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 \(\mathbb{L}_2 \)(ℝ2) and oriented wavelet φ ∈ \(\mathbb{L}_2 \)(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 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.
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