A Geometric Algebra Co-Processor for Color Edge Detection

Mishra, Biswajit; Wilson, Peter; Wilcock, Reuben

doi:10.3390/electronics4010094

Cited by 13 publications

(7 citation statements)

References 31 publications

(48 reference statements)

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“…Definition 2. The real CA C𝓁 r, s is the algebra generated by the orthonormal basis {e 1 , e 2 , • • • , e n } in R n , where r + s = n with [16][17][18][19] :…”

Section: Fundamentals Of Clifford Algebrasmentioning

confidence: 99%

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A generalization of Euler's formula in Clifford algebras

Akar

2022

Math Methods in App Sciences

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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.

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“…Definition 2. The real CA C𝓁 r, s is the algebra generated by the orthonormal basis {e 1 , e 2 , • • • , e n } in R n , where r + s = n with [16][17][18][19] :…”

Section: Fundamentals Of Clifford Algebrasmentioning

confidence: 99%

“…The wedge product of k vectors is a k-vector, or k-blade, and such a quantity is said to have a grade k. The highest grade element in a space is named the pseudoscalar. 7,17,18 For ∀x = (x 1 , x 2 , x 3 ) ∈ R 3 , y = (𝑦 1 , 𝑦 2 , 𝑦 3 ) ∈ R 3 , the wedge product:…”

Section: Fundamentals Of Clifford Algebrasmentioning

confidence: 99%

A generalization of Euler's formula in Clifford algebras

Akar

2022

Math Methods in App Sciences

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“…Definition 1. The real CA C𝓁 r, s is the algebra generated by the orthonormal basis {e 1 , e 2 , … , e n } in R n where r + s = n with [26][27][28][29] e…”

Section: Essentials Of Casmentioning

confidence: 99%

“…Definition The real CA Cℓ r , s is the algebra generated by the orthonormal basis

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

ℝ^{n}

where r + s = n with 26–29

e_{i}^{2} = 1, i = 1, \dots, r, e_{i}^{2} = - 1, i = r + 1, \dots, n,

e_{i} e_{j} = - e_{j} e_{i}, i, j = 1, \dots, n, i \neq j .

…”

Section: Essentials Of Casmentioning

confidence: 99%

Clifford algebra multivectors and kernels for melanoma classification

Akar

Sirakov

Mete

2021

Math Methods in App Sciences

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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.

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“…In [3], the above ideas have been extended further by introducing color vector transformations and a different subspace within GA. The paper presents an overview of the geometric algebra fundamentals and convolution operations involving rotors for image processing applications.…”

Section: Geometric Algebra Methods In Edge Detectionmentioning

confidence: 99%