# Measures of edge image continuity and connectivity

Suppose I have a scalar image $$I:\mathbb{Z}_{1,m} \times \mathbb{Z}_{1,n}\rightarrow [0,1]\subset \mathbb{R}$$, where $$\mathbb{Z}_{1,m} = \{1,\ldots, m\}$$. (For instance, computed as the (scaled) gradient magnitude of a natural image $$N$$, e.g. $$I=s||\nabla N||_2$$).

I would like a measure $$E$$ of the "edge continuity", "connectivity", or "contour continuity" of $$I$$. We can call $$E(I)$$ the "connectivity energy" of the image.

In other words, I am assuming $$I$$ is a scalar image of edges/curves, and I want to know how "well-attached" the curves are. For instance, Gaussian noise or some number of isolated pixels would do poorly, as would the entire image being 1 or a smooth gradient. A bunch of lines would score highly.

I do not really care about the curvature of the curves. For circles, for instance, any radius is equally fine (though of course as $$r\rightarrow 0$$ it becomes a point which should score poorly!). This means I don't want to do corner detection. Similarly, I am also not picky about thickness being more than one pixel, but of course beyond some number of pixels a "curve" becomes a "region" and should be penalized (perhaps the measure requires a parameter for this).

Lastly, ideally, the computation of $$E$$ should be cheap (fast), with as few parameters as possible.

Unfortunately I don't have image processing expertise to figure this out. I was thinking about using the structure tensor, but we already kind of have a gradient-magnitude-like image, so I'm not sure it makes sense here. Another idea is to convolve a bunch of oriented edge filters over the image and then try to have it match one and only one of them? Surely there must be better methods in the literature. :)