplitude bias. Brackets report 95% subject-bootstrap confidence intervals. Best results are bold and second-best results are underlined.

Figure 17 : Example beat-centered ecg waveforms comparing the reference, noisy observation, bmnd , and sscf . The vertical dotted line marks the expert beat annotation.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Stage	BM3D	BM4D
General and scheduling
Reference schedule	Both	generated	generated
Shift density	Both	2.0	2.0
Schedule density	Both	2	2
Covariance and matching
Local-variance domain $n_{f}$	Both	$32\times 32$	$16\times 16\times 16$

Appendix

Table 4 : Standard Gaussian BM3D and BM4D profile values used by BMND.

Profile parameter	Candidate values or fixed setting	Levels
Varied Cartesian factors
HT matching distance / policy	poisson_deviance / fixed , poisson_deviance / reference_finite_count , poisson_deviance / candidate_standardized , pearson / fixed , pearson / reference_finite_count , pearson / candidate_standardized , anscombe_ssd / fixed , anscombe_ssd / reference_finite_count , anscombe_ssd / candidate_standardized , ssd / fixed	10
Exact covariance planes $k$	$\{0,1,4,32\}$	4
Wiener gain	$\{\,$ classic , noise-floor , variance-scaled $\,\}$	3
HT aggregation-weight model	$\{\,$ classic , variance $\,\}$	2
HT aggregation-weight domain	$\{\,$ coefficient , windowed_synthesis $\,\}$	2

Appendix

Table 5: Profile space evaluated in the Cartesian ablation. The level count is the number of independently crossed options.

Figure 18 : Marginal effects of the number of covariance planes

k

on psnr relative to

k=0

. Error bars denote 95% source-unit bootstrap intervals.

Figure 19 : Marginal effects of the hard-thresholding aggregation-weight domain on psnr relative to coefficient-domain weighting. Error bars denote 95% source-unit bootstrap intervals.

Figure 20 : Marginal effects of the hard-thresholding aggregation-weight scope on psnr relative to group-level weighting. Error bars denote 95% source-unit bootstrap intervals.

Figure 21 : Marginal effects of the Wiener-stage aggregation-weight domain on psnr relative to coefficient-domain weighting. Error bars denote 95% source-unit bootstrap intervals.

Figure 22 : Marginal effects of the Wiener-stage aggregation-weight scope on psnr relative to group-level weighting. Error bars denote 95% source-unit bootstrap intervals.

Figure 23 : Marginal effects of group mass conservation on psnr relative to denoising without mass conservation. Conservation is applied in the hard-thresholding stage, the Wiener stage, or both stages. Error bars denote 95% source-unit bootstrap intervals.

Method and parameter	Candidate values or fixed setting	Candidates
1D BMND		240
Patch length $L$	$\{8,16,32,64,128\}$ samples
Reference step	$L/4$ samples
One-sided search radius	$\{1,2,3,4\}\,\mathrm{s}$
Maximum group size	$\{8,16,32,64\}$
Matching-threshold scale $c_{m}$	$\{0.5,1,2\}$

Appendix

Table 6: Parameter spaces evaluated in the controlled ECG experiment. Candidate counts include only combinations varied during development, fixed settings are reported for reproducibility.

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