Invertible orientation scores as an application of generalized wavelet theory

Abstract

Inspired by the visual system of many mammals, we consider the construction of-and reconstruction from-an orientation score of an image, via a wavelet transform corresponding to the left-regular representation of the Euclidean motion group in double-struck L sign2(ℝ2) and oriented wavelet φ ∈ double-struck L sign2(ℝ 2). Because this representation is reducible, the general wavelet reconstruction theorem does not apply. By means of reproducing kernel theory, we formulate a new and more general wavelet theory, which is applied to our specific case. As a result we can quantify the well-posedness of the reconstruction given the wavelet φ and deal with the question of which oriented wavelet φ is practically desirable in the sense that it both allows a stable reconstruction and a proper detection of local elongated structures. This enables image enhancement by means of left-invariant operators on orientation scores. © Nauka/Interperiodica 2007.