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It is not obvious how to extend Shannon's original information entropy tohigher dimensions, and many different approaches have been tried. We replacethe English text symbol sequence originally used to illustrate the theory by adiscrete, bandlimited signal. Using Shannon's later theory of sampling wederive a new and symmetric version of the second order entropy in 1D. The newtheory then naturally extends to 2D and higher dimensions, where by naturallywe mean simple, symmetric, isotropic and parsimonious. Simplicity arises fromthe direct application of Shannon's joint entropy equalities and inequalitiesto the gradient (del) vector field image embodying the second order relationsof the scalar image. Parsimony is guaranteed by halving of the vector data rateusing Papoulis' generalized sampling expansion. The new 2D entropy measure,which we dub delentropy, is underpinned by a computable probability densityfunction we call deldensity. The deldensity captures the underlying spatialimage structure and pixel co-occurrence. It achieves this because each scalarimage pixel value is nonlocally related to the entire gradient vector field.Both deldensity and delentropy are highly tractable and yield many interestingconnections and useful inequalities. The new measure explicitly defines arealizable encoding algorithm and a corresponding reconstruction. Initial testsshow that delentropy compares favourably with the conventional intensity-basedhistogram entropy and the compressed data rates of lossless image encoders(GIF, PNG, WEBP, JP2K-LS and JPG-LS) for a selection of images. The symmetricapproach may have applications to higher dimensions and problems concerningimage complexity measures.

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