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A polarimetric radar pixel can contain one dominant scattering mechanism or a mixture whose components vary across looks. Eigenvalue decomposition separates the averaged coherency matrix into orthogonal contributions. Entropy summarises how evenly their powers are distributed, while mean alpha summarises their eigenvector-derived scattering character. The pair is useful for classification, but it is not a unique physical inventory of the resolution cell.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. Start from an averaged coherency matrix
After calibration and multilooking, a polarimetric observation can be represented by a Hermitian positive-semidefinite covariance or coherency matrix. Its eigenvalues are nonnegative powers associated with orthogonal eigenvectors.
The result depends on polarization basis, reciprocity assumptions, noise correction and the averaging ensemble. An eigen-decomposition cannot repair a badly calibrated or heterogeneous input by itself.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Normalise the eigenvalue powers
Let λ1≥λ2≥λ3≥0 be the eigenvalues and define pi=λi/(λ1+λ2+λ3). The pi values sum to one and describe the relative contribution of the three eigenchannels.
Scaling all channels by a common gain changes total power but not pi. Unequal channel gain, cross-talk or noise subtraction can change both the eigenvalues and their interpretation.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Calculate polarimetric entropy
The three-component entropy is H=−Σpi log3(pi), with zero terms interpreted by continuity. H=0 when one eigenvalue carries all power and H=1 when the three powers are equal.
| Eigenvalue shares | Entropy H | Reading |
|---|---|---|
| 1, 0, 0 | 0 | One dominant eigenchannel |
| 0.70, 0.20, 0.10 | about 0.73 | A dominant mechanism with substantial mixture |
| 1/3, 1/3, 1/3 | 1 | No dominant eigenvalue |
For shares 0.70, 0.20 and 0.10, the calculation is −[0.70ln0.70+0.20ln0.20+0.10ln0.10]/ln3≈0.73.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Build the mean alpha angle
Each eigenvector is parameterised by a scattering-type angle αi under the chosen decomposition convention. The mean angle is ᾱ=Σpiαi.
| Eigenchannel | pi | αi | Weighted contribution |
|---|---|---|---|
| 1 | 0.70 | 10° | 7° |
| 2 | 0.20 | 50° | 10° |
| 3 | 0.10 | 80° | 8° |
| Total | ᾱ=25° |
Mean alpha is not an incidence angle or a direct building-orientation estimate. It is a coordinate in a model-based scattering feature space.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. See why entropy and alpha form zones
Low entropy with a stable ᾱ supports a relatively clear dominant mechanism. High entropy indicates that several eigenchannels contribute, so a single mechanism label becomes less secure.
Entropy–alpha diagrams divide the feature space into interpretation zones. Zone boundaries are classification aids, not physical phase boundaries; neighbouring pixels can cross them because of speckle, averaging or geometry.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Expose eigenvalue ambiguity
When λ2 and λ3 are nearly equal, their associated eigenvectors can rotate strongly under a small perturbation even while the matrix changes little. Entropy may remain stable while individual α2 and α3 values become unstable.
The changed-context check is to perturb the coherency matrix within an estimated noise level and repeat the decomposition. Report whether the class and ᾱ remain stable, not only the nominal eigenvectors.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Transfer from pure targets to mixed pixels
A resolution cell containing roof, road and vegetation can combine coherent and distributed returns. The averaged matrix may have the same H and ᾱ as a different mixture, because two summary coordinates cannot uniquely reconstruct every component.
Compare window sizes, spatial neighbours, total power and complementary features. If the label changes whenever the support shifts by one pixel, mixture and boundary effects are more plausible than a sharp land-cover change.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect orientation, coherence and calibration
Use polarimetric deorientation and urban misclassification when target alignment moves power among channels. Use polarimetric coherence and calibration drift when channel relationships or instrument stability are uncertain.
Entropy–alpha features should be recomputed only after the selected compensation and calibration steps are documented. Different preprocessing chains need not yield directly comparable zones.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Know the limits and common misconceptions
High entropy does not automatically mean vegetation, randomness in nature or poor data. It can also arise from deterministic mixtures, orientation diversity, temporal change or noise.
Low entropy does not prove a pure material. A strong dominant scatterer can hide weaker components, and eigenvectors remain basis- and convention-dependent.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Use an evidence and source routine
Record wavelength, incidence angle, orbit heading, basis, calibration, noise correction, look count, averaging support, eigenvalue ordering, logarithm normalisation, α convention, zone boundaries and stability tests.
ESA’s SAR Polarimetry: Basic Concepts develops eigenvalue-based entropy and alpha parameters. ESA’s Polarimetric Remote Sensing: Advanced Concepts presents entropy–alpha classification and anisotropy for complex scenes.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Is entropy a measure of image noise?
Not by itself. It measures the balance of coherency eigenvalues, which can reflect signal mixture as well as noise.
Can H and ᾱ identify a unique material?
No. Different mixtures can occupy the same two-dimensional feature region.
Why do eigenvectors become unstable?
When eigenvalues are close, small matrix perturbations can rotate the corresponding eigenspace substantially.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue with a changed-mixture check
Recalculate H after changing the shares from 0.70, 0.20, 0.10 to 0.55, 0.30, 0.15. Keep the three αi values fixed, compute ᾱ and explain which summary changes more.
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