Clifford‐Valued Bendlet Transforms: Theory and Localization Operators

ABSTRACT To achieve an efficient time‐frequency representation of higher‐dimensional signals, we introduce the notion of the Clifford‐valued bendlet transform, utilizing the geometric and algebraic properties intrinsic to bendlets and Clifford algebras. The bendlet system, a second‐order shearlet with bent elements, enables the precise characterization of the curvature of discontinuities. Clifford algebras, meanwhile, convert geometric objects into fundamental computational elements and define universal operators applicable to all types of geometric elements. In this work, we rigorously investigate the essential properties of the proposed transform using operator theory and Clifford‐valued Fourier transforms. We present a detailed analysis of the convergence of inversion formulae and the boundedness of the associated localization operators. Furthermore, we thoroughly explore certain classes of uncertainty principles for the Clifford‐valued bendlet transform. Our findings establish a robust theoretical foundation for the practical application of this transform in multidimensional signal processing.