Fast computation methods for estimation of image spatial entropy

Abstract

Computation of image spatial entropy (ISE) is prohibitive in many applications of image processing due to its high computational complexity. Four fast or computationally efficient methods for estimation of ISE are thus introduced in this paper. Three of these estimation methods are parametric and the fourth one is non-parametric. The reduction in the computational complexity from the original formulation of ISE is made possible by making use of the Markovianity constraint which causes the joint histograms of neighboring pixels to become dense around their main diagonal. It is shown that by tolerating merely 1% estimation error, the order of complexity is significantly reduced and for applications that can tolerate 6% estimation error, the complexity is reduced to that of the classical monkey model entropy.

Semiotic Aggregation in Deep Learning

“…Some computational improvements for calculating the spatial entropy were proposed by Razlighi et al [ 52 , 53 ]. The computational overhead can be significantly reduced if we accept a reduction of the approximation accuracy.…”

Spatial Mutual Information as Similarity Measure for 3-D Brain Image Registration

“…In this approach, a causal MRF model, called Quadrilateral Markov Random Field (QMRF), was used to compute image spatial information under the definition of Shannon entropy. The spatial entropy defined in [6] was further simplified for homogenous but anisotropic QMRF in [29] for nearest neighboring structures as per the following equation: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$$\eqalignno{&H({{\bf X}})=mn(H(X,X_{u})+H(X,X_{l})\cr&~~\quad\qquad-H(X))-{{mn}\over{2}}(H(X_{u},X_{l})+H(X_{u},X_{r}))&{\hbox{(5)}}}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$m\times n$\end{document} denotes the image size, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$H(X,X_{u})$\end{document} the joint entropy of a voxel with its upper neighbor, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$H(X,X_{l})$\end{document} the joint entropy of a voxel with its left neighbor, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$H(X_{u},X_{l})$\end{document} the joint entropy of the left and upper neighbors, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$H(X_{u},X_{r})$\end{document} the joint entropy of the right and upper neighbors; see Fig. 2 .…”

“…Finally, it should be added that in the case of 3D images all the existing cliques in the neighboring structure of the first order QMRF need to be included. Even though these additional cliques would not change the final outcome for the spatial mutual information of a homogeneous random field (as described in [29] ) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}${SMI}_{3D}$\end{document} , it is necessary to include them to ensure the positivity constraint of this similarity measure. This way, the new SMI is given by the following equation, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$$\eqalignno{&\!\!\!\!SMI\cr &\!\!\!=\!-mnH(X,Y)\!+\!{{mn}\over{4}}\left\{\!\!{\matrix{H(X,Y_{u})\!+\!H(X_{u},Y)\!+\!H(X,Y_{d})\hfill\cr\!+H(X_{d},Y)\!+\!H(X,Y_{l})\!+\!H(X_{l},Y)\hfill\cr\!+H(X,Y_{r})\!+\!H(X_{r},Y)\hfill}}\!\!\right\}\cr&-{{mn}\over{8}}\left\{\!…”

Evaluating similarity measures for brain image registration

“…are the same as in (13), with their symmetric counterparts. It should be noted that H ( X l , Y ) and its symmetric counterpart H ( X , Y l ) are omitted from Equation (14) since H ( X l , Y ) = H ( X r ,Y ) from a computational standpoint [33]. To prevent confusion we denote the original SMI defined by equation (13) as SMI NS .…”