One-Step Is Optimal: Unconditional Rectified Flows are Noise2Noise Denoisers, and Multi-Step Integration Provably Hurts---A Benchmark and Task-Based Detectability Study on Low-Dose CT

Timothy Sereda; Debesh Jha

cs.CVSep 15, 2026

One-Step Is Optimal: Unconditional Rectified Flows are Noise2Noise Denoisers, and Multi-Step Integration Provably Hurts---A Benchmark and Task-Based Detectability Study on Low-Dose CT

Authors: Timothy Sereda, Debesh Jha

Organizations: Department of Computer Science Biomedical Perception & Intelligence Lab University of South Dakota Vermillion, South Dakota, USA

Abstract

Iterative and generative denoisers are increasingly used under the assumption that multi-step refinement outperforms a single regression pass. We show the opposite for label-free denoising. An unconditional rectified flow trained on two noisy observations of the same signal, as in Noise2Noise, has a minimiser whose one-step readout is exactly the MMSE denoiser without requiring clean targets. In contrast, multi-step integration provably departs from the MMSE solution because the flow terminates at the noisy data distribution rather than the clean-signal distribution. This departure is exact in a tractable Gaussian model and is confirmed experimentally: one-step flow matches a direct regressor, whereas multi-step Euler integration progressively reduces fidelity. Counterintuitively, the degradation increases with training quality, as a better velocity field more faithfully transports samples toward the noisy terminal law. The key ingredient is therefore the decorrelated pairing, not the flow machinery: a one-step regressor trained on matched noisy pairs gives our best label-free result ( $+1.99$ ,dB). We evaluate these findings on CTDenoiser, a controlled low-dose CT benchmark spanning five architectures and supervised, similarity-based, blind-spot, and per-image methods. Among label-free approaches, only correlated-noise-aware Noise2Sim improves over the noisy baseline, while Noise2Void is flat-to-negative because CT noise violates its pixel-independence assumption. Finally, although supervised denoisers gain approximately $4$ ,dB PSNR, a channelized Hotelling observer shows reduced low-contrast lesion detectability, revealing clinically relevant degradation missed by PSNR and SSIM.

Figures & tables

Model	Supervision	Params	PSNR $\uparrow$ ( $\Delta$ )	SSIM $\uparrow$	RMSE $\downarrow$	GMSD $\downarrow$	RSR $\downarrow$
LDCT input	— (no denoising)	—	$30.25$	$0.876$	$0.0320$	$0.174$	$0.0115$
RED-CNN [ 2 ]	Supervised	$1.85\mathrm{M}$	34.50 ( $+4.25{\pm}0.23$ )	$0.927{\pm}0.013$	$0.0194$	$0.167$	$0.0041$
U-Net	Supervised	$1.95\mathrm{M}$	34.26 ( $+4.02{\pm}0.22$ )	$0.925{\pm}0.014$	$0.0199$	$0.188$	$0.0044$
DnCNN [ 3 ]	Supervised	$0.56\mathrm{M}$	34.23 ( $+3.99{\pm}0.18$ )	$0.924{\pm}0.014$	$0.0200$	$0.198$	$0.0044$
CFM [ 8 ]	Supervised	$2.11\mathrm{M}$	33.09 ( $+2.84{\pm}0.22$ )	$0.915{\pm}0.013$	$0.0228$	$0.192$	$0.0057$
CTformer [ 4 ]	Supervised	$0.40\mathrm{M}$	32.01 ( $+1.77{\pm}0.35$ )	$0.904{\pm}0.013$	$0.0255$	$0.249$	$0.0070$

Table 1: Main benchmark ( sweep.yml ): validation results on TCIA LDCT (abdomen; patient-level

3

-seed splits,

50

epochs), grouped by supervision regime. Mean

\pm

s.d. over

3

seeds on full slices via overlapped tiling (corrected border handling). The PSNR column gives the absolute value with the gain over each split’s LDCT input floor in parentheses (mean floor

30.25

dB). All five architectures are complete over

3

seeds. CFM requires paired targets, so it has no blind-spot or similarity variant. Per-image methods (ZS-N2N, Filter2Noise) are in Table 9 ; the label-free SSFlow and its one- vs. multi-step ablation are in Section 4 . Bold : best

\Delta

PSNR per supervision block.

\uparrow

\downarrow

: higher/lower is better.

Figure 1: Qualitative comparison on a held-out abdomen slice (top: full slice with the ROI boxed; bottom: $2\times$ ROI). The supervised model removes the most noise but imparts the blotchy, low-frequency “waxy” texture that the NPS-centroid shift quantifies (Section 3.3 ); Noise2Sim preserves finer texture; and multi-step SSFlow visibly drifts back toward the noisy input relative to its one-step readout, as the departure result predicts.

Figure 2: Task-based detectability on the benchmark CNNs (mean over

3

architectures

\times

3

seeds). (a) CHO detectability preserved

d^{\prime}_{\text{den}}/d^{\prime}_{\text{in}}

vs lesion contrast: the erosion concentrates at the high-contrast operating point—precisely where PSNR/SSIM report their largest, most confident gains. (b) The denoised NPS radial centroid falls below the input (dashed)—the low-frequency “waxy” texture—most for the supervised models. (c) Fabrication

d^{\prime}

stays at the signal-absent input floor (dashed) for every regime: detectability is eroded without inventing structure.

Reference $d^{\prime}$ (method-independent)
	lesion contrast (HU)			NPS	fab.
	$40$	$80$	$160$	centroid $\downarrow$	$d^{\prime}\,\downarrow$
input (floor)	$1.51$	$3.03$	$6.06$	$0.112$	$0.028$
clean (ceiling)	$1.54$	$3.09$	$6.18$	—	—
Detectability preserved $d^{\prime}_{\text{den}}/d^{\prime}_{\text{in}}$
Supervised	$1.024\,{\pm}\,0.008$	$1.016\,{\pm}\,0.007$	$0.934\,{\pm}\,0.033$	$0.087\,{\pm}\,0.004$	$0.022\,{\pm}\,0.003$

Table 2: Task-based detectability on the completed benchmark models (Table 1 ); mean over

3

CNN architectures

\times

3

seeds (

27

runs; CTformer, now complete, reproduces the same erosion and is held out to keep the aggregate CNN-only). Top: CHO detectability index

d^{\prime}

for the noisy input (floor) and clean reference (ceiling) at three lesion contrasts; these are method-independent and carry a large between-split (patient-cohort) s.d. (

\pm 1.3

160

HU), which is why we report the ratio below. Bottom: detectability preserved ,

d^{\prime}_{\text{denoised}}/d^{\prime}_{\text{input}}

(mean

\pm

s.d.;

1.0=

unchanged,

<1=

eroded); the denoised noise-power-spectrum radial centroid (input

0.112

; lower

=

more low-frequency “waxy” texture); and the fabrication

d^{\prime}

from a CHO run on signal-absent ROIs (input floor

0.028

;

>

floor

=

invented structure). Architecture differences are within s.d., so rows are aggregated by supervision regime.

d^{\prime}_{\text{denoised}}

is below

d^{\prime}_{\text{input}}

in all

27

runs and never exceeds the clean ceiling, while fabrication

d^{\prime}

never rises above the floor.

$r$	flow	regression	$1$	$4$	$8$	$20$
	one-step $\Delta$ PSNR		flow, Euler steps
$1$	$+3.91$	$+4.12$	$+3.91$	$+1.94$	$+1.59$	$+1.37$
$2$	$+3.90$	$+4.09$	$+3.90$	$+1.89$	$+1.52$	$+1.30$
$3$	$+4.05$	$+4.20$	$+4.05$	$+1.92$	$+1.54$	$+1.30$

Table 3: The result generalises beyond CT (BSDS500 natural images, white Gaussian noise

\sigma{=}0.1

;

\Delta

PSNR over the noisy input, mean over

3

seeds). One-step flow ties the direct regressor at every decorrelation radius

r

, and multi-step Euler integration erodes PSNR monotonically—the same two findings as on CT (Table 5 ), now off the medical domain and under i.i.d. noise, where

r

(the correlated-noise knob) is inert.

Figure 3: The finite-step departure, predicted and confirmed. (a) In the jointly-Gaussian model the relative denoising risk

R(t)/R(0)

of the flow’s readout rises monotonically with integration time

t

from the one-step optimum (

t{=}0

) — iteration provably increases error (Eq. ( 7 )). (b) Empirically, PSNR gain falls monotonically with the Euler step count at every decorrelation radius

r

( sweep_hallucination.yml ,

3

seeds); the one-step values sit at the matched-pairing one-step regression baseline (dashed), which one-step flow ties. (c) The same monotone erosion on natural images (BSDS500, i.i.d. Gaussian noise; sweep_natural_hallucination.yml ,

3

seeds), with the natural one-step regression reference (dashed) — the departure is not an artefact of CT or of correlated noise.

Reference
Steps	Radius $r$	PSNR $\uparrow$ ( $\Delta$ )	SSIM $\uparrow$	RMSE $\downarrow$	GMSD $\downarrow$	RSR $\downarrow$	fab. $d^{\prime}$ $\downarrow$
LDCT input	—	30.25	0.876	0.0320	0.174	0.0115	0.028
One-step posterior mean
$1$	$1$	32.00 (+1.75)	0.899	0.0261	0.215	0.0077	0.030
$1$	$2$	32.12 (+1.88)	0.902	0.0256	0.213	0.0073	0.029
$1$	$3$	31.93 (+1.69)	0.896	0.0261	0.240	0.0076	0.028

Table 4: SSFlow inference-steps ablation ( sweep_hallucination.yml ): one-step posterior mean vs. multi-step Euler refinement across the decorrelation radius

r

, on TCIA LDCT (abdomen;

3

seeds,

50

epochs). PSNR gives the absolute value with the gain over the

30.25

dB input floor. PSNR degrades monotonically with the step count at every

r

—the

20

-step gain is roughly a third of the one-step gain—exactly as the finite-step departure of Eq. ( 7 ) predicts, while the fabrication

d^{\prime}

holds at the input floor (

\approx

0.028

) at every step count: iteration erodes fidelity without inventing structure. Bold : best cell (one-step,

r{=}2

\uparrow

\downarrow

: higher/lower is better.

Reference
Estimator	Radius $r$	PSNR $\uparrow$ ( $\Delta$ )	SSIM $\uparrow$	RMSE $\downarrow$	GMSD $\downarrow$	RSR $\downarrow$	fab. $d^{\prime}$ $\downarrow$
LDCT input	—	30.25	0.876	0.0320	0.174	0.0115	0.028
Regression (Noise2Sim, RED-CNN) — same pairs, direct MMSE
Regression	$1$	32.02 ( $+1.77{\pm}0.14$ )	$0.900{\pm}0.015$	$0.0260$	$0.163$	$0.0076$	$0.029$
Regression	$2$	32.24 ( $+1.99{\pm}0.09$ )	$0.904{\pm}0.015$	$0.0253$	$0.160$	$0.0071$	$0.030$
Regression	$3$	32.12 ( $+1.87{\pm}0.11$ )	$0.904{\pm}0.014$	$0.0256$	$0.162$	$0.0072$	$0.029$

Table 5: Flow-vs-regression control ( sweep_flow_vs_reg.yml

+

sweep_hallucination.yml ): does the one-step theorem (Section 4.1 ) hold empirically? We fix the similarity pairing and vary only the estimator. The regression arm is a direct Noise2Sim regressor (RED-CNN,

1.85\mathrm{M}

params, capacity-comparable to SSFlow’s

2.11\mathrm{M}

velocity net); the flow arm is one-step SSFlow. Both sweep

r\in\{1,2,3\}

over

3

seeds (

9

runs each) on the identical pairs, so the arms overlay cell-for-cell; PSNR gain is over each split’s own input floor. The theorem holds: at matched

r

the one-step flow and the regressor agree to within

\leq 0.2

dB (e.g.

r{=}2

+1.88

vs.

+1.99

), both tracing the same inverted-U that peaks at

r{=}2

—so the decorrelated pairing, not the estimator, is what helps. A small residual favours regression (consistent with the finite-capacity caveat), so the decorrelated-pairing one-step regressor at

r{=}2

(

+1.99

dB) is the best label-free result and the estimator of choice; the large cost is iteration , not the objective—multi-step flow falls to

+0.5

dB (Table 4 ). Fabrication

d^{\prime}

confirms neither estimator invents structure (

\approx

input floor). Bold : best cell.

\uparrow

\downarrow

: higher/lower is better.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Supervision	Params	PSNR $\uparrow$ ( $\Delta$ )	SSIM $\uparrow$	RMSE $\downarrow$	GMSD $\downarrow$	RSR $\downarrow$
Simulated LDCT input	— (no denoising)	—	$28.26$	$0.597$	$0.0388$	$0.418$	$0.0153$
RED-CNN [ 2 ]	Supervised	$1.85\mathrm{M}$	33.17 ( $+4.91{\pm}0.13$ )	$0.937$	$0.0221$	$0.169$	$0.0050$
DnCNN [ 3 ]	Supervised	$0.56\mathrm{M}$	32.90 ( $+4.64{\pm}0.14$ )	$0.934$	$0.0228$	$0.222$	$0.0053$
U-Net	Supervised	$1.95\mathrm{M}$	32.78 ( $+4.52{\pm}0.11$ )	$0.933$	$0.0231$	$0.222$	$0.0055$
CTformer † [ 4 ]	Supervised	$0.40\mathrm{M}$	30.22 ( $+1.96{\pm}0.04$ )	$0.860$	$0.0309$	$0.361$	$0.0098$
CFM [ 8 ]	Supervised	$2.11\mathrm{M}$	31.07 ( $+2.81{\pm}0.17$ )	$0.914$	$0.0281$	$0.226$	$0.0081$

Appendix

Table 6: Robustness to a second LDCT source ( sweep_sim_ldct.yml ): the Table 1 grid re-run with the low-dose arm simulated from the abdomen full-dose volumes (correlated, signal-dependent noise;

3

seeds,

50

epochs, matched splits). PSNR gives the absolute value with the gain over the

28.26

dB simulated input floor in parentheses. The supervision-regime ordering matches the real-source benchmark—supervised

\sim

4.5

–

4.9

dB, Noise2Sim the only label-free method above the floor, Noise2Void flat-to-negative—so it is not an artefact of one acquisition; the label-free margin compresses under the more stationary simulated noise. Bold : best

\Delta

PSNR per supervision block. † Two CTformer runs crashed near convergence (epoch

47

–

49

), leaving supervised at

2/3

and Noise2Void at

1/3

seeds (shown without s.d.); all other cells are

3

seeds.

\uparrow

\downarrow

: higher/lower is better.

Model	Supervision	Params	PSNR $\uparrow$ ( $\Delta$ )	SSIM $\uparrow$	RMSE $\downarrow$	GMSD $\downarrow$	RSR $\downarrow$
Chest LDCT input	— (no denoising)	—	$39.68$	$0.932$	$0.0110$	$0.235$	$0.0004$
RED-CNN [ 2 ]	Supervised	$1.85\mathrm{M}$	42.97 ( $+3.29$ )	$0.970$	$0.0075$	$0.209$	$0.0002$
CFM [ 8 ]	Supervised	$2.11\mathrm{M}$	41.97 ( $+2.29$ )	$0.965$	$0.0084$	$0.206$	$0.0002$
DnCNN [ 3 ]	Supervised	$0.56\mathrm{M}$	41.70 ( $+2.02$ )	$0.959$	$0.0088$	$0.216$	$0.0002$
U-Net	Supervised	$1.95\mathrm{M}$	41.01 ( $+1.33$ )	$0.965$	$0.0092$	$0.207$	$0.0002$
RED-CNN [ 2 ]	Noise2Sim [ 20 ]	$1.85\mathrm{M}$	41.20 ( $+1.52$ )	$0.954$	$0.0092$	$0.219$	$0.0003$

Appendix

Table 7: Robustness to a second anatomy ( sweep_chest.yml ): the Table 1 grid on a chest LDCT cache (real low/full-dose pairs,

10

patients, single seed ) instead of abdomen. PSNR gives the absolute value with the gain over the

39.68

dB chest input floor in parentheses. The CNN supervision ordering reproduces—supervised leads, Noise2Sim the only label-free method above the floor, Noise2Void flat-to-negative—though gains are compressed by the high chest floor and noisier at single seed. CTformer diverged on the

8

-patient chest cohort (all modes below the floor) and is omitted; CFM requires paired targets and so appears only under supervision. Bold : best

\Delta

PSNR per supervision block.

\uparrow

\downarrow

: higher/lower is better.

Model	Supervision	$\Delta$ PSNR $\uparrow$ (dB)	PSNR $\uparrow$	SSIM $\uparrow$	RMSE $\downarrow$	RSR $\downarrow$
RED-CNN [ 2 ]	Supervised	+4.19 $\pm$ 0.22	34.43	0.927 $\pm$ 0.013	0.0195	0.0042
DnCNN [ 3 ]	Supervised	+4.04 $\pm$ 0.22	34.29	0.924 $\pm$ 0.015	0.0199	0.0044
RED-CNN [ 2 ]	Noise2Sim [ 20 ]	+1.74 $\pm$ 0.09	31.99	0.900 $\pm$ 0.015	0.0261	0.0077
DnCNN [ 3 ]	Noise2Sim [ 20 ]	+1.42 $\pm$ 0.28	31.66	0.895 $\pm$ 0.015	0.0271	0.0082
RED-CNN [ 2 ]	Noise2Void [ 18 ]	+0.32 $\pm$ 0.10	30.57	0.883 $\pm$ 0.019	0.0306	0.0105
DnCNN [ 3 ]	Noise2Void [ 18 ]	$-0.02$ $\pm$ 0.06	30.23	0.875 $\pm$ 0.018	0.0320	0.0114

Appendix

Table 8: Three-fold patient-level cross-validation on TCIA LDCT (abdomen; 15 epochs).

\Delta

PSNR and SSIM are the mean (

\pm

s.d.) across the three folds; the PSNR gain is measured against each fold’s own LDCT-input floor (

31.44/30.04/29.27

dB for folds

1/2/3

). PSNR, RMSE, and RSR are fold means. Only DnCNN and RED-CNN completed all three folds and are shown; U-Net had incomplete fold coverage at export. Filter2Noise and ZS-N2N are per-image and model-agnostic (single row each). Bold : best per column. Underline : best label-free PSNR gain.

Method	Type	Params	PSNR $\uparrow$ ( $\Delta$ )	SSIM $\uparrow$	RMSE $\downarrow$	GMSD $\downarrow$	RSR $\downarrow$
Filter2Noise [ 22 ]	Zero-shot	—	31.58 (+0.14)	0.896	0.0269	0.176	0.0076
ZS-N2N [ 21 ]	Zero-shot	$21.3\mathrm{K}$	31.17 ( $-0.27$ )	0.892	0.0281	0.182	0.0083

Appendix

Table 9: Per-image sweep ( sweep_periimage.yml ): test-time, data-free methods, model-agnostic. Values shown are single-split seed-0 (floor

31.44

dB;

\Delta

over that floor); the

3

-seed means (

2

modes

\times

3

seeds) are pending and will match the benchmark’s

30.25

dB floor. Both methods sit essentially at the noisy floor—consistent with the label-free ordering in Table 1 .

\uparrow

\downarrow

: higher/lower is better.

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