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A polarimetric coherency matrix can be written as a weighted sum of orthogonal eigenchannels. Cloude decomposition uses that spectral structure to expose dominant and secondary scattering directions. The largest eigenchannel can be informative, but it is not automatically the whole target. Its rendered colour also depends on basis, eigenvector convention, averaging and display choices—especially when eigenvalues are close.
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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 a coherency matrix
After calibration and a stated multilook or averaging operation, a reciprocal full-polarimetric observation is often represented by a 3×3 Hermitian positive-semidefinite coherency matrix T. Its trace is the total span under the chosen normalisation.
Positive semidefiniteness gives real non-negative eigenvalues. Calibration errors, incomplete data or numerical processing can produce small negative values; these require diagnosis rather than physical interpretation.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Write the eigenchannel decomposition
Spectral decomposition gives T=λ1u1u1ᴴ+λ2u2u2ᴴ+λ3u3u3ᴴ with λ1≥λ2≥λ3≥0 and orthonormal eigenvectors ui. Each rank-one term is an eigenchannel with weight λi.
ESA SNAP describes the Cloude option as an eigenvector-based decomposition that extracts the dominant scattering mechanism through the largest eigenvalue. The output convention and band names must still be read from the exact software version.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Work a three-channel ledger
Take T=diag(6,3,1) in a stated orthonormal basis. The span is 10 and the normalised eigenvalue weights are 0.6, 0.3 and 0.1.
| Channel | Eigenvalue | Share of span | If retained alone |
|---|---|---|---|
| 1 | 6 | 60% | Dominant rank-one summary |
| 2 | 3 | 30% | Largest omitted component |
| 3 | 1 | 10% | Smallest omitted component |
| Total | 10 | 100% | Full coherency matrix |
Calling this pixel “rank one” would discard substantial mixed scattering. A dominant channel is a summary, not a certificate that λ2 and λ3 are zero.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Change the eigenvalue gap
Compare diag(6,3,1) with a matrix having eigenvalues 6,3.1 and 2.9. The first remains separated, but the second and third form a nearly degenerate pair.
| Spectrum | Leading share | λ2−λ3 | Stability reading |
|---|---|---|---|
| 6, 3, 1 | 60% | 2 | Secondary directions separated |
| 6, 3.1, 2.9 | 50% | 0.2 | Secondary eigenvectors sensitive |
| 4, 3.9, 3.8 | 34.2% | 0.1 and 0.1 | No clear dominant rank-one channel |
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Understand eigenvector ambiguity
Multiplying an eigenvector by a complex phase leaves uiu iᴴ unchanged. When two eigenvalues are exactly equal, any orthonormal rotation inside their shared eigenspace is equally valid.
Near equality, small perturbations from speckle, calibration or window choice can rotate displayed eigenvectors sharply. The sum of the nearly degenerate rank-one terms may stay stable while their individual colours change.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Separate eigenchannels from RGB display
An RGB product assigns selected eigenchannel quantities to red, green and blue, then applies scaling, clipping or gamma. The colour is therefore a rendering of an algorithmic coordinate, not a universal material label.
Archive the numerical eigenvalues, eigenvectors, basis, colour assignment and stretch. A screenshot alone cannot show whether a hue change came from scattering, eigenvector rotation or display settings.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Connect the polarimetric decomposition routes
Use the Pauli decomposition guide to separate scattering-vector coordinates from colour meaning. Use entropy-alpha and eigenvalue ambiguity to summarise the full normalised spectrum.
Use Freeman-Durden decomposition when fitting explicit surface, double-bounce and volume models, and polarimetric coherence for channel-consistency and calibration checks.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Test rank assumptions against scale
A point target may be close to rank one, while a distributed pixel can mix independent mechanisms. Increasing the averaging window often stabilises estimates but can add land-cover boundaries and increase effective rank.
Compare eigenvalue fractions across several defensible window sizes. If the “dominant mechanism” changes with support, report that scale dependence rather than selecting the most vivid product.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Know the limits and common misconceptions
The largest eigenvalue is not automatically a named physical mechanism. Its eigenvector still requires a convention-aware interpretation, and multiple geometries can produce similar coordinates.
Basis invariance of eigenvalues under a consistent unitary transformation does not make every rendered eigenvector colour invariant. Degeneracy, non-unitary preprocessing, orientation effects and channel ordering all matter.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Use an evidence and source routine
Record sensor, wavelength, incidence geometry, calibration, reciprocity test, speckle filter, averaging support, basis, matrix normalisation, eigenvalue ordering, gap ratios, rank threshold, colour mapping, stretch and sensitivity to window choice.
ESA SNAP’s Polarimetric Decomposition Operator documentation identifies Cloude as an eigenvector-based decomposition. NASA Earthdata’s SAR polarimetry training material develops the coherency-matrix and eigenvalue framework used in polarimetric interpretation.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Does a rank-one reconstruction preserve total span?
Only when the omitted eigenvalues are zero. Otherwise it preserves λ1 and drops λ2+λ3.
Why can colours flip between similar pixels?
Close eigenvalues make eigenvectors sensitive to small perturbations, and RGB mapping amplifies the coordinate change.
Are the eigenvalues basis invariant?
They are invariant under a consistent unitary change of basis, but preprocessing and normalisation must match.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue with a perturbation check
Study the lower 2×2 block [[3,ε],[ε,3]]. Its eigenvalues are 3+ε and 3−ε, while the eigenvectors switch orientation when the sign of ε changes. Explain why the two-dimensional subspace is unchanged even though two channel colours may exchange.
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