Support vector machine skin lesion classification in Clifford algebra subspaces

Math Methods in App Sciences

Sirakov

Mete

2021

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Melanoma is a deadly skin disease. Availability of digital skin lesion datasets ease the exploration of ample classification studies. Both theoretical and heuristics improvements are achieved thanks to these new datasets. Being one of many high‐level feature‐driven classification methods, support vector machines (SVMs) are widely used in the literature as melanoma classifiers. Almost all of these studies are using a limited set of predefined kernels. In this study, we propose a newly developed Clifford kernel for the classification of dermoscopic skin lesions. We develop Clifford‐based linear, polynomial, and exponential kernels in the Clifford algebra (CA) Cℓ5, 0 0‐, 2‐, and 4‐vector subspaces. CAs are noncommutative but associative and distributive over addition. We showed that the newly developed Clifford kernels are embedded into SVM classifiers to successfully identify malignant skin lesions in a binary classification settings. Clifford kernel results are compared with mostly used gaussian and polynomial kernels with real‐valued SVM classifiers. Accuracy of all classifiers are assessed with cross‐validation using imbalanced and balanced datasets of 112, 162, and 192 lesions. SVM kernels in comparison are parameterized to scan wide range of possibilities. We show that Clifford‐based polynomial kernels outperforms in all, balanced and imbalanced, datasets having average accuracy of 83%. The consistence of high accuracies obtained with Clifford polynomial kernel shows that skin lesion features are logically designed and Clifford‐based SVM is able to model class separations in the feature space.

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

“…We consider the vector

U = ((), u_{1}, u_{2}, \dots, u_{n}) \in ℝ^{n}

in the basis

({}, e_{1}, e_{2}, \dots, e_{n})

and construct the formula below filling up the left factor with

u_{i_{1}} e_{i_{1}}, ((), i_{1} = 1, \dots, n - 1)

and the right factor with

u_{i_{2}} e_{i_{2}}, ((), i_{2} = 2, \dots, n)

. Thus, we generate bivectors using Clifford product:

u_{i_{1}} u_{i_{2}} e_{i_{1}} e_{i_{2}} f o r i_{1} < i_{2} .

Analogously to the work in Akar and Sirakov, 25 we employ the notation e 12 … n = e 1 e 2 … e n 25,26 . Further, we calculate the coefficients of the k ‐vectors

e_{i_{1} i_{2} \dots i_{k}}

of the CA Cℓ n , 0 .…”

Section: Multivector In Cℓnmentioning

confidence: 99%

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Clifford algebra multivectors and kernels for melanoma classification

Math Methods in App Sciences

Sirakov

Mete

2021

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“…given as direct sums of subspaces pp  ,   0 pn  given by Eq. (6) equation, can be written as in [5,6] 01…”

mentioning

confidence: 99%

“…where fg is the inner or dot product while  fg is the wedge or outer (or exterior) product. Hence [3,5]  …”

mentioning

confidence: 99%

The Essentials of Clifford Algebras with Maple Programming

Sakarya University Journal of Science

2021

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Clifford algebra (geometric algebra) which has many applications in physics, robotics, Computer-Aided Manufacture, computer graphics, image processing, Computer-Aided Design etc. is one of the important subjects in mathematics. In this paper, after we give the definition of Clifford algebras, introduce their subspaces. Firstly, we develop an algorithm which obtains some concepts of Clifford algebras using Maple programming. Secondly, another algorithm calculates the norm of the multivector obtained by finding the Clifford product of any two vectors of the same finite dimension.

A generalization of Euler's formula in Clifford algebras

Math Methods in App Sciences

2022

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In this paper, the equation ‖‖eboldx∧boldy=1, such that boldx,boldy∈ℝn, is proved using defined group homomorphism and Euler's formula, where x ∧ y is the outer product between x and y. Firstly, this equation is verified for n=2,3,4,5 using Clifford product. Then, it turns out that it is difficult to maintain the proof in this way since the outer product is anticommutative, the size increases, and the calculation becomes a lot of work. For this reason, a group of homomorphism from the group Cℓn, 0 to the group ℝn,0 is described and used Euler's formula. Eventually, it is proved for any n∈ℤ+()n≥2 by generalizing this equation.