A mathematical analysis of the Tensorial Morphological Gradient approach

• A qualitative analysis of the Tensorial Morphological Gradient approach is presented.
• Frobenius norm leads the Tensorial Morphological Gradient approach to the best results.
• Some (dis)similarity measures used in Tensorial Morphological Gradient are not well defined.

This paper presents a mathematical analysis of the Tensorial Morphological Gradient, an approach to estimate color image gradients based on a color representation scheme that establishes a correspondence between different color models and tensors. We detail and provide the necessary theoretical explanations for six (dis)similarity measures used to compute the gradient considering the HSV color model and conclude that only one is pertinent to the Tensorial Morphological Gradient approach. The other measures present different types of limitations, including the computation of unbounded values, that disturb the results. The experimental results are consistent with the theoretical conclusions.