Identifying PSL(2, ") as a projective group for patterns in the conformal camera model, the projective harmonic analysis on its double covering group SL(2, ") is presented in the noncompact and compact pictures * the pictures used to study di!erent aspects of irreducible unitary representations of semisimple Lie groups. Bypassing technicalities of representation theory, but stressing the motivation and similarities with Euclidean Fourier analysis, each constructed picture of the projective Fourier analysis includes the Fourier transform, Plancherel's theorem and convolution property. Projectively covariant characteristics of the analysis in the noncompact picture allow rendering any of image projective transformations of a pattern (after removing conformal distortions) by using only one projective Fourier transform of the original pattern, what is demonstrated in a computer simulation. The convolution properties in both pictures must by used to develop algorithms for projectively invariant matching of patterns. Work in progress on fast algorithms for computing with projective Fourier transforms and for rendering image projective transformations is discussed. E$cient computations of the convolutions would follow from the both fast projective Fourier transforms and their inverses.