This paper presents a novel and unreported approach developed to filter T2-weighetd Magnetic Resonance Imaging (MRI). The MRI data is fitted with a parametric bivariate cubic Lagrange polynomial, which is used as the model function to build the continuum into the discrete samples of the two-dimensional MRI images. On the basis of the aforementioned model function, the Classic-Curvature (CC) and the Signal Resilient to Interpolation (SRI) images are calculated and they are used as filter masks to convolve the two-dimensional MRI images of the pathological human brain. The pathologies are human brain tumors. The result of the convolution provides with filtered T2-weighted MRI images. It is found that filtering with the CC and the SRI provides with reliable and faithful reproduction of the human brain tumors. The validity of filtering the T2-weighted MRI for the quest of supplemental information about the tumors is also found positive.

This paper solves the biomedical engineering problem of the extraction of complementary and/or additional information related to the depths of the anatomical structures of the human brain tumor imaged with Magnetic Resonance Imaging (MRI). The combined calculation of the signal resilient to interpolation and the Intensity-Curvature Functional provides with the complementary and/or additional information. The steps to undertake for the calculation of the signal resilient to interpolation are: (i) fitting a polynomial function to the signal, (ii) the calculation of the classic-curvature of the signal, (iii) the calculation of the Intensity-Curvature term before interpolation of the signal, (iv) the calculation of the Intensity-Curvature term after interpolation of the signal, (v) the solution of the equation of the two aforementioned Intensity-Curvature terms of the signal provides with the signal resilient to interpolation. The Intensity-Curvature Functional is the result of the ratio between the two Intensity-Curvature terms before and after interpolation. Because of the fact that the signal resilient to interpolation and the Intensity-Curvature Functional are derived through the process of re-sampling the original signal, it is possible to obtain an immense number of images from the original MRI signal. This paper shows the combined use of the signal resilient to interpolation and the Intensity-Curvature Functional in diagnostic settings when evaluating a tumor imaged with MRI. Additionally, the Intensity-Curvature Functional can identify the tumor contour line.

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.

A Signal-Image fitted with a model function, embeds the property of the intensity-curvature content, which is defined through the math formulae merging together the signal intensity with the second order derivatives of the model function. This work presents one of the measures of the intensity-curvature content, which is called the Intensity-Curvature Functional along with qualitative results obtained with Magnetic Resonance Imaging (MRI) of the human brain and also with a sample contextual image. The Intensity-Curvature Functional is calculated in three dimensions while re-sampling the signal-image with the trivariate cubic Lagrange interpolation formula and also in two dimensions while re-sampling using the bivariate cubic Lagrange interpolation formula. The Intensity-Curvature Functional is defined as the ratio between the numerator called intensity-curvature term before interpolation and the denominator called intensity-curvature term after interpolation. The intensity-curvature term before interpolation is calculated through the multiplication between: (i) the signal intensity and (ii) the sum of the second order partial derivatives of the model function, both of them calculated at the grid point. The intensity-curvature term after interpolation is calculated through the multiplication between: (i) the signal intensity and (ii) the sum of second order partial derivatives of the model function, both of them calculated at the intra-pixel location chosen to re-sample the signal. Two most relevant properties are discernible through the Intensity-Curvature Functional. One property is the intensity-curvature content, and the other property is that the signal-image is re-imaged so to create a novel mapping of the original signal-image from which the Intensity-Curvature Functional is calculated. The novel mapping highlights and portraits the original image features under a different perspective.

The biomedical engineering problem addressed in this work is the one of finding a novel signal-image content measure called intensity-curvature functional making use of all of the second order derivatives of the model function fitted to the data. Given a signal-image made of a sequel of discrete samples and given a model function which embeds the property of second order differentiability, it is possible to quantify the content of the signal-image through a novel approach based on both of the intensity and of the total curvature of the signal-image. The signal-image is fitted with the model function. The total curvature can be calculated through the sum of all of the second order derivatives of the Hessian of the model function fitted to the data. The intensity-curvature functional is defined as the ratio between: (i) the integral of the multiplication between the value of the signal modeled through an interpolation function and the total curvature of the signal-image; both of them at the temporal-spatial location of its sampling (the grid nodes) and, (ii) the integral of the value of the multiplication between the signal modeled through an interpolation function and the total curvature of the signal-image; both of them at any given temporal-spatial location of its re-sampling (intra-pixel location). This manuscript shows both of the formulae and the qualitative results of: the intensity-curvature functional and the intensity-curvature measures which are conceptually linked to the intensity-curvature functional. The formulations here presented make the engineering innovation. The intensity-curvature functional depends on both of the model function fitting the signal-image and the magnitude of re-sampling employed to calculate the second order derivatives of the Hessian of the model function.