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Clifford algebra; multivector; subspace; classification; skin lesion
The present study develops the Clifford algebra ${\rm Cl}_{5,0}$ within a dermatological task to diagnose skin melanoma using images of skin lesions, which are modeled here by means of 5D lesion feature vectors (LFVs). The LFV is a numerical approximation of the most used clinical rule for melanoma diagnosis - ABCD. To generate the ${\rm Cl}_{5,0}$ we develop a new formula that uses the entries of a 5D vector to calculate the entries of a 32D multivector. This vector provides a natural mapping of the original 5D vector onto the 2-, 3-, 4-vector ${\rm Cl}_{5,0}$ subspaces. We use a sample set of 112 5D LFVs and apply the new formula to calculate 112 32D multivectors in the ${\rm Cl}_{5,0}$. Next we map the 5D LFVs onto the 2-, 3-, 4-vector subspaces of the ${\rm Cl}_{5,0}$. In every subspace we apply a binary support vector machine to classify the mapped 112 LFVs. With the obtained results we calculate six metrics and evaluate the effectiveness of the diagnosis in every subspace. At the end of the paper we compare the classification results, obtained in every subspace, with the results obtained by the four diagnosing rules most used in clinical practice and contemporary machine learning methods. This way we reveal the potential of using Clifford algebras in the analysis and classification of medical images.
[1] Agresti, A., Coull, B. A.: Approximate is better than ``exact'' for interval estimation of binomial proportions. Am. Stat. 52 (1998), 119-126. DOI 10.2307/2685469 | MR 1628435
[2] Society, American Cancer: Cancer Facts & Figures. American Cancer Society, Atlanta (2010), Available at \brokenlink{{all-cancer-facts-figures/cancer-facts-figures-2010.html}}
[3] Aragón, J. L., Aragon-Camarasa, G., Aragón-González, G., Rodríguez-Andrade, M. A.: Clifford algebra with mathematica. Available at (2018), 10 pages.
[4] Argenziano, G., Soyer, H. P., Giorgi, V. De, Piccolo, D.: Dermoscopy: A Tutorial. Edra Medical Publishing & New Media, Milan (2000).
[5] Barata, C., Celebi, M. E., Marques, J. S.: A survey of feature extraction in dermoscopy image analysis of skin cancer. IEEE J. Biomedical and Health Inf. 23 (2018), 1096-1109. DOI 10.1109/jbhi.2018.2845939
[6] Bayro-Corrochano, E. J., Arana-Daniel, N.: Clifford support vector machines for classification, regression, and recurrence. IEEE Trans. Neural Networks 21 (2010), 1731-1746. DOI 10.1109/tnn.2010.2060352
[7] N. C. F. Codella, D. Gutman, M. E. Celebi, B. Helba, M. A. Marchetti, S. W. Dusza, A. Kalloo, K. Liopyris, N. Mishra, H. Kittler, A. Halpern: Skin lesion analysis toward melanoma detection: A Challenge at the 2017 International Symposium on Biomedical Imaging (ISBI), Hosted by the International Skin Imaging Collaboration (ISIC). Available at (2017), 5 pages.
[8] Dolianitis, C., Kelly, J., Wolfe, R., Simpson, P.: Comparative performance of 4 dermoscopic algorithms by nonexperts for the diagnosis of melanocytic lesions. Arch. of Dermatology 141 (2005), 1008-1014. DOI 10.1001/archderm.141.8.1008
[9] Esteva, A., Kuprel, B., Novoa, R. A., Ko, J., Swetter, S. M., Blau, H. M., Thrun, S.: Dermatologist-level classification of skin cancer with deep neural networks. Nature 542 (2017), 115-118. DOI 10.1038/nature21056
[10] Guy, G. P., Ekwueme, D. U., Tangka, F. K., Richardson, L. C.: Melanoma treatment costs: A systematic review of the literature, 1990-2011. Am. J. Prev. Med. 43 (2012), 537-545. DOI 10.1016/j.amepre.2012.07.031
[11] Harangi, B.: Skin lesion classification with ensembles of deep convolutional neural networks. J. Biomedical Inf. 86 (2018), 25-32. DOI 10.1016/j.jbi.2018.08.006
[12] Jafari, M. H., Samavi, S., Karimi, N., Soroushmehr, S. M. R., Ward, K., Najarian, K.: Automatic detection of melanoma using broad extraction of features from digital images. Engineering in Medicine and Biology Society (EMBC), Annual International Conference of the IEEE IEEE, New York (2016), 1357-1360. DOI 10.1109/EMBC.2016.7590959
[13] Korotkov, K.: Automatic Change Detection in Multiple Skin Lesions. Ph.D. Thesis, Universitat de Girona, Girona (2014).
[14] Lounesto, P.: Clifford Algebras and Spinors. London Mathematical Society Lecture Note Series 286, Cambridge University Press, Cambridge (2001). DOI 10.1017/CBO9780511526022 | MR 1834977 | Zbl 0973.15022
[15] Masood, A., Al-Jumaily, A. A.: Computer aided diagnostic support system for skin cancer: a review of techniques and algorithms. Int. J. Biomedical Imaging 2013 (2013), Article ID 323268, 22 pages. DOI 10.1155/2013/323268
[16] Mete, M., Sirakov, N. M.: Dermoscopic diagnosis of melanoma in a 4D feature space constructed by active contour extracted features. Computerized Medical Imaging and Graphics 36 (2012), 572-579. DOI 10.1016/j.compmedimag.2012.06.002
[17] Mete, M., Sirakov, N. M., Griffin, J., Menter, A.: A novel classification system for dysplastic nevus and malignant melanoma. IEEE International Conference on Image Processing (ICIP) IEEE, New York (2016), 3414-3418. DOI 10.1109/icip.2016.7532993
[18] Mishra, B., Wilson, P. R., Wilcock, R.: A geometric algebra co-processor for color edge detection. Electronics 4 (2015), 94-117. DOI 10.3390/electronics4010094
[19] F. Nachbar, W. Stolz, T. Merkle, A. B. Cognetta, T. Vogt, M. Landthaler, P. Bilek, O. Braun Falco, G. Plewig: The ABCD rule of dermatoscopy. High prospective value in the diagnosis of doubtful melanocytic skin lesions. J. Am. Acad. Dermatol. 30 (1994), 551-559. DOI 10.1016/S0190-9622(94)70061-3
[20] Roy, S., Mitra, A., Setua, S. K.: Color image representation using multivector. 5th International Conference on Intelligent Systems, Modelling and Simulation IEEE, Langkawi (2014), 27-29. DOI 10.1109/isms.2014.66
[21] Schott, R., Staples, G. S.: Reductions in computational complexity using Clifford algebras. Adv. Appl. Clifford Algebr. 20 (2010), 121-140. DOI 10.1007/s00006-008-0143-2 | MR 2601894 | Zbl 1191.68335
[22] Singh, N., Gupta, S. K.: Recent advancement in the early detection of melanoma using computerized tools: An image analysis perspective. Skin Res. Technol. 25 (2019), 129-141. DOI 10.1111/srt.12622
[23] Sirakov, N. M., Mete, M., Selvaggi, R., Luong, M.: New accurate automated melanoma diagnosing systems. International Conference on Healthcare Informatics IEEE, Dallas 374-379 (2015). DOI 10.1109/ICHI.2015.53
[24] Sultana, N. N., Mandal, B., Puhan, N. B.: Deep residual network with regularized fisher framework for detection of melanoma. IET Computer Vision 12 (2018), 1096-1104. DOI 10.1049/iet-cvi.2018.5238
[25] J. Vaz, Jr., R. da Rocha, Jr.: An Introduction to Clifford Algebras and Spinors. Oxford University Press, Oxford (2016). DOI 10.1093/acprof:oso/9780198782926.001.0001 | MR 3931311 | Zbl 1347.15001
[26] Wahba, M. A., Ashour, A. S., Guo, Y., Napoleon, S. A., Elnaby, M. M. Abd: A novel cumulative level difference mean based GLDM and Modified ABCD features ranked using eigenvector centrality approach for four skin lesion types classification. Computer Methods and Programs in Biomedicine 165 (2018), 163-174. DOI 10.1016/j.cmpb.2018.08.009
[27] Yu, Z., Jiang, X., Zhou, F., Qin, J., Ni, D., Chen, S., Lei, B., Wang, T.: Melanoma recognition in dermoscopy images via aggregated deep convolutional features. IEEE Trans. Biomed. Eng. 66 (2019), 1006-1016. DOI 10.1109/TBME.2018.2866166
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