Harmonic analysis on \(SL(2,\mathbb{C})\) and projectively adapted pattern representation (Q1271503)

The author develops a projective analogue of Fourier analysis, i.e. a noncommutative \(SL(2,\mathbb{C})\)-harmonic analysis on the complex line which decomposes the \(L^2\)-space of functions into irreducible invariant subspaces. First, a projectively invariant classification of patterns is constructed in terms of orbits of the group \(PSL(2,\mathbb{C}) =SL(2, \mathbb{C})/ \{\pm \text{Id}\}\) acting on the complex image plane by linear fractional transforms. Then, \(SL(2,\mathbb{C})\)-harmonic analysis in the noncompact picture of induced representations is used to decompose patterns into the components invariant under irreducible representations of the principle series of \(SL (2, \mathbb{C})\). By constructing a camera model, the author shows that the orbit of a pattern under the group \(PSL(2,\mathbb{C})\) acting on the (complex) image plane by linear fractional transforms contains all ``generic'' projective distortions of the pattern. The projectively adapted properties of \(SL(2,\mathbb{C})\)-harmonic analysis are confirmed by computational tests.