Normalized difference

The bounded ratio (a-b)/(a+b) behind Michelson contrast, chi-squared histogram distance, and Fresnel reflection.

Normalized difference: $\frac{𝑎-𝑏}{𝑎+𝑏}$

Applications:

• Michelson index: $\frac{{𝐼}_{max}-{𝐼}_{min}}{{𝐼}_{max}+{𝐼}_{min}}$
• Chi-squared distance for two histograms: ${𝜒}^2={\sum }_{𝑖=1}^{𝑘}\frac{{\left({𝑎}_{𝑖}-{𝑏}_{𝑖}\right)}^2}{{𝑎}_{𝑖}+{𝑏}_{𝑖}}$
• Fresnel reflection coefficient: $𝑛=\frac{{𝑛}_2}{{𝑛}_1}$, $𝑟=\frac{1-𝑛}{1+𝑛}=\frac{{𝑛}_1-{𝑛}_2}{{𝑛}_1+{𝑛}_2}$

For $𝑎,𝑏\ge 0$ and $𝑎+𝑏>0$, the normalized difference satisfies $-1\le \frac{𝑎-𝑏}{𝑎+𝑏}\le 1$. By the triangle inequality,

Dividing by $𝑎+𝑏>0$ gives the desired bound.