## A New Ring Theory Based Algorithm and Stopping Criterion for Image Segmentation

April 20, 2019
Alisa Rahim

Abstract: Ring theory is most widely known as a branch of pure mathematics under the field of abstract algebra. Some of the uses of Ring Theory in the modern world involve cryptography, computer vision, and image segmentation. As of now, finite cyclic rings have been incorporated into performing image segmentations for the Mean Shift Iterative Algorithm. This paper analyzes the Mean Shift Iterative Algorithm and devises an improved algorithm and stopping criterion using finite cyclic rings and matrices in Ring Theory that perform high-quality image segmentations for images that can be used in computer vision and possibly the segmentation(s) of grayscale (d = 1), colored (d = 3), and multispectral (d ≥ 3) images.

Keywords: ring theory, Mean Shift, and Iterative Algorithm

Introduction: Based on the concepts of Group Theory and the field of abstract algebra, Ring Theory is a concept where a “ring” is a set of elements with two binary factors: addition and multiplication. To subtract within a ring would essentially mean to add an element to its additive inverse. Likewise, to divide would mean to multiply an element by its multiplicative inverse. A ring also satisfies the following axioms:

• The ring, under addition, is an abelian group.
• The multiplication operation is associative, and therefore closed.
• All operations satisfy the distributive law of multiplication over addition.

An example of a ring includes the set of real polynomials. Within this ring, you can freely add, subtract, and multiply one polynomial, essentially an element within the ring, to get another polynomial - another element. The additive identity is presented as zero. Since zero is a constant polynomial, it is also considered to be an element in the ring of real polynomials. The multiplicative identity is presented as one. Since multiplication is always commutative among all polynomials, the ring of real polynomials is deduced as a commutative ring with an identity element.

Some rings are finite, meaning that the amount and type of elements may be limited. Some rings may not have the additive identity zero or the multiplicative identity one. Upon adding, subtracting, or multiplying two even numbers, the result is always another even number. The value of 1 does not fall within the set of even numbers. Therefore, the set of even integers does not have the multiplicative identity of one - and is only a commutative ring. Image segmentation is the practice of breaking a picture up into pixels and assigning each pixel a value based on a given class. The purpose of image segmentation is to partition images into more meaningful, easy to examine, sections. The segmentation of images is primarily applied to image editing/compression, as well as the recognition of certain objects or another relevant aspects of a taken image. The Mean Shift Iterative Algorithm uses finite cyclic rings to detect specific features of an image (i.e. eyes of a face, abnormalities in an MRI scan of a heart, tumors in a brain) and the probability of there being a specific part of an image. A finite cyclic ring is any ring where the elements derive from a single element (hence, they are limited in regards to what elements may be present within the ring and, when brought back within range of said ring, the elements repeat in a cycle).

A primary factor in determining the stopping criteria for a segmentation algorithm is the entropy, or number of consistent microscopic configurations, of an image. The number of consistent microscopic configurations is significant to constructing a stopping criterion for an algorithm for image segmentation because while images may interchangeably be weakly and strongly equivalents, images that are strongly equivalent are not weakly equivalent. Images are defined in a finite cyclic ring when the Mean Shift Iterative Algorithm is used for image segmentation. However, an established stopping criterion for the Mean Shift Iterative Algorithm has not been formulated thus far; instead, the entropy formula has been in place as the stopping criterion for Mean Shift for stability purposes.

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