The present study develops the Clifford algebra Cl5,0 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 Cl5,0 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 Cl5,0 subspaces. We use a sample set of 112 5D LFVs and apply the new formula to calculate 112 32D multivectors in the Cl5,0. Next we map the 5D LFVs onto the 2-, 3-, 4-vector subspaces of the Cl5,0. 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.

中文翻译：

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Clifford 代数子空间中的支持向量机皮肤病变分类

本研究在皮肤病学任务中开发了 Clifford 代数 Cl5,0，以使用皮肤病变图像诊断皮肤黑色素瘤，此处通过 5D 病变特征向量 (LFV) 建模。LFV 是最常用的黑色素瘤诊断临床规则 - ABCD 的数值近似值。为了生成 Cl5,0，我们开发了一个新公式，该公式使用 5D 向量的条目来计算 32D 多向量的条目。该向量提供了原始 5D 向量到 2、3、4 向量 Cl5,0 子空间的自然映射。我们使用 112 个 5D LFV 的样本集并应用新公式计算 Cl5,0 中的 112 个 32D 多向量。接下来，我们将 5D LFV 映射到 Cl5,0 的 2、3、4 向量子空间。在每个子空间中，我们应用二进制支持向量机对映射的 112 个 LFV 进行分类。根据获得的结果，我们计算六个指标并评估每个子空间中诊断的有效性。在论文的最后，我们将在每个子空间中获得的分类结果与通过临床实践中最常用的四种诊断规则和当代机器学习方法获得的结果进行了比较。通过这种方式，我们揭示了在医学图像分析和分类中使用 Clifford 代数的潜力。