. Author manuscript; available in PMC: 2018 Jun 29.

Published in final edited form as: IEEE Signal Process Mag. 2017 Jul 11;34(4):43–59. doi: 10.1109/MSP.2017.2695801

TABLE I.

Cumulative Distribution Transform pairs. Note that the composition holds for all strictly monotonically increasing functions g.

Cumulative Distribution Transform pairs

Property

Signal domain

CDT domain

I (x)

\tilde{I} (x)

Translation

I (x − τ)

\tilde{I} (x) + τ \sqrt{I_{0} (x)}

Scaling

aI (ax)

\frac{\tilde{I} (x)}{a} - x \frac{(a - 1)}{a} \sqrt{I_{0} (x)}

Composition

g′ (x)I (g(x))

(g^{- 1} (\frac{\tilde{I} (x)}{\sqrt{I_{0} (x)}} + x) - x) \sqrt{I_{0} (x)}