Spectral Angle Mapper (SAM)

Spectral Angle Mapper quantifies the shape similarity between a pixel’s spectrum and a reference spectral signature by treating each as a vector in an n-dimensional space (one dimension per band).  Because the comparison is purely angular, SAM is highly robust to illumination differences and overall brightness scaling.

How it works #

1. Vector definition
   • Pixel spectrum $\mathbf{R}={[R_1,R_2,\dots ,R_n]}^{\mathsf{T}}$
   • Reference spectrum (endmember) $\mathbf{S}={[S_1,S_2,\dots ,S_n]}^{\mathsf{T}}$
2. Angle computation $\theta =\arccos \left(\frac{\mathbf{R}·\mathbf{S}}{\Vert \mathbf{R}\Vert \Vert \mathbf{S}\Vert }\right),0\le \theta \le \frac{\pi }{2}$
   • $\mathbf{R}·\mathbf{S}$ Dot product (spectral shape agreement)
   • $\Vert ·\Vert$ Euclidean norm (vector length)
3. Normalization (0–1 scale) To provide an intuitive “similarity” output, the raw angle is mapped to the unit interval: ${\text{SAM}}_{\text{norm}}=1-\frac{\theta }{\pi /2}$
   • 1 → identical spectral shape
   • 0 → orthogonal (maximally different) spectral shape
4. Per-pixel evaluation The calculation is applied to every image pixel, producing a SAM band where higher values indicate closer spectral conformity to the chosen reference.

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Key properties #

Property	Benefit
Illumination‐invariant	Ignores overall brightness differences, focusing on spectral shape.
Unit-less	Comparable across instruments and scenes once wavelengths are aligned.
Normalised output	Consistent 0–1 scale simplifies thresholding and visualisation.

When to use SAM #

• Mineral prospection: Differentiate minerals with similar albedo but distinct absorption shapes.
• Vegetation studies: Match species or health states under varying sun–sensor geometries.
• Change detection: Highlight spectral‐shape changes while suppressing mere brightness shifts.