Enhancing a Kernel Method for Shape Analysis in Kendall Space

Abstract In learning methods defined on a manifold, it is often important to embed the manifold into a Hilbert space to apply kernel methods, as it enables the utilization of linear techniques in the transformed space, thereby facilitating complex data analysis and classification tasks with nonlinear structures [10]. In [6], the authors define an embedding of the $$\textit{m}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> </mml:math> -dimensional Kendall shape space into an $$\mathbb {R}^{\textit{N}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> space, from which kernel methods are introduced in the Kendall shape space. In this paper, we significantly simplify this result by achieving an embedding of the Kendall shape space into a Euclidean space $$\mathbb {R}^{\textit{n}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> of much lower dimension $$\textit{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> than $$\textit{N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> , greatly simplifying calculations and computational costs of applications. Additionally, we characterize the image of the Kendall shape space in $$\mathbb {R}^{\textit{n}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> , allowing us to define a new extrinsic mean of shapes in the Kendall shape space.