Zoom-Invariant Vision of Figural Shape: The Mathematics of Cores

Believing that figural zoom invariance and the cross-figural boundary linking implied by medial loci are important aspects of object shape, we present the mathematics of and algorithms for the extraction of medial loci directly from image intensities. The medial loci called cores are defined as generalized maxima in scale space of a form of medial information that is invariant to translation, rotation, and in particular, zoom. These loci are very insensitive to image disturbances, in strong contrast to previously available medial loci, as demonstrated in a companion paper. Core-related geometric properties and image object representations are laid out which, together with the aforementioned insensitivities, allow the core to be used effectively for a variety of image analysis objectives. 3 SYMBOLS σ ρ θ , , Greek letter, standing for a scalar variable. Italic. Roman (italic) letters, upper case or lower case, may also be used for variables. r r x, ξ Roman or Greek letter with arrow over it, standing for a vector variable. Italic. ξ ξ ξ i i j i ij k ijkl R , , , , Γ Variable subscripted, superscripted, or both. The variable may be upper or lower case and Greek or Roman (italic), but the subscripts and superscripts will be one or more digits or lower case Roman (italic) P k , or P i k , Variable with subscript with comma in it, with the comma possibly not preceded by any symbol, indicating a differentiated tensor σ blur variable subscripted with a word 1 2 Upper-case letter with upper-case subscript, subscripted subscript r p t Vector (or matrix) with superscript t, indicating transpose ℑ Script upper case letter, indicating operator T Bold upper case letter, indicating set f x () letter followed by parenthesized expression, indicating function r u θ () letter with arrow over it followed by parenthesized variable(s), standing for vector function M x y K x , , , ; σ σ () () r letter followed by list of parenthesized variables (scalar or vector) separated by commas, with possibly one comma replaced by a semicolon ln , cos f () () −1 θ Specific functions named by multiple letters and possibly a superscript, here the natural logarithm and the inverse cosine p V Θ () () , det Probability function of the set upper-case theta, determinant of the matrix upper-case V ∀ ∈ → , , Mathematical symbols for …