# Statistical measure for if an image consists of spatially connected separate regions

Consider these two grayscale images:  The first image shows a meandering river pattern.
The second image shows random noise.

I am looking for a statistical measure that I can use to determine if it is likely that an image shows a river pattern.

The river image has two areas: river = high value and everywhere else = low value.

The result is that the histogram is bimodal: Therefore an image with a river pattern should have a high variance.

However so does the random image above:

River_var = 0.0269, Random_var = 0.0310


On the other hand the random image has low spatial continuity, whereas the river image has high spatial continuity, which is clearly shown in the experimental variogram: In the same way that the variance “summarizes” the histogram in one number,
I am looking for a measure of spatial contiuity that “summarizes” the experimental variogram.

I want this measure to “punish” high semivariance at small lags harder than at large lags, so I have come up with:

$\ svar = \sum_{h=1}^n \gamma(h)/h^2$

If I only add up from lag = 1 to 15 I get:

River_svar = 0.0228, Random_svar = 0.0488


I think that a river image should have high variance, but low spatial variance so I introduce a variance ratio:

$\ ratio = var/svar$

The result is:

River_ratio = 1.1816, Random_ratio = 0.6337


My idea is to use this ratio as a decision criteria for if an image is a river image or not; high ratio (e.g. > 1) = river.

Any ideas on how I can improve things?

EDIT: Following the advice of whuber and Gschneider here are the Morans I of the two images calculated with a 15×15 inverse distance weight matrix using Felix Hebeler’s Matlab function:  I need to summarize the results into one number for each image.
According to wikipedia: “Values range from −1 (indicating perfect dispersion) to +1 (perfect correlation). A zero value indicates a random spatial pattern.”
If I sum up the square of the Morans I for all pixels I get:

River_sumSqM = 654.9283, Random_sumSqM = 50.0785


There is a huge difference here so Morans I seem to be a very good measure of spatial continuity :-).

And here is a histogram of this value for 20 000 permutations of the river image: Clearly the River_sumSqM value (654.9283) is unlikely and the River image is therefore not spatially random.