Harmonic Smoothing of Digital Images

Abstract

The problem of smoothing monochromic digital images is considered. The proposed approach can be attributed to the diffusion smoothing method. Digital images are modeled using functions from the space $L_{2}(Q)$, $Q$ is the field of vision. The diffusion transformation is introduced as a solution of the heat conductivity equation and the problem of stabilization of the solution is considered with an unlimited increase in time. We study the extension of the classical Laplace operator with the allocation of a subspace of its one-to-one action and then the construction the inverse operator - the harmonic smoothing operator. Such an operator is the convolution of the original (sharp) image and the fundamental solution of the Laplace equation minus the projection of this convolution to the harmonic subspace. A method for the approximate calculation of the convolution integral is given. The decomposition of the space $L_{2}(Q)$ into the orthogonal sum of a harmonic and Novikov subspace is analyzed. The algorithm of the method of basic potentials of the allocation of the harmonic component of a digital image is presented; the method is based on the completeness of systems of basis potentials. A discrete case and a one-parameter family of smoothing transformations for which the parameter acts as a measure of blur is considered. The results of computational experiments for different values of the spectral parameter is presented.