The D-Dimensional Inverse Vector-Gradient Operator and Its Application for Scale-Free Image Enhancement

A class of enhancement techniques is proposed for images in arbitrary dimension D. They are free from either space/frequency (scale) or grey references. There are two subclasses. One subclass is the chain: {Negative Laplace operator, Multiplication by Power (γ−1) of modulus-Laplace value, Inverse Negative Laplace operator} together with generalized versions. The generalization of the Negative Laplace operator consists of replacing its isotropic frequency square transfer function by an (equally isotropic) modulusfrequency to-the-power-p transfer-function. The inverse is defined accordingly. The second subclass is the chain: {Vector-Gradient operator, Multiplication by Power (γ−1) of modulus-gradient value, Inverse Vector-Gradient operator} together with generalized versions. We believe the Inverse Vector-Gradient operator (and its generalized version) to be a novel operation in image processing. The generalization of the Vector-Gradient operator consists of multiplying its transfer functions by an isotropic modulus-frequency to-thepower-(p−1) transfer-function. The inverse is defined accordingly. Although the generalized (Inverse) Negative Laplace and Vector-Gradient operators are best implemented via the frequency domain, their point spread functions are checked for too large footprints in order to avoid spatial aliasing in practical (periodic image) implementations. The limitations of frequency-power p for given dimension D are studied.