Reflections on Shannon Information: In search of a natural information-entropy for images

It is not obvious how to extend Shannon's original information entropy to higher dimensions, and many different approaches have been tried. We replace the English text symbol sequence originally used to illustrate the theory by a discrete, bandlimited signal. Using Shannon's later theory of sampling we derive a new and symmetric version of the second order entropy in 1D. The new theory then naturally extends to 2D and higher dimensions, where by naturally we mean simple, symmetric, isotropic and parsimonious. Simplicity arises from the direct application of Shannon's joint entropy equalities and inequalities to the gradient (del) vector field image embodying the second order relations of the scalar image. Parsimony is guaranteed by halving of the vector data rate using Papoulis' generalized sampling expansion. The new 2D entropy measure, which we dub delentropy, is underpinned by a computable probability density function we call deldensity. The deldensity captures the underlying spatial image structure and pixel co-occurrence. It achieves this because each scalar image pixel value is nonlocally related to the entire gradient vector field. Both deldensity and delentropy are highly tractable and yield many interesting connections and useful inequalities. The new measure explicitly defines a realizable encoding algorithm and a corresponding reconstruction. Initial tests show that delentropy compares favourably with the conventional intensity-based histogram entropy and the compressed data rates of lossless image encoders (GIF, PNG, WEBP, JP2K-LS and JPG-LS) for a selection of images. The symmetric approach may have applications to higher dimensions and problems concerning image complexity measures.