Hyperspectral Analysis
Technical & Scientific Documentation

1. Scope & Final Hyperspectral Toolbox

This module starts after radiometric and atmospheric preprocessing. Its preferred input is a calibrated reflectance cube with reliable wavelength metadata. The tools are intentionally independent: users do not have to run every tool in sequence.

Hyperspectral
├─ Spectral Smoothing
├─ Spectral Derivative
├─ Continuum Removal
├─ Minimum Noise Fraction (MNF)
├─ Endmember Extraction
├─ Spectral Library Matching
│    ├─ SAM — Spectral Angle Mapper
│    └─ SID — Spectral Information Divergence
└─ Spectral Unmixing

2. Scientific Foundation

2.1 Hyperspectral pixel as a vector

A hyperspectral pixel is not a single brightness value. It is a vector containing measurements across many narrow spectral bands. For a pixel with N bands:

If the image contains 200 bands, each pixel is a point in a 200-dimensional spectral space. A material such as vegetation, water, soil, kaolinite, calcite, or iron oxide can have a characteristic spectral shape or “spectral fingerprint”.

2.2 Reflectance domain

Reflectance is dimensionless. Physically, a reflectance of 0.20 corresponds to 20% reflectance under the measurement definition. In real corrected remote-sensing products, valid values can occasionally fall slightly below 0 or above 1 because of noise, correction artifacts, adjacency effects, scaling, or retrieval uncertainty. Therefore the analysis engine should not automatically clip all values to [0,1] unless the selected method specifically requires it.

2.3 Wavelength alignment is fundamental

For spectral matching or endmember-based analysis, the numerical band index is not enough. The engine should know each band's center wavelength and preferably its spectral response or FWHM. Two spectra can only be compared meaningfully when they represent the same wavelength space. Reference libraries therefore normally require spectral resampling to the image wavelengths.

3. Recommended Workflow

The seven tools form a toolbox, not a mandatory linear pipeline.

3.1 Two important final branches

Pixel spectrum
     +
Reference library
     ↓
SAM / SID
     ↓
1-band class map

Pixel spectrum
     +
K endmembers
     ↓
Unmixing
     ↓
K abundance bands

4. Recommended Input Data Contract

4.1 Required or strongly recommended metadata

4.2 Preferred data state

Preferred:
Surface Reflectance
+ wavelength metadata
+ bad-band mask
+ valid scale/offset already applied or declared

Accepted with warning:
Radiance / TOA Reflectance
Unknown wavelength metadata
Generic band-index-only raster

Band-index-only mode can still support some mathematical processing, but physically meaningful spectral matching should be marked as limited or rejected when wavelength alignment cannot be established.

5. Spectral Smoothing

Spectral smoothing reduces high-frequency noise along the spectral axis. It does not blur the image spatially. For a 200-band cube, the preferred output remains 200 bands so that the wavelength grid remains unchanged.

5.1 Moving-average smoothing

5.2 Weighted/Gaussian smoothing

5.3 Savitzky–Golay smoothing

Savitzky–Golay smoothing fits a low-order polynomial to a moving local spectral window using least squares. The center value of the fitted polynomial becomes the smoothed value. It is attractive for spectroscopy because it can preserve peak shape and can also provide analytical derivative estimates.

5.4 Recommended parameters

• Method: Savitzky–Golay / Moving Average / Gaussian.
• Window size: odd number such as 5, 7, 9, 11 bands.
• Polynomial order: commonly 2 or 3 for Savitzky–Golay.
• Wavelength-aware mode: preferred when band spacing is irregular.
• Bad bands: should be masked or interpolated before smoothing.

5.5 Output

Input : 200-band surface reflectance
Output: 200-band smoothed spectral cube

Band 1  → same λ1
Band 2  → same λ2
...
Band 200→ same λ200

6. Spectral Derivative

Spectral derivatives emphasize changes in the shape of a spectrum. They can suppress broad baseline effects and reveal absorption edges, inflection points, and subtle spectral differences. The output is not reflectance; its units depend on wavelength units.

6.1 First derivative

6.2 Second derivative

For approximately uniform spectral spacing Δλ:

6.3 Savitzky–Golay derivative

The same local polynomial used for Savitzky–Golay smoothing can be differentiated analytically, often producing a more stable derivative than applying a finite difference directly to noisy spectra.

6.4 Why smoothing often precedes derivative

Differentiation amplifies high-frequency noise. Therefore:

Surface Reflectance
      ↓
Spectral Smoothing
      ↓
1st / 2nd Derivative

6.5 Output dimension

Mathematically, naïve differencing may yield N−1 or N−2 samples. For a raster-analysis platform, the recommended contract is to preserve N output bands by using centered differences for interior wavelengths, one-sided differences or declared NoData at edges, and retaining original wavelength metadata.

7. Continuum Removal

Continuum removal normalizes a spectrum relative to an upper continuum, usually represented by a convex hull over the spectrum. It is especially useful for comparing the position, shape and depth of absorption features.

7.1 Basic equation

At wavelengths where the measured spectrum touches the continuum, R_CR is close to 1. Inside an absorption feature, it is typically less than 1.

7.2 Absorption depth

7.3 Output

Input : 200-band reflectance
Output: 200-band continuum-removed cube

Original wavelength dimension is preserved.
Output values are normalized spectral response, not original surface reflectance.

7.4 Spectral subset matters

The continuum depends on the wavelength interval used to construct it. The UI should therefore allow all valid bands, a wavelength range, or selected spectral windows.

8. Minimum Noise Fraction (MNF)

MNF is a noise-aware linear transformation designed to order transformed components by image quality / signal-to-noise information. It is commonly used to determine effective data dimensionality, separate noisy components, and reduce subsequent computational cost.

8.1 Data model

Let Σ_n be the estimated noise covariance matrix.

8.2 Noise whitening

8.3 Second rotation

Compute the covariance of the noise-whitened data:

Then solve the eigenproblem:

The ordered eigenvectors define the MNF transform:

8.4 Why output can be much smaller

Input: 200 spectral bands
      ↓
Forward MNF
      ↓
MNF 1    strong information
MNF 2    strong information
...
MNF 20   usable
MNF 21+  increasingly noisy
      ↓
Output selected M = 20 components

A full MNF transform can contain N components, but the practical benefit is often obtained by retaining only the informative first M components. The output bands are MNF components, not wavelengths.

8.5 Noise estimation

When a dedicated dark/noise image is unavailable, a practical approach estimates noise from local spatial differences, assuming adjacent pixels share similar signal but have independent noise. The method used should be recorded in output metadata because MNF results depend strongly on the noise model.

8.6 Recommended parameters

• Noise estimation: spatial shift difference / external noise covariance / user-supplied noise raster.
• Number of components: Auto or integer M.
• Selection criterion: eigenvalue curve, estimated SNR, cumulative information, or manual QC threshold.
• Whitening regularization: needed if the noise covariance matrix is ill-conditioned.

9. Endmember Extraction

An endmember is a spectral signature representing a relatively pure material or an extreme spectral component in the scene. Endmember extraction is not a classification raster. Its primary result is a set of spectral signatures and, where possible, the source pixel locations.

9.1 Output concept

Input image: 200 bands
Requested endmembers: K = 6

Output spectral library:
Endmember 1 → 200 spectral values
Endmember 2 → 200 spectral values
Endmember 3 → 200 spectral values
Endmember 4 → 200 spectral values
Endmember 5 → 200 spectral values
Endmember 6 → 200 spectral values

If the search is performed in reduced MNF space, the preferred workflow is to retain the pixel locations and sample the original/smoothed reflectance cube at those locations so that the final endmember library remains in physical wavelength space.

9.2 Geometric interpretation

Under a linear-mixture model, image pixels occupy a cloud in high-dimensional spectral space. Pure materials tend to occur near the extreme vertices of the data cloud, while mixtures lie between them.

9.3 PPI — Pixel Purity Index

PPI repeatedly projects spectra onto random directions. Pixels occurring most often at the extreme ends of those projections receive a high purity count.

PPI is commonly applied to informative MNF components to reduce noise and dimensionality before searching for pure pixels.

9.4 N-FINDR concept

N-FINDR searches for a set of pixels that maximizes simplex volume in spectral space.

9.5 VCA — Vertex Component Analysis

VCA is an unsupervised endmember-extraction method based on the geometry of linear mixtures. It searches for vertices of the data simplex by repeated projections under assumptions that include the presence of sufficiently pure signatures.

9.6 Recommended platform methods

9.7 Recommended output object

{
  "analysis": "endmember_extraction",
  "method": "VCA",
  "source_raster": "...",
  "wavelength_unit": "nm",
  "wavelengths": [400.0, 410.0, "..."],
  "endmembers": [
    {
      "id": 1,
      "name": "Endmember 1",
      "pixel": {"row": 812, "col": 455},
      "coordinate": {"x": "...", "y": "..."},
      "spectrum": [0.081, 0.084, "..."],
      "purity_score": 0.97
    }
  ]
}

Optional auxiliary outputs include point/vector source locations, PPI/purity raster, CSV, spectral plot preview, and user-editable material labels.

10. Spectral Library Matching

Spectral Library Matching is the hyperspectral material-identification / whole-pixel classification tool. The user supplies or selects a reference spectral library, and each image pixel is compared against candidate reference spectra.

10.1 Library preprocessing requirements

• Reference wavelengths must overlap image wavelengths.
• Wavelength units must be known and harmonized.
• Library spectra should be resampled using wavelength/FWHM information when possible.
• If image spectra are continuum-removed, derivative-transformed, or otherwise normalized, the reference library must receive the equivalent transformation.
• Bad bands should be excluded from both image and reference spectra.

10.2 SAM — Spectral Angle Mapper

SAM treats each spectrum as a vector in N-dimensional band space and computes the angle between image spectrum x and reference spectrum s.

• θ ≈ 0: highly similar spectral direction.
• Larger θ: less similar.
• Best reference is typically the class with the smallest valid angle.
• A maximum-angle threshold should allow pixels to remain unclassified.

Because SAM focuses on vector direction rather than magnitude, it can be relatively insensitive to overall albedo/illumination scaling when used with calibrated data; this does not remove the need for correct radiometric/atmospheric preprocessing.

Recommended SAM outputs

classification.tif
  0 = Unclassified
  1 = Material A
  2 = Material B
  ...

sam_best_angle.tif
  pixel value = minimum spectral angle

optional rule images:
  one angle band per reference material

10.3 SID — Spectral Information Divergence

SID converts nonnegative spectral vectors into probability-like distributions and measures their divergence using information theory.

Step 1 — normalize spectra

Step 2 — directional divergence

Step 3 — symmetric SID

• SID close to 0: highly similar.
• Larger SID: greater spectral divergence.
• A maximum SID threshold should support an unclassified class.

Recommended SID outputs

classification.tif
  best material/reference class

sid_best_score.tif
  minimum SID divergence

optional rule images:
  one SID score band per reference material

10.4 SAM vs SID

11. Spectral Unmixing

Spectral Unmixing estimates how much of each reference material/endmember contributes to a pixel. It is therefore a sub-pixel composition analysis, not a conventional hard classification.

11.1 Linear mixing model

For N spectral bands and K endmembers:

Expanded by wavelength:

11.2 Unconstrained least squares

When M has full column rank:

Unconstrained solutions can contain negative abundances or sums different from 1.

11.3 Non-negative constraint

11.4 Fully constrained linear unmixing

The constraints are often physically intuitive for fractional surface mixtures, but they are assumptions and may not fit every radiative process.

11.5 Output band count

Input hyperspectral raster = 200 bands
Endmembers = 6

Spectral Unmixing
        ↓

Output abundance raster = 6 bands

Band 1 = abundance Endmember 1
Band 2 = abundance Endmember 2
Band 3 = abundance Endmember 3
Band 4 = abundance Endmember 4
Band 5 = abundance Endmember 5
Band 6 = abundance Endmember 6

Example pixel:

Vegetation  = 0.62
Soil        = 0.23
Dry grass   = 0.08
Shadow      = 0.05
Other       = 0.02
-------------------
Total       = 1.00

11.6 Residual and RMSE

A high residual/RMSE can indicate missing endmembers, nonlinear mixing, poor spectral alignment, uncorrected atmosphere, bad bands, or an endmember library that does not adequately represent the scene.

11.7 Unmixing after MNF

Unmixing can be performed in MNF space, but image and endmembers must be transformed by the same fitted MNF transformation. For platform defaults, physically interpretable abundance mapping is clearer in calibrated reflectance space, while MNF is used to improve endmember discovery and computational efficiency.

12. Relationship Between the Seven Tools

12.1 Valid workflows

Surface Reflectance → Smoothing → SAM/SID

Surface Reflectance → Smoothing → Continuum Removal → SAM/SID

Surface Reflectance → Smoothing → Derivative → Random Forest

Surface Reflectance → MNF → Random Forest

Surface Reflectance → MNF → Endmember Extraction
                    → sample original spectra
                    → Spectral Unmixing

Surface Reflectance → Endmember Extraction → SAM/SID

External Spectral Library → resample to image wavelengths → SAM/SID

12.2 Workflow to avoid by default

Spectral Unmixing → SAM/SID spectral-library matching   ✕

After unmixing, the bands represent abundance of endmembers rather than wavelength samples.

13. Recommended Platform Output Contract

13.1 Suggested endmember JSON fields

• analysis/method/version;
• source raster identifier;
• spectral domain and wavelength units;
• wavelength array;
• endmember identifier and optional material label;
• source pixel row/column and geospatial coordinate;
• spectral vector;
• purity/quality score where available;
• preprocessing history;
• bad-band list;
• creation timestamp.

13.2 Suggested classification legend

{
  "classes": [
    {"value": 0, "name": "Unclassified"},
    {"value": 1, "name": "Kaolinite", "reference_id": "lib-001"},
    {"value": 2, "name": "Calcite", "reference_id": "lib-021"},
    {"value": 3, "name": "Vegetation", "reference_id": "lib-104"}
  ],
  "method": "SAM",
  "score_semantics": "lower_is_better"
}

13.3 Suggested abundance metadata

{
  "bands": [
    {"band": 1, "endmember": "Vegetation", "unit": "fraction"},
    {"band": 2, "endmember": "Soil", "unit": "fraction"},
    {"band": 3, "endmember": "Shadow", "unit": "fraction"}
  ],
  "constraint": "FCLS",
  "sum_to_one": true,
  "nonnegative": true
}

14. Validation & Quality Control

14.1 Common validation

• Reject empty rasters and invalid band dimensions.
• Verify NoData behavior and finite-value proportion.
• Verify wavelength array length equals raster band count.
• Check wavelengths are monotonically ordered or explicitly reorder data and metadata together.
• Detect duplicate wavelengths.
• Apply declared scale/offset before physics-based spectral analysis.
• Exclude known bad bands and strong atmospheric-absorption regions when appropriate.

14.2 Smoothing QC

• Window cannot exceed valid spectral range.
• Savitzky–Golay window must be compatible with polynomial order.
• Report edge policy.

14.3 Derivative QC

• Reject zero wavelength spacing.
• Handle irregular wavelength spacing explicitly.
• Record derivative order and units.

14.4 MNF QC

• Noise covariance must be numerically valid.
• Regularize near-singular covariance matrices.
• Return eigenvalues/component-quality information.
• Do not label MNF components as wavelengths.

14.5 Endmember QC

• K must be smaller than or consistent with effective spectral dimensionality.
• Report duplicate/highly similar extracted endmembers.
• Report source pixels and validity masks.
• Allow user to rename “Endmember 1” after interpretation.

14.6 Matching QC

• Require spectral overlap.
• Resample library to image spectral grid.
• Use threshold to avoid forced classification.
• Return best score and optionally second-best score/margin.
• For SID, handle nonpositive values explicitly.

14.7 Unmixing QC

• Endmember count K must normally be less than number of spectral dimensions used.
• Endmember and image spectral dimensions must match exactly after resampling/transformation.
• Return residual or RMSE.
• For constrained methods, report abundance-sum deviation and negative-value violations.

15. Implementation Notes for an Analysis Server

15.1 Processing strategy

A hyperspectral cube can be hundreds of bands and several gigabytes. Algorithms should use block/window processing where mathematically possible. Do not read a full cube into RAM unless required and size-checked.

15.2 Dtype

• Use Float32 for transformed spectral/abundance outputs unless higher precision is specifically required.
• Use integer type for class raster, choosing UInt8/UInt16/UInt32 based on class count.
• Do not encode transformed physical values into integers unless scale/offset is explicit.

15.3 Raster statistics

A statistics sidecar should describe each output band correctly. For a 200-band cube this can be large; the implementation may store per-band min/max/mean/std plus sampled histograms. MNF and abundance outputs should have semantic band descriptions.

15.4 Provenance

{
  "input_domain": "surface_reflectance",
  "operations": [
    {"tool": "spectral_smoothing", "method": "savitzky_golay", "window": 7, "order": 2},
    {"tool": "continuum_removal", "range_nm": [2000, 2400]}
  ]
}

Provenance is essential because a reference library must receive compatible preprocessing before matching.

15.5 Do not silently change spectral semantics

16. Example Analytical Workflows

16.1 Mineral identification from reflectance

Surface Reflectance
      ↓
Remove bad bands
      ↓
Spectral Smoothing
      ↓
Continuum Removal (target SWIR range)
      ↓
Resample mineral spectral library to image wavelengths
      ↓
Apply same continuum removal to library
      ↓
Spectral Library Matching — SAM or SID
      ↓
Mineral Class Map + Match Score

16.2 Endmember discovery and abundance mapping

Surface Reflectance (200 bands)
      ↓
MNF (retain 20 informative components)
      ↓
PPI / VCA / N-FINDR
      ↓
K pure-pixel locations
      ↓
Sample original 200-band reflectance
      ↓
K endmember spectra
      ↓
Fully Constrained Linear Spectral Unmixing
      ↓
K abundance bands + RMSE

16.3 Vegetation stress feature engineering

Surface Reflectance
      ↓
Spectral Smoothing
      ↓
First Derivative
      ↓
Selected spectral features / ML
      ↓
Random Forest / SVM / Deep Learning

16.4 Direct spectral-library classification

Surface Reflectance
      +
Reference Library
      ↓
Spectral resampling
      ↓
SAM / SID
      ↓
Class Raster
      +
Best-Match Score Raster

16.5 Why Unmixing and Matching are parallel

Pixel 100,200
→ Kaolinite

Pixel 100,200
Kaolinite = 0.58
Calcite   = 0.27
Soil      = 0.15

17. Scientific & Technical References

Primary papers and official technical documentation used to ground the definitions and implementation notes.

1. Savitzky, A. & Golay, M. J. E. (1964). Smoothing and Differentiation of Data by Simplified Least Squares Procedures. Analytical Chemistry 36(8), 1627–1639. ACS.
2. NV5 Geospatial / ENVI. Continuum Removal. Official documentation.
3. NV5 Geospatial / ENVI. Minimum Noise Fraction Transform. Official documentation.
4. NV5 Geospatial / ENVI. Forward MNF Transform Task. Official documentation.
5. NV5 Geospatial / ENVI. Pixel Purity Index. Official documentation.
6. Nascimento, J. M. P. & Bioucas-Dias, J. M. (2005). Vertex Component Analysis: A Fast Algorithm to Unmix Hyperspectral Data. IEEE Transactions on Geoscience and Remote Sensing 43(4), 898–910. IEEE.
7. NV5 Geospatial / ENVI. SMACC Endmember Extraction. Official documentation.
8. NV5 Geospatial / ENVI. Spectral Angle Mapper. Official documentation.
9. Chang, C.-I. (2000). An Information-Theoretic Approach to Spectral Variability, Similarity, and Discrimination for Hyperspectral Image Analysis. IEEE Transactions on Information Theory 46(5), 1927–1932. IEEE.
10. NV5 Geospatial / ENVI. Material Identification Tool — SAM, SID and spectral-library resampling. Official documentation.
11. NV5 Geospatial / ENVI. Linear Spectral Unmixing. Official documentation.
12. U.S. Geological Survey. USGS Spectral Library. Official USGS resource.
13. U.S. Geological Survey. USGS Spectral Library Version 7 Data. Official data release.
14. NV5 Geospatial / ENVI. Spectral Hourglass Workflow. Official documentation.