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An Analysis of Brain MRI Segmentation
Published in Siddhartha Bhattacharyya, Anirban Mukherjee, Indrajit Pan, Paramartha Dutta, Arup Kumar Bhaumik, Hybrid Intelligent Techniques for Pattern Analysis and Understanding, 2017
Tuhin Utsab Paul, Samir Kumar Bandhyopadhyay
To indicate the number of classes we set a variable. Here we use three classes to perform the task. In order to implement the fuzzy logic, we have to set the fuzzy exponent to a value that must be greater than one. Now, to make this algorithm run for a finite number of times it demands a value that would highlight the highest number of iterations of the proposed algorithm. As the algorithm mainly deals with the medical images, it is necessary for it to deal with high-precision value. Hence, we have introduced a parameter or variable whose main job is to control the limit of error or to specify the maximum error value that can be accepted. Here we have evaluated the Mahalanobis distance wherever necessary. The Mahalanobis distance is the multidimensional generalization of the idea of measuring the amount of standard deviation of P from the mean of D. Zero distance denotes that P is the mean of D. As the distance increases, the deviation also increases along with the principal component axis. Rescaling the axes to have unit variance makes the Mahalanobis distance correspond to the standard Euclidean distance. In geometry, measurements of variables are done in the same unit of length. In modeling problems, variables have different scales. The Mahalanobis distance uses the covariance among variables to calculate distances. Hence, the problem of scale and correlation of the Euclidean distance has been overcome. While using the Euclidean distance, all equidistant points from a particular location form a sphere. The Mahalanobis distance corrects the corresponding scales of the variables and accounts for them. In Euclidean geometry, the data is considered to be isotropic Gaussian, treating each of them equally. Whereas, the Mahalanobis distance measures the correlation between the variables, assuming anisotropic Gaussian distribution. If the priorities are known among the features, then using the Mahalanobis distance is a better option. The Z-score feature scaling can overcome the usefulness of choosing a Mahalanobis distance over Euclidean distance. Lastly a variable is used to control the number of cases to find a perfect solution. We have introduced a parameter which determines the nature of the input image and the quantity of noise that is present in it. To mark a low-contrast object another variable or parameter is needed. This parameter is tuned to a low value. In order to control noise we utilize a parameter whose value is one for noisy image, or else it can be initialized to zero.
The concept of invariance in school mathematics
Published in International Journal of Mathematical Education in Science and Technology, 2018
Shlomo Libeskind, Moshe Stupel, Victor Oxman
In fact, every mathematical theorem expresses one or several invariant properties. For example, any point located on a perpendicular bisector of a segment is equidistant from its ends. The invariant property here is the fact that even when one changes the location of the point on the perpendicular bisector, the distances to the endpoints are equal. Another invariant property is the fact that the segments that connect any point on the perpendicular bisector to the endpoints of the segment form equal angles with the perpendicular bisector.