The Crispness Index (C) provides a means of quantifying the crispness, or fuzziness, of a membership probability (MP) image. MP images describe the probability of each grid cell belonging to some feature or class. MP images contain values ranging from 0 to 1.

The index, as described by Lindsay (2006), is the ratio between the sum of the squared differences (from the image mean) in the MP image divided by the sum of the squared differences for the Boolean case in which the total probability, summed for the image, is arranged crisply.

C is closely related to a family of relative variation coefficients that measure variation in an MP image relative to the maximum possible variation (i.e. when the total probability is arranged such that grid cells contain only 1s or 0s). Notice that 0 < C < 1 and a low C-value indicates a nearly uniform spatial distribution of any probability value, and C = 1 indicates a crisp spatial probability distribution, containing only 1's and 0's.

C is calculated as follows:

> C = SS_mp ∕ SS_B = [∑(pij − p-bar)^2] ∕ [ ∑pij(1 − p-bar)^2 + p2(RC − ∑pij)]

Note that there is an error in the original published equation. Specifically, the denominator read:

> ∑pij(1 - p_bar)^2 + p_bar^2 (RC - ∑pij)

instead of the original:

> ∑pij(1 - p_bar^2) - p_bar^2 (RC - ∑pij)

## References

Lindsay, J. B. (2006). Sensitivity of channel mapping techniques to uncertainty in digital elevation data. International Journal of Geographical Information Science, 20(6), 669-692.

## Function Signature

`def crispness_index(self, raster: Raster, output_html_file: str) -> None: ...`

## Project Links

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