'Pends on whether it represents a fractal set or not...realistically, yes, it'll likely actually have a value at or near the Euclidean dimensionality, but the query as posed is, I think, indeterminate.

I thought it was something like, if it was a straight line the fractal dimension would be 1. Then as the line wiggles and gets more tortuous, the fractal dimension gets larger and larger until it reaches 2 when the line is so tortuous that it completely fills the 2D plane. But I'm not an expert on fractals so that's just my very rudimentary understanding of it.

That's the basic idea, yeah...since the OP did limit his question to 2D I reacted too quickly I suppose. The image could have been from a higher dimension construct, but once it was projected into 2D its maximum is restricted to the topological dimension.

Thank you both. i am in the process of offline handwritten signature verification. i tried the code given by Alceu Ferraz Costa. it Returns the Haussdorf fractal dimension of an object represented by the binary image. i get values like 0.48 0.05 for various signatures like the one in the aatchment. please help. will the fractal dimension be less than 0.

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