Spectral Unmixing

Decompose mixed spectral signals into constituent endmember abundances using linear unmixing methods.

Model: Textbook approximation

Endmember 1Share of photons

Endmember 2Share of photons

Endmember 3Share of photons

Endmember 4Share of photons

Detected photons per pixelRead noise per band (e⁻)Spectral bandsNoise seedRange start (nm)Range end (nm)

Method

Results

Dye	True	Estimate	±σ
EGFP	50.0 %	50.3 %	2.4 %
mVenus	30.0 %	29.2 %	2.5 %
mCherry	20.0 %	20.5 %	1.2 %

Condition number κ(E)2.23

Residual rms6.57 e⁻/band

χ²/(B − K)0.99

Band width7.50 nm

The ±σ column is the OLS standard error of each share, from (EᵀE)⁻¹EᵀVE(EᵀE)⁻¹ by the delta method (FCLS errors are smaller); χ²/(B − K) ≈ 1 when the model fits within the noise.

Measured spectrum and unmixed components

Relative error σ/x vs photon budget

Model

y = E·x + n, E_bj = ∫_band b F_j dλ / ∫_range F_j dλ, var(n_b) = (E·x)_b + σ_read²

OLS x̂ = (EᵀE)⁻¹Eᵀy; NNLS min‖y − Ex‖² with x ≥ 0; FCLS adds Σx = Σy

Cov(x̂) = (EᵀE)⁻¹ Eᵀ diag(var) E (EᵀE)⁻¹, κ(E) = √(λ_max/λ_min of EᵀE)

Endmembers are the sampled FPbase emission spectra (CC BY-SA 4.0) of the fluorescence-spectra page, binned into equal bands between the range limits; x is the number of detected photons from each dye. One noisy pixel is simulated from a seeded generator, with shot noise in its Gaussian limit and read noise in every band; change the seed for another noise draw. Background, autofluorescence and errors in the reference spectra are not included: in practice those, not shot noise, often limit unmixing.

Model and references

Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers

Sampled FPbase emission spectra in equal bands; one pixel with shot and read noise from a seeded generator, solved by OLS, NNLS or FCLS, with OLS error bars. Background and reference-spectrum errors are left out.

Lambert T. J. (2019). FPbase: a community-editable fluorescent protein database. Nat. Methods 16, 277–278. Emission spectra from the FPbase API, CC BY-SA 4.0 (read 2026-11-05). doi:10.1038/s41592-019-0352-8
Zimmermann T., Rietdorf J., Pepperkok R. (2003). Spectral imaging and its applications in live cell microscopy. FEBS Lett. 546, 87–92. Linear unmixing of fluorescent proteins. doi:10.1016/S0014-5793(03)00521-0
Lawson C. L., Hanson R. J. (1995). Solving Least Squares Problems. SIAM, ch. 23. The NNLS active-set algorithm. doi:10.1137/1.9781611971217
Heinz D. C., Chang C.-I (2001). Fully constrained least squares linear spectral mixture analysis. IEEE Trans. Geosci. Remote Sens. 39, 529–545. doi:10.1109/36.911111

Report a problem