A mixed pixel is a spectrum to explain, not a label to force
One remote-sensing pixel can contain water, vegetation, soil, roof and shadow. Linear spectral unmixing represents its measured spectrum as a weighted combination of endmember spectra. The fitted weights are often called abundances, but they deserve that physical interpretation only when the mixing model, endmembers and constraints match the scene.
The useful workflow is therefore more than solving a least-squares equation. Build defensible endmembers, examine conditioning, fit under appropriate constraints, inspect the residual spectrum, and transfer the model to surfaces whose spectra vary.
This guide is a specialist extension of the Gold Standard of Remote Sensing. The sequence is deliberate: explain the model, calculate one case, change the context, and check what remains valid.
Find your route through this guide
| If you want to… | Start | Continue |
|---|---|---|
| understand the mechanism | Chapter 1 | Chapters 2–3 |
| work through the calculation | Chapter 4 | Chapter 5 |
| interpret evidence and diagnostics | Chapter 6 | Chapter 7 |
| transfer the model to a changed case | Chapter 8 | Chapter 9 |
| check limits, FAQs and practice | Chapter 10 | Chapters 11–12 |
Chapter contents
Foundations and setup
Calculation and evidence
1. The linear mixing model
Write a pixel spectrum y with B bands as y=Ma+e. The columns of M are p endmember spectra, a contains their coefficients, and e is the residual. Each band supplies one equation, but solvability depends on the number and independence of the endmembers.
The model is most plausible when materials occupy separate subpixel areas and photons interact predominantly with one material before reaching the sensor. Multiple scattering and intimate mineral mixtures can be nonlinear.
| Symbol | Role |
|---|---|
| y | Observed B-band pixel spectrum |
| M | B×p endmember matrix |
| a | p fitted coefficients or candidate abundances |
| e | Residual spectrum y−Ma |
2. Endmembers define the explanation
An endmember is a representative spectrum for one component of the mixture. It can come from field or laboratory spectra, a spectral library, or relatively pure pixels in the image.
Endmember selection is often the weak link because real surfaces vary with moisture, species, grain size, illumination and sensor response. A mathematically excellent fit to poor endmembers is still a poor physical explanation.
Store the source, date, processing level and wavelength grid for every endmember. These are part of the model.
3. Unconstrained, non-negative and fully constrained fits
Unconstrained least squares minimises squared residuals but may return negative coefficients or a sum unlike one. Non-negative least squares enforces aⱼ≥0. Fully constrained least squares adds Σaⱼ=1.
Non-negativity supports an abundance interpretation. Sum-to-one is appropriate when the included endmembers are meant to exhaust the pixel and the radiometric model has no unresolved shade or scale effect.
| Fit | Constraints | Diagnostic value |
|---|---|---|
| UCLS | None | Can expose poor or missing endmembers through negative values |
| NNLS | aⱼ≥0 | Avoids negative fractions but need not close the mixture |
| FCLS | aⱼ≥0 and Σaⱼ=1 | Enforces a closed convex mixture |
4. Worked example: a three-endmember pixel
Use three bands and endmembers vegetation v=(0.05,0.10,0.50), soil s=(0.30,0.40,0.35) and water w=(0.02,0.03,0.01). Suppose the pixel contains coefficients (0.50,0.30,0.20).
| Band | 0.50 vegetation | 0.30 soil | 0.20 water | Mixed y |
|---|---|---|---|---|
| 1 | 0.025 | 0.090 | 0.004 | 0.119 |
| 2 | 0.050 | 0.120 | 0.006 | 0.176 |
| 3 | 0.250 | 0.105 | 0.002 | 0.357 |
Solving with the same matrix recovers the coefficients in the noise-free case. In real imagery, noise, spectral response and endmember mismatch produce residuals and uncertainty.
5. Simplex geometry and conditioning
With sum-to-one and non-negativity, all modelled mixtures lie inside the convex hull of the endmembers. For three endmembers, that hull is a triangle in spectral space after accounting for dimensionality.
If two endmember spectra are nearly parallel, the matrix is ill-conditioned: many coefficient combinations produce similar spectra. Small measurement noise can then cause large abundance changes.
Report conditioning or sensitivity rather than trusting a stable-looking map. Smooth colours can hide unstable inverse problems.
6. Read the residual spectrum
A single residual norm compresses useful information. Plot e by band. A structured residual around a water absorption, red edge or thermal feature can indicate a missing endmember, band misregistration, atmospheric residual or nonlinear interaction.
Compare residuals across land-cover types and brightness. If error grows with illumination, a shade or scaling model may be needed. If error is concentrated at boundaries, spatial mixing or geolocation may dominate.
Residuals diagnose the current model; they do not automatically identify the correct replacement.
7. Endmember variability is part of the scene
Vegetation is not one spectrum, nor is soil or water. Phenology, water content, particle size, turbidity, sun angle and sensor geometry can shift an endmember within a spectral family.
One fixed column per material can force natural variability into false abundance change. Libraries, bundles, multiple-endmember models or probabilistic descriptions can represent that variability, but they also increase identifiability demands.
Separate ‘more material’ from ‘different spectrum of the same material’ whenever the decision depends on change.
8. Transfer check: change the vegetation spectrum
Keep the worked pixel coefficients but change vegetation from (0.05,0.10,0.50) to (0.08,0.15,0.45), representing a plausible spectral-state change. The new mixed spectrum becomes (0.134,0.201,0.332).
Fit that new pixel with the old endmember matrix. The solver must absorb the mismatch by changing coefficients or leaving a structured residual. If the map is interpreted as abundance change, spectral variability becomes a false ecological story.
What transfers is the matrix-and-residual workflow. The original vegetation column does not transfer without testing.
9. Spatial support, shade and nonlinear mixing
Linear area mixing is most defensible when materials form macroscopic patches within the sensor footprint. Intimate mixtures, layered canopies and repeated scattering can depart from linearity.
Changing spatial resolution also changes the mixture population. Resampling a fine image before unmixing is not always equivalent to unmixing first and aggregating the coefficients, especially when preprocessing or constraints are nonlinear.
Shade can be an explicit endmember, an illumination factor or a source of closure failure. Choose one interpretation and test it.
10. Limits and common misconceptions
- A low residual does not prove the endmembers are physically correct.
- Non-negative coefficients are not automatically area fractions.
- Sum-to-one is an assumption about closure, not a universal law.
- More endmembers can worsen conditioning and create non-unique explanations.
- A spectral library must be convolved and processed consistently with the sensor product.
- Endmember variability can masquerade as abundance change.
- Linear unmixing may fail for intimate or multiple-scattering mixtures.
When the intended claim is physical fraction, validate against independent field or high-resolution reference data.
11. Frequently asked questions
How many endmembers can I fit?
No more independent components than the effective spectral dimension supports, and usually fewer once noise and similarity are considered.
Should abundances always sum to one?
Only when the model includes all relevant components and the chosen radiometric representation supports a closed mixture.
Why are some unconstrained coefficients negative?
They can signal noise, collinearity, missing components or endmember mismatch. They are a diagnostic, not a negative physical area.
12. A practical unmixing checklist
| Stage | Evidence to retain |
|---|---|
| Preprocessing | Calibration, atmosphere, wavelength and bad-band choices |
| Endmembers | Source, variability and sensor convolution |
| Fit | Solver, constraints and conditioning |
| Diagnostics | Residual spectra by surface class |
| Validation | Independent fractions or controlled mixtures |
| Transfer | Season, sensor, scale and illumination tests |
The gold standard is a defensible explanation of the spectrum, not merely a colourful coefficient image.
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