Spectral Information Divergence (SID)

Spectral Information Divergence measures how different the information content of a pixel’s spectrum is from a reference signature. Instead of comparing raw reflectance values, SID treats each spectrum as a probability distribution across bands, allowing it to capture subtle, band-to-band variations that shape-based methods can miss. This makes SID particularly powerful for hyperspectral data where fine spectral structure matters.

How it works #

1. Convert spectra to probability distributions For a spectrum $\mathbf{R}=[R_1,\dots ,R_n]$ and a reference $\mathbf{S}=[S_1,\dots ,S_n]$: $p_i=\frac{R_i}{\sum _{k=1}^n{R}_k}$, $q_i=\frac{S_i}{\sum _{k=1}^n{S}_k}$ $p_i$ and $q_i$ now sum to 1 and represent the relative contribution of each band.
2. Kullback–Leibler divergence in both directions
   • $D(p\Vert q)=\sum _{i=1}^n{p}_i\ln \left(\frac{p_i}{q_i}\right)$
   • $D(q\Vert p)=\sum _{i=1}^n{q}_i\ln \left(\frac{q_i}{p_i}\right)$
3. Symmetric SID distance ${\text{SID}}_{\text{raw}}=D(p\Vert q)+D(q\Vert p)$${\text{SID}}_{\text{raw}}$ is always non-negative and equals 0 only when the two spectra are identical in information content.
4. Normalization (0–1 scale) The raw SID values in a scene are empirically scaled so that the 99ᵗʰ-percentile distance maps to 0, and perfect matches map to 1: ${\text{SID}}_{\text{norm}}=1-\mathrm{clamp}(\frac{{\text{SID}}_{\text{raw}}}{{\text{SID}}_{99\%}},0,1)$ where ${\text{SID}}_{99\%}$ is the 99th-percentile SID of all pixels in the area of interest. Result: 1 → best match, 0 → poorest match.

Key properties #

Property	Benefit
Direction-sensitive	Detects asymmetric differences that angle-based methods may ignore.
Band-detail aware	Responds to subtle shifts in individual bands—ideal for hyperspectral analyses.
Robust scaling	Scene-adaptive 0–1 normalization simplifies thresholding and comparison.

When to use SID #

• Mineral exploration: Differentiate minerals with similar overall shape but distinct fine-band features (e.g., subtle OH or Fe²⁺ absorptions).
• Chemical residue mapping: Detect slight compositional changes where each band’s contribution is critical.
• Precision agriculture: Identify plant stress signatures that manifest as nuanced spectral tweaks rather than broad shape changes.