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 (R2) and oriented wavelet f ¿ (R2). 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 f and deal with the question of which oriented wavelet f 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.
Duits, R., Duits, M., Almsick, van, M. A., & Haar Romenij, ter, B. M. (2007). Invertible orientation scores as an application of generalized wavelet theory. Pattern Recognition and Image Analysis, 17(1), 42-75. https://doi.org/10.1134/S1054661807010063