This is not quite right---that's just counting the density of the noise but says nothing about distribution, which is critical to fractal dimension. But there are well-known ways to get an approximate fractal dimension from an image. (I say "approximate" because of course the dots on an autochrome aren't really fractal; they're a finite number of discrete points scattered on the area of the image. You're not trying to find the fractal dimension of the actual object, but the fractal dimension of a theoretical object of which the real one is a good approximation.)

It's easy to cook up noise of a specified fractal dimension using fractional Brownian motion (Mandelbrot sketches the algorithm in _The Fractal Geometry of Nature_, I believe, and one of the coffee-table fractal books from the 1980s has actual pseudocode).

What I'm trying to understand is, is noise with the right fractal dimension "Perlin noise" by definition, or is there some more specific characterisation?

-NT