λSplit: Self-Supervised Content-Aware Spectral Unmixing for Fluorescence Microscopy

Federico Carrara; Talley Lambert; Mehdi Seifi; Florian Jug

cs.CVMar 24, 2026

λSplit: Self-Supervised Content-Aware Spectral Unmixing for Fluorescence Microscopy

Authors: Federico Carrara, Talley Lambert, Mehdi Seifi, Florian Jug

Organizations: Fondazione Human Technopole, Milan, Italy · Università Campus Bio-Medico, Rome, Italy · Harvard Medical School, Boston, MA, USA

Abstract

In fluorescence microscopy, spectral unmixing aims to recover individual fluorophore concentrations from spectral images that capture mixed fluorophore emissions. Since classical methods operate pixel-wise and rely on least-squares fitting, their performance degrades with increasingly overlapping emission spectra and higher levels of noise, suggesting that a data-driven approach that can learn and utilize a structural prior might lead to improved results. Learning-based approaches for spectral imaging do exist, but they are either not optimized for microscopy data or are developed for very specific cases that are not applicable to fluorescence microscopy settings. To address this, we propose λSplit, a physics-informed deep generative model that learns a conditional distribution over concentration maps using a hierarchical Variational Autoencoder. A fully differentiable Spectral Mixer enforces consistency with the image formation process, while the learned structural priors enable state-of-the-art unmixing and implicit noise removal. We demonstrate λSplit on 3 real-world datasets that we synthetically cast into a total of 66 challenging spectral unmixing benchmarks. We compare our results against a total of 10 baseline methods, including classical methods and a range of learning-based methods. Our results consistently show competitive performance and improved robustness in high noise regimes, when spectra overlap considerably, or when the spectral dimensionality is lowered, making λSplit a new state-of-the-art for spectral unmixing of fluorescent microscopy data. Importantly, λSplit is compatible with spectral data produced by standard confocal microscopes, enabling immediate adoption without specialized hardware modifications.

Figures & tables

Figure 2: Proposed architecture of

\lambda

Split. The model builds on an LVAE backbone [ 35 ] , where a bottom-up encoder produces features

h_{i}

at multiple hierarchy levels. At the highest hierarchy level we employ a default multivariate Gaussian prior, followed by learnable top-down priors

p_{i}

(conditioned from above) that, combined with the respective

h_{i}

, form the final posterior distributions

(\mu_{q,i},\sigma_{q,i})

. Latent samples are generated by drawing from the topmost Gaussian and successively propagating and resampling down the hierarchy through a deterministic decoder to, finally, generate FP concentration maps. The last decoder layer comprises an unmixing convolutional block (blue) that maps the final activation into the desired unmixed concentration map tensor. To render our method fully self-supervised, a differentiable Spectral Mixer multiplies the predicted concentration maps by the transposed mixing matrix

M^{T}\in\mathbb{R}^{F\times L}

, obtainable by discretizing the emission spectra of the FPs contained in the imaged sample (bottom). This models the physical image formation process and, given that the predicted concentration maps are correct, reconstructs the spectral image we received as input, rendering this setup an end-to-end trainable variational autoencoder. The model is trained by minimizing the sum of the hierarchical KL divergence (computed from latent samples) and a spectral MSE reconstruction loss.

Figure 3: Qualitative results on high-noise BioSR data (exposure 10 ms). Compared to $7$ baselines, $\lambda$ Split produces the cleanest results, also benefiting from its implicit noise removal capabilities. Objects and details (e.g., puncta in the CCPs channel or F-Actin structures) are well recovered, enabling best possible downstream analyses.

Exp	SNR sp	Method	PSNR $\uparrow$	MS3IM $\uparrow$	$\mu$ MS3IM $\uparrow$	LPIPS $\downarrow$	Pearson $\uparrow$	SNR u $\uparrow$
2 ms	5.33	LU [ 7 ]	23.96	0.799	0.647	0.485	0.779	8.44
		NNLU [ 10 ]	24.06	0.803	0.647	0.439	0.785	8.59
		FCLU [ 9 ]	20.70	0.520	0.619	0.511	0.540	5.04
		HyU [ 12 ]	24.28	0.803	0.500	0.442	0.795	11.35
		RLU [ 11 ]	23.98	0.799	0.647	0.442	0.781	8.47
		NMF-RI [ 15 ]	24.09	0.803	0.474	0.440	0.790	8.69

Table 1: Quantitative results on the CellAtlas data for various noise levels. Columns, from left to right, report:

(i)

exposure times (less causing more noise);

(ii)

the resulting SNR of the spectral input images (aggregated over all spectral bands);

(iii)

baselines considered for the set of experiments;

(iv)

quality metrics, i.e., PSNR, MS-SSIM (MS3IM), MicroMS-SSIM (

\mu

MS3IM), LPIPS, Pearson correlation coefficient, and average SNR of unmixed images (see main text for details). Higher values are better, except for LPIPS. In each metric column, the best result is shown in bold , while the second best is underlined .

Figure 4: Spectral unmixing with controlled spectral overlap (32 bands). For this experiment, we use the Microtubules and CCPs structures from the BioSR dataset, with 32 spectral bands imaged. a,e Synthetically shifted EGFP and mTurquoise emission spectra, respectively. While we fix the spectrum associated with Microtubules, the emission spectrum of CCPs is rigidly shifted with increasing $\Delta\lambda$ . b,f Quantitative metrics over $\Delta\lambda$ , showing that $\lambda$ Split (blue) consistently outperforms all other baselines and that the performance gap widens when spectral overlap is larger (hence, when $\Delta\lambda$ is smaller). c,d,g,h Representative crops from unmixing results for $\Delta\lambda=2$ nm by $\lambda$ Split and the best-performing baseline (w.r.t. PSNR), plus the corresponding ground truth (GT) in the center. In c,d, we show Microtubules, while in g,h, we show CCPs.

		# bands Same per-band SNR				# bands Same total photon budget
Metric	Method	3	4	5	32	3	4	5	32
PSNR $\uparrow$	LU [ 7 ]	23.69	26.76	27.77	35.91	23.72	27.77	27.77	25.81
	NNLU [ 10 ]	26.21	27.30	28.56	36.02	26.79	28.36	28.56	25.83
	HyU [ 12 ]	23.69	26.87	27.19	30.37	23.72	27.61	27.19	25.66
	RLU [ 11 ]	26.39	27.56	28.91	36.68	27.13	28.63	28.91	26.15
	NMF-RI [ 15 ]	26.31	27.76	28.38	36.49	26.77	28.70	28.38	26.13

Table 2: Quantitative results on CellAtlas for varying spectral dimensionality. We evaluate spectral unmixing performance when reducing the number of acquisition bands. Two experimental settings are considered: ( left ) the number of bands is reduced while keeping the per-band SNR constant, isolating the effect of spectral dimensionality; ( right ) the total photon budget is fixed, such that fewer bands lead to higher per-band SNR. Find more details in section 6.3 . Best results are shown in bold , second best underlined .

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Figure S.1: Overview of the spectral fluorescence microscopy simulation pipeline based on microsim [ 42 ] . Multi-channel inputs are interpreted as fluorophore concentration maps $U_{\mathrm{GT}}$ , where each channel corresponds to a fluorescent probe (FP) with a known emission spectrum. First, for each FP, a noise-free spectral emission volume $S_{\mathrm{emission}}$ is generated by combining concentration maps with their discretized emission spectra $M$ , while modeling tunable illumination power and fluorophore-specific emission and excitation efficiency. Next, microscope-specific optical effects, such as point-spread functions (PSFs), are applied, and per-fluorophore emission volumes are combined to obtain an optical spectral volume $S_{\mathrm{opt}}$ consistent with the chosen imaging modality. Finally, the detection process introduces realistic noise sources, including photon and detector noise, producing the final digital spectral image $S_{\mathrm{dig}}$ .

Figure S.2: Ablation on the number of hierarchical levels

K

. Reconstruction quality as a function of exposure time (shorter exposure

\rightarrow

lower SNR) on CellAtlas (top) and BioSR (bottom), for

K\in\{2,3,4,5\}

stochastic LVAE levels; our default

K=4

is shown in blue. We report PSNR, MS-SSIM, LPIPS, and the SNR of the unmixed images (

\mathrm{SNR_{u}}

, log scale); higher is better except for LPIPS. On both datasets, deeper hierarchies perform best at high SNR while shallower ones are more robust under noise, confirming the fidelity–denoising tradeoff.

Figure S.3: Ablation on the KL loss weight

\beta

. Reconstruction quality as a function of exposure time (shorter exposure

\rightarrow

lower SNR) on BioSR, for

\beta\in\{0.1,1,2,5\}

; our default

\beta=1

is shown in blue. We report PSNR, MS-SSIM, LPIPS, and the SNR of the unmixed images (

\mathrm{SNR_{u}}

, log scale); higher is better except for LPIPS. Weak regularization (

\beta=0.1

) preserves detail at high SNR but fits noise at low SNR, whereas stronger regularization denoises more aggressively at the cost of high-SNR fidelity.

Figure S.4: Four FP setup for CellAtlas and BioSR datasets for the Spectral Unmixing and Noisy Data experiment. On the left, we report the excitation spectra of the four fluorophores together with the positions of the four excitation lasers. The number of lasers, their placement, and their relative power are selected by solving an optimization problem to achieve balanced excitation across all fluorophores, accounting for their excitation and emission efficiency (excitation cross-sections and quantum yields). On the right, we display the corresponding emission spectra alongside the placement of the 32 spectral detection bands used for imaging.

Figure S.5: Five FP setup for HHMI25 dataset for the Spectral Unmixing and Noisy Data experiment. On the left, we report the excitation spectra of the five fluorophores together with the positions of the three excitation lasers. The number of lasers, their placement, and their relative power are selected by solving an optimization problem to achieve balanced excitation across all fluorophores, accounting for their excitation and emission efficiency (excitation cross-sections and quantum yields). On the right, we display the corresponding emission spectra alongside the placement of the 32 spectral detection bands used for imaging.

Exp	SNR sp	Method	PSNR $\uparrow$	MS3IM $\uparrow$	$\mu$ MS3IM $\uparrow$	LPIPS $\downarrow$	Pearson $\uparrow$	SNR u $\uparrow$
10 ms	5.60	LU [ 7 ]	30.33	0.922	0.837	0.548	0.849	9.86
		NNLU [ 10 ]	30.42	0.923	0.842	0.503	0.852	10.69
		FCLU [ 9 ]	26.63	0.752	0.736	0.638	0.607	5.51
		HyU [ 12 ]	30.34	0.900	0.845	0.476	0.852	15.17
		RLU [ 11 ]	30.54	0.925	0.846	0.502	0.854	11.45
		NMF-RI [ 15 ]	30.56	0.925	0.842	0.497	0.854	10.59

Appendix

Table S.1: Quantitative results on the BioSR dataset for various noise levels. Columns, from left to right, report:

(i)

exposure times (less causing more noise);

(ii)

the resulting SNR of the spectral input images (aggregated over all spectral bands);

(iii)

baselines considered for the set of experiments;

(iv)

quality metrics, i.e., PSNR, MS-SSIM (MS3IM), MicroMS-SSIM (

\mu

Exp	SNR sp	Method	PSNR $\uparrow$	MS3IM $\uparrow$	LPIPS $\downarrow$	Pearson $\uparrow$	SNR u $\uparrow$
2 ms	4.49	LU [ 7 ]	27.60	0.878	0.419	0.665	4.88
		NNLU [ 10 ]	27.62	0.878	0.333	0.666	4.91
		FCLU [ 9 ]	26.01	0.772	0.398	0.524	4.39
		HyU [ 12 ]	27.94	0.873	0.402	0.687	9.49
		RLU [ 11 ]	27.77	0.882	0.332	0.679	5.57
		NMF-RI [ 15 ]	27.83	0.885	0.347	0.684	5.84

Appendix

Table S.2: Quantitative results on the HHMI25 dataset for various noise levels. Columns, from left to right, report:

(i)

exposure times (less causing more noise);

(ii)

the resulting SNR of the spectral input images (aggregated over all spectral bands);

(iii)

baselines considered for the set of experiments;

(iv)

quality metrics, i.e., PSNR, MS-SSIM (MS3IM), LPIPS, Pearson correlation coefficient, and average SNR of unmixed images (see main text for details). MicroMS-SSIM is not reported as not implemented for 3D images. Higher values are better, except for LPIPS. In each metric column, the best result is shown in bold , while the second best is underlined .

Figure S.6: Qualitative results on high-noise CellAtlas data (exposure 2 ms). Compared to $7$ baselines, $\lambda$ Split produces the cleanest results, also benefiting from its implicit noise removal capabilities. Still, in this example, we see that very high levels of noise cannot be fully denoised solely by relying on $\lambda$ Split’s implicit capabilities, suggesting that future work should also focus on incorporating explicit denoising modules [ 31 , 33 , 32 ] . Objects and details (e.g., puncta and roundish structures in the mitochondria and ER channels or elongated microtubule structures) are well recovered, facilitating possible downstream analyses.

Figure S.7: Qualitative results on high-noise HHMI25 data (exposure 2 ms). Compared to $6$ baselines, $\lambda$ Split produces the cleanest results, also benefiting from its implicit noise removal capabilities. Still, in this example, we see that very high levels of noise cannot be fully denoised solely by relying on $\lambda$ Split’s implicit capabilities, suggesting that future work should also focus on incorporating explicit denoising modules [ 31 , 33 , 32 ]

Figure S.8: Analysis of the SNR computation and the implicit denoising capability of $\lambda$ Split. For each dataset used in the increasing noise set of experiments, we show a representative structure and exposure level. In each top row, from left to right, we report example crops from: $(i)$ the spectral band corresponding to the emission peak of the FP associated with the structure; $(ii)$ the ground truth (GT) concentration map; $(iii)$ the concentration map predicted by $\lambda$ Split; $(iv)$ the baseline methods achieving the highest SNR in that setting; $(v)$ the concentration map obtained from linear unmixing. All crops are min-max normalized to the same intensity range and displayed using a standard grayscale colormap. For each image, a background patch used for SNR estimation is highlighted (red square); the content of the extracted patch is displayed in a corner, together with its mean ( $\mu$ ) and standard deviation ( $\sigma$ ); the resulting SNR value is also reported. Notice that the SNR values differ from those reported in the result tables, as they are computed only on a small sample crop rather than the entire dataset. The orange profiles show intensity along the annotated 10-pixel thick line and highlight the denoising effect of $\lambda$ Split. Indeed, we observe a reduction in noise, as evidenced by fewer oscillations in the line histogram and a larger gap separating the signal peaks from the background level.

EGFP – ER/CCPs
Metric	Method	5 bands					32 bands
		$\Delta\lambda=2$	$\Delta\lambda=5$	$\Delta\lambda=10$	$\Delta\lambda=20$	$\Delta\lambda=50$	$\Delta\lambda=2$	$\Delta\lambda=5$	$\Delta\lambda=10$	$\Delta\lambda=20$	$\Delta\lambda=50$
PSNR $\uparrow$	LU [ 7 ]	31.73	38.45	43.77	49.13	53.31	31.44	38.35	43.75	48.52	52.52
	NNLU [ 10 ]	32.76	38.98	44.17	49.44	53.38	31.53	37.15	43.65	48.52	52.52
	HyU [ 12 ]	31.88	38.56	43.66	49.31	53.12	32.50	39.63	44.64	48.74	52.16
	RLU [ 11 ]	31.69	37.68	44.07	49.68	53.95	32.02	38.05	43.79	48.55	52.75

Appendix

Table S.3: Quantitative results for different spectral overlaps. For these experiments, we use the ER and CCP structures from the BioSR dataset and associate them with the EGFP fluorophore. We report the results for both 5- and 32-band input spectral data. For each metric, the best result is shown in bold , while the second best is underlined .

EGFP – Microtubules/CCPs
Metric	Method	5 bands					32 bands
		$\Delta\lambda=2$	$\Delta\lambda=5$	$\Delta\lambda=10$	$\Delta\lambda=20$	$\Delta\lambda=50$	$\Delta\lambda=2$	$\Delta\lambda=5$	$\Delta\lambda=10$	$\Delta\lambda=20$	$\Delta\lambda=50$
PSNR $\uparrow$	LU [ 7 ]	32.76	39.59	44.95	50.36	54.53	32.34	39.33	44.77	49.57	53.44
	NNLU [ 10 ]	33.78	40.15	45.33	50.65	54.60	32.16	38.06	44.67	49.56	53.44
	HyU [ 12 ]	32.98	39.76	44.86	50.54	54.12	33.34	40.49	45.53	49.64	52.75
	RLU [ 11 ]	32.33	38.59	45.14	50.83	55.13	32.66	38.86	44.60	49.16	53.60

Appendix

Table S.4: Quantitative results for different spectral overlaps. For these experiments, we use the Microtubules and CCPs structures from the BioSR dataset and associate them with the EGFP fluorophore. We report the results for both 5- and 32-band input spectral data. For each metric, the best result is shown in bold , while the second best is underlined .

mTurquoise – ER/CCPs
Metric	Method	5 bands					32 bands
		$\Delta\lambda=2$	$\Delta\lambda=5$	$\Delta\lambda=10$	$\Delta\lambda=20$	$\Delta\lambda=50$	$\Delta\lambda=2$	$\Delta\lambda=5$	$\Delta\lambda=10$	$\Delta\lambda=20$	$\Delta\lambda=50$
PSNR $\uparrow$	LU [ 7 ]	29.96	36.21	41.30	45.18	50.55	29.57	35.78	40.83	45.20	49.68
	NNLU [ 10 ]	31.50	37.05	42.19	45.63	50.69	30.91	34.63	40.53	45.20	49.68
	HyU [ 12 ]	29.85	36.41	41.53	45.01	50.34	30.54	37.45	42.32	45.87	49.52
	RLU [ 11 ]	31.07	35.65	41.82	46.00	51.29	31.32	35.91	40.88	45.14	49.77

Appendix

Table S.5: Quantitative results for different spectral overlaps. For these experiments, we use the ER and CCP structures from the BioSR dataset and associate them with the mTurquoise fluorophore. We report the results for both 5- and 32-band input spectral data. For each metric, the best result is shown in bold , while the second best is underlined .

mTurquoise – Microtubules/CCPs
Metric	Method	5 bands					32 bands
		$\Delta\lambda=2$	$\Delta\lambda=5$	$\Delta\lambda=10$	$\Delta\lambda=20$	$\Delta\lambda=50$	$\Delta\lambda=2$	$\Delta\lambda=5$	$\Delta\lambda=10$	$\Delta\lambda=20$	$\Delta\lambda=50$
PSNR $\uparrow$	LU [ 7 ]	30.95	37.33	42.48	46.38	51.76	30.36	36.67	41.76	46.17	50.57
	NNLU [ 10 ]	32.46	38.22	43.39	46.80	51.89	31.52	35.41	41.47	46.17	50.57
	HyU [ 12 ]	30.97	37.65	42.81	46.25	51.55	31.35	38.18	43.08	46.66	50.30
	RLU [ 11 ]	31.70	36.52	42.88	47.08	52.42	31.96	36.67	41.63	45.60	50.41

Appendix

Table S.6: Quantitative results for different spectral overlaps. For these experiments, we use the Microtubules and CCPs structures from the BioSR dataset and associate them with the mTurquoise fluorophore. We report the results for both 5- and 32-band input spectral data. For each metric, the best result is shown in bold , while the second best is underlined .

Figure S.9: Spectral unmixing with controlled spectral overlap (32 bands). For this experiment, we use the ER and CCPs structures from the BioSR dataset, with 32 spectral bands imaged. a,e Synthetically shifted EGFP and mTurquoise emission spectra, respectively. While we fix the spectrum associated with ER, the emission spectrum of CCPs is rigidly shifted with increasing $\Delta\lambda$ . b,f Quantitative metrics over $\Delta\lambda$ , showing that $\lambda$ Split (blue) consistently outperforms all other baselines and that the performance gap widens when spectral overlap is larger (hence, when $\Delta\lambda$ is smaller). c,d,g,h Representative crops from unmixing results for $\Delta\lambda=2$ nm by $\lambda$ Split and the best-performing baseline (w.r.t. PSNR), plus the corresponding ground truth (GT) in the center. In c,d, we show ER, while in g,h, we show CCPs.

Figure S.10: Spectral unmixing with controlled spectral overlap (5 bands). For this experiment, we use the ER and CCPs structures from the BioSR dataset, with 5 spectral bands imaged. a,e Synthetically shifted EGFP and mTurquoise emission spectra, respectively. While we fix the spectrum associated with ER, the emission spectrum of CCPs is rigidly shifted with increasing $\Delta\lambda$ . b,f Quantitative metrics over $\Delta\lambda$ , showing that $\lambda$ Split (blue) consistently outperforms all other baselines and that the performance gap widens when spectral overlap is larger (hence, when $\Delta\lambda$ is smaller). c,d,g,h Representative crops from unmixing results for $\Delta\lambda=2$ nm, by $\lambda$ Split and the best-performing baseline (w.r.t. PSNR), plus the corresponding ground truth (GT) in the center. In c,d, we show ER, while in g,h, we show CCPs.

Figure S.11: Spectral unmixing with controlled spectral overlap (5 bands). For this experiment, we use the Microtubules and CCPs structures from the BioSR dataset, with 5 spectral bands imaged. a,e Synthetically shifted EGFP and mTurquoise emission spectra, respectively. While we fix the spectrum associated with Microtubules, the emission spectrum of CCPs is rigidly shifted with increasing $\Delta\lambda$ . b,f Quantitative metrics over $\Delta\lambda$ , showing that $\lambda$ Split (blue) consistently outperforms all other baselines and that the performance gap widens when spectral overlap is larger (hence, when $\Delta\lambda$ is smaller). c,d,g,h Representative crops from unmixing results for $\Delta\lambda=2$ nm, by $\lambda$ Split and the best-performing baseline (w.r.t. PSNR), plus the corresponding ground truth (GT) in the center. In c,d, we show Microtubules, while in g,h, we show CCPs.

		# bands Same per-band SNR				# bands Same total photon budget
Metric	Method	3	4	5	32	3	4	5	32
$\mu$ MS3IM $\uparrow$	LU [ 7 ]	0.516	0.548	0.597	0.696	0.564	0.726	0.597	0.770
	NNLU [ 10 ]	0.728	0.691	0.620	0.696	0.773	0.750	0.620	0.770
	HyU [ 12 ]	0.516	0.460	0.781	0.873	0.564	0.751	0.781	0.660
	RLU [ 11 ]	0.733	0.693	0.813	0.907	0.631	0.752	0.813	0.713
	NMF-RI [ 15 ]	0.724	0.700	0.544	0.703	0.757	0.747	0.544	0.596

Appendix

Table S.7: Quantitative results on CellAtlas for varying spectral dimensionality. We evaluate spectral unmixing performance when reducing the number of acquisition bands. Two experimental settings are considered: ( left ) the number of bands is reduced while keeping the per-band SNR constant, isolating the effect of spectral dimensionality; ( right ) the total photon budget is fixed, such that fewer bands lead to higher per-band SNR. Find more details in section 6.3 . Best results are shown in bold , second best underlined . Note that SNR u is not available for LUMoS, as the algorithm performs hard clustering and thus produces a flat, uniform background with a standard deviation of

0

, rendering our SNR computation ill-posed (see also eq. S.12 ).

Figure S.12: Qualitative results for the reduced spectral dimensionality experiments (CellAtlas, 5-band acquisition). We compare $\lambda$ Split against six baselines on representative structures. Although $\lambda$ Split achieves the best quantitative fidelity, its predictions do not always appear as high-contrast as those of some baselines. This behavior can be attributed to two design choices. First, the network is trained solely with an MSE objective and does not include perceptual losses, which could be explored in future work to improve visual appearance. Second, $\lambda$ Split does not impose constraints on the unmixed images (e.g., non-negativity or sparsity), allowing the model to fully exploit learned structural priors. While this flexibility improves reconstruction fidelity, spectral overlap can induce intensity redistribution across channels. For instance, bright structures in one channel may appear as faint negative counterparts in another, reducing perceived contrast in our figure while still yielding the best MSE.

				Same per-band SNR			Same tot. photon budget
Bands	Method	SSL	Params	PSNR $\uparrow$	MS3IM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	MS3IM $\uparrow$	LPIPS $\downarrow$
3	AutoUnmix [ 19 ]	✗	167M	32.24	0.961	0.106	33.05	0.973	0.086
	UNet [ 39 ]	✗	31M	19.96	0.621	0.631	20.11	0.606	0.496
	$\lambda$ Split (ours)	✓	3M	27.25	0.878	0.283	27.45	0.868	0.273
4	AutoUnmix	✗	293M	34.01	0.980	0.088	34.50	0.982	0.081
	UNet	✗	31M	20.22	0.642	0.526	19.38	0.638	0.549

Appendix

Table S.8: Quantitative comparison of

\lambda

Split vs. supervised spectral unmixing baselines for low spectral dimensionality . The first four columns, from left to right, report:

(i)

the number of bands in the input spectral data;

(ii)

the considered baselines;

(iii)

whether the method is self-supervised;

(iv)

the number of parameters in the architecture used by each baseline. The following columns report a subset of metrics for experiments in which: ( left ) the number of bands is reduced while keeping the per-band SNR constant, isolating the effect of spectral dimensionality; ( right ) the total photon budget is fixed, such that fewer bands lead to higher per-band SNR.

EGFP – ER/CCPs – 5 bands
Metric	Method	SSL	#params	$\Delta\lambda=2$	$\Delta\lambda=5$	$\Delta\lambda=10$	$\Delta\lambda=20$	$\Delta\lambda=50$
PSNR $\uparrow$	AutoUnmix [ 19 ]	✗	450M	46.82	50.14	51.10	50.50	52.62
	UNet [ 39 ]	✗	31M	25.40	27.06	25.85	26.43	25.87
	$\lambda$ Split (ours)	✓	3M	37.61	42.91	47.55	51.40	54.29
MS3IM $\uparrow$	AutoUnmix	✗	450M	0.998	0.999	0.999	0.999	0.999
	UNet	✗	31M	0.689	0.799	0.758	0.760	0.672

Appendix

Table S.9: Quantitative comparison of

\lambda

Split vs. supervised spectral unmixing baselines for different spectral overlaps. In these experiments, we consider the ER and CCPs structures from the BioSR dataset and associate them with the EGFP fluorophore. The datasets used here comprise 5-band spectral images.

EGFP – Microtubules/CCPs – 5 bands
Metric	Method	SSL	#params	$\Delta\lambda=2$	$\Delta\lambda=5$	$\Delta\lambda=10$	$\Delta\lambda=20$	$\Delta\lambda=50$
PSNR $\uparrow$	AutoUnmix [ 19 ]	✗	450M	48.08	49.70	51.30	52.88	53.62
	UNet [ 39 ]	✗	31M	26.52	28.52	27.00	26.59	27.33
	$\lambda$ Split (ours)	✓	3M	37.91	42.81	47.31	51.27	52.65
MS3IM $\uparrow$	AutoUnmix	✗	450M	0.998	0.999	0.999	1.000	1.000
	UNet	✗	31M	0.728	0.836	0.726	0.736	0.746

Appendix

Table S.10: Quantitative comparison of

\lambda

Split vs. supervised spectral unmixing baselines for different spectral overlaps. In these experiments, we consider the Microtubules and CCPs structures from the BioSR dataset and associate them with the EGFP fluorophore. The datasets used here comprise 5-band spectral images.

mTurquoise – ER/CCPs – 5 bands
Metric	Method	SSL	#params	$\Delta\lambda=2$	$\Delta\lambda=5$	$\Delta\lambda=10$	$\Delta\lambda=20$	$\Delta\lambda=50$
PSNR $\uparrow$	AutoUnmix [ 19 ]	✗	450M	43.18	48.58	49.39	49.67	52.27
	UNet [ 39 ]	✗	31M	27.84	28.67	28.31	29.29	25.76
	$\lambda$ Split (ours)	✓	3M	36.88	42.24	47.32	49.75	52.60
MS3IM $\uparrow$	AutoUnmix	✗	450M	0.993	0.998	0.999	0.999	0.999
	UNet	✗	31M	0.817	0.861	0.832	0.879	0.747

Appendix

Table S.11: Quantitative comparison of

\lambda

Split vs. supervised spectral unmixing baselines for different spectral overlaps. In these experiments, we consider the ER and CCPs structures from the BioSR dataset and associate them with the mTurquoise fluorophore. The datasets used here comprise 5-band spectral images.

mTurquoise – Microtubules/CCPs – 5 bands
Metric	Method	SSL	#params	$\Delta\lambda=2$	$\Delta\lambda=5$	$\Delta\lambda=10$	$\Delta\lambda=20$	$\Delta\lambda=50$
PSNR $\uparrow$	AutoUnmix [ 19 ]	✗	450M	46.09	48.47	49.85	51.69	52.52
	UNet [ 39 ]	✗	31M	26.01	26.32	26.56	28.36	27.54
	$\lambda$ Split (ours)	✓	3M	37.64	43.17	46.52	49.26	53.27
MS3IM $\uparrow$	AutoUnmix	✗	450M	0.997	0.998	0.999	0.999	1.000
	UNet	✗	31M	0.662	0.704	0.716	0.827	0.801

Appendix

Table S.12: Quantitative comparison of

\lambda

Split vs. supervised spectral unmixing baselines for different spectral overlaps. In these experiments, we consider the Microtubules and CCPs structures from the BioSR dataset and associate them with the mTurquoise fluorophore. The simulated datasets used here comprise 5-band spectral images.

Figure S.13: Qualitative results for supervised baselines for the reduced spectral dimensionality experiments (CellAtlas, 5-band acquisition). We compare $\lambda$ Split against two supervised baselines, AutoUnmix and UNet. AutoUnmix produces high-quality unmixing results, benefiting from strong supervision through paired noise-free GT unmixed targets and a large-capacity architecture. In contrast, the standard UNet baseline fails to reliably separate the fluorophore signals, producing visibly mixed structures. $\lambda$ Split predictions remain structurally consistent and faithfully recover the underlying biological structures, although they may appear perceptually less contrasted than AutoUnmix.

Figure S.14: Qualitative comparison with AutoUnmix for increasing spectral overlap experiments. Representative unmixed concentration maps of ER and CCP structures on the BioSR dataset with 5 spectral bands are shown for $\lambda$ Split and AutoUnmix at $\Delta\lambda=50$ . Because of the absence of "negative" structures in this case, we display the unmixed images using the standard grayscale colormap. At this level of spectral separation, $\lambda$ Split produces results comparable to AutoUnmix both visually and according to the corresponding intensity histograms, despite operating without supervision or explicit denoising.

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