Cross-Metric Normalization (0 – 1 Similarity Scale)

When each metric is computed in its native form, it produces values on very different scales:

Metric	Raw output range	Meaning of a “small” value
SAM	0 – π/2 radians	Spectra point in nearly the same direction (shape match)
SID	0 – ∞ (bits)	Information content is nearly identical
SED	0 – ∞ (reflectance²)	Little absolute radiance difference
EMD	0 – ∞ (mass-distance)	Minimal effort to realign bands

If you fed those raw numbers straight into a classifier or colour ramp you would need a different threshold—or even a different colour bar—for every metric.

What the 0-to-1 normalisation does #

• SAM Angle is inherently bounded, so we convert it to a similarity: ${\text{SAM}}_{\text{norm}}=1-\frac{\theta }{\pi /2}$
  • 1 = identical spectral shape 
  • 0 = orthogonal shape
• SID, SED, EMD Their raw distances are unbounded, so we:
  1. Find a robust upper cap (99-percentile of all pixels)
  2. Scale each pixel distance by that cap
  3. Flip distance → similarity: ${\text{Metric}}_{\text{norm}}=1-\frac{\text{distance}}{\text{cap}}[\text{clamped }0-1]$

Result: every metric now outputs 1 for a perfect match and 0 for the worst practical mismatch within the scene.

Why this helps you #

• Cross-comparison – You can display several similarity layers side-by-side without tweaking legends; a value of 0.8 always means “high similarity,” whichever metric produced it.
• Unified thresholds – Rules like “keep pixels ≥ 0.7” work across all four metrics; no need to remember that for SAM the cut-off is 0.1 rad, for SED it’s 0.004, etc.
• Consistent analytics – Aggregating or stacking metrics (e.g., averaging SAM and SID similarities) is safe because they live on the same numeric scale.

Think of it like converting temperatures from °F, K, and °C to a single “percentage of boiling point” scale—once everything is in 0–1 you can compare, combine, and threshold effortlessly.