Computational Intelligence in Magnetic Resonance Imaging of the Human Brain: The Classic-Curvature and the Intensity-Curvature Functional in a Tumor Case Study

Abstract

Abstract — This research solves the computational intelligence problem of devising two mathematical engineering tools called Classic-Curvature and Intensity-Curvature Functional. It is possible to calculate the two mathematical engineering tools from any model polynomial function which embeds the property of second-order differentiability. This work presents results obtained with bivariate and trivariate cubic Lagrange polynomials. The use of the Classic-Curvature and the Intensity-Curvature Functional can add complementary information in medical imaging, specifically in Magnetic Resonance Imaging (MRI) of the human brain. Index Terms — Classic-Curvature, Computational Intelligence, Intensity-Curvature Functional, Magnetic Resonance Imaging (MRI), Model Polynomial Function, Second-Order Derivative, Second-Order Differentiability. I. I NTRODUCTION Since the inception of signal-image interpolation through the works of Isaac Newton [1], an enormous amount of studies corroborated with formulations of improved interpolation paradigms have been reported in the literature. The most representative studies [2-4] have guided the discovery of the unifying framework [5] and the unified framework [6] for the improvement of the interpolation error. From the two frameworks [5, 6], two mathematical engineering tools have been conceived. The two mathematical engineering tools are: (i) the Classic-Curvature [6] and (ii) the Intensity-Curvature Functional [5-8]. The methodological characteristics for the calculation of the two aforementioned mathematical engineering tools are the same regardless of the model polynomial function fitted to the signal-image data. The only requirement is that the model polynomial function benefits of the property of second-order differentiability in its interval of definition, which is that property that makes the model polynomial function to have non null and continuous second-order derivatives. The key to the formulation of the mathematical engineering tools presented here is thus the computational intelligence of the Classic-Curvature, which is calculated summing up all of the second-order derivatives of Hessian of the model polynomial function fitted to the signal-image data. The calculation of the Classic-Curvature is also beneficial to the calculation of the Intensity-Curvature Functional, which is defined through the ratio between two terms, both of them inclusive of the intensity-curvature content of the signal-image. The two terms are: (i) the integral of the product between the Classic-Curvature and the value of the signal-image intensity, both of them calculated at the grid node, and (ii) the integral of the product between the Classic-Curvature and the value of the signal-image intensity, both of them calculated at the intra-pixel coordinate used to re-sample the signal. Thus, re-sampling, which is inherent to interpolation, is also another piece of computational intelligence, which allows the calculation of both of the Classic-Curvature and the Intensity-Curvature Functional. Specifically, the intra-pixel coordinate chosen to calculate the aforementioned mathematical engineering tools is such to determine the information content of the resulting images and thus, in the present research, it is relevant to the information extracted from the Magnetic Resonance Images. In this paper, emphasis is given to two model polynomial functions, namely: the bivariate and the trivariate cubic Lagrange polynomials [6, 8]. It is here shown that from the aforementioned two model polynomial functions it is possible to calculate the Classic-Curvature and the Intensity-Curvature Functional, which extract information from the original MRI images. Therefore, through the use of the Classic-Curvature and the Intensity-Curvature Functional of the model polynomial functions it is possible to highlight characteristics of the signal-image which are seen in the MRI domain. In the theory section, the procedure for the calculation of the mathematical engineering tools will be