Computation of Image Spatial Entropy Using Quadrilateral Markov Random Field

Abstract

Abstract-Shannon entropy is a powerful tool in image analysis, but its reliable computation from image data faces an inherent dimensionality problem that calls for a low-dimensional and closed form model for the pixel value distributions. The most promising such models are Markovian, however, the conventional Markov random field is hampered by noncausality and its causal versions are also not free of difficulties. For example, the Markov mesh random field has its own limitations due to the strong diagonal dependency in its local neighboring system. A new model, named quadrilateral Markov random field (QMRF) is introduced in this paper in order to overcome these limitations. A property of QMRF with neighboring size of 2 is then used to decompose an image prior into a product of 2-D joint pdfs in which they are estimated using a joint histogram under the homogeneity assumption. In addition, the paper includes an extension of the introduced method to the computation of image spatial mutual information. Comparisons on synthesized images as well as two applications with real images are presented to motivate the developments in this paper and demonstrate the advantages in the performance of the introduced method over the existing ones. Index Terms-Image histogram, image spatial entropy, Markov mesh random field (MMRF), Markov random field (MRF), quadrilateral Markov random field (QMRF).

Multi-resolution classification of urban areas using hierarchical symmetric Markov mesh models

“…Unlike conventional MRFs, MMRF can be used not only to incorporate spatial information at each scale of the quad-tree but also to keep the causality of the resulting hierarchical model. However, MMRFs and their lattice models exhibit a well-known weakness, i.e., they may favor artifacts aligned with a direction departing from one corner of the image (they are "corner-dependent") [9]. Consequently, the integration of a hierarchical MRF and a spatial Markov mesh model does not necessarily exhibit an anisotropic behavior and can be affected by the use of a non-regular (non-symmetric) neighborhood.…”

“…To mitigate this drawback several techniques have been recently introduced in the literature. Quadrilateral MRFs were introduced in [9], in which the non-regularity problem is avoided by using four Markov meshes related to the different corners of the lattice and enforcing them into a unique field definition; however, the model is still non-symmetric. To overcome these limitations from both mathematical and practical points of view, in [10] a new random field was established: a symmetric, corner-independent, and isotropic model that incorporates the dependency of a pixel on all its neighbors using a symmetric MMRF.…”

Fast computation methods for estimation of image spatial entropy

“…In this paper, four methods of estimating ISE (three parametric and one non-parametric) have been introduced that reduce the order of complexity of the original formulation of ISE discussed in [3]. It has been shown that one of the three parametric methods reduces the complexity to that of the classical MME at the expense of a 6% decrease in accuracy, while the other two parametric methods lower the complexity with higher accuracy when the number of off diagonals becomes [25.…”

“…In [3], we introduced a new method of computing ISE which was shown to be more accurate than the other existing methods. The computational complexity of this new method is of the order of O(mn ?…”

Breast cancer detection using MRF-based probable texture feature and decision-level fusion-based classification using HMM on thermography images

“…In some of the related works, Shannon entropy, mutual Information, and also different similarity measures have been used as the energy functions for segmentation or classification based upon MRF model [36]. Thus, we have used Tsallis entropy to extract the rate of energy changes between cliques in each CM1 and CM2 matrixes.…”