Ridge Functions and Orthonormal Ridgelets

Orthonormal ridgelets are a specialized set of angularly-integrated ridge functions which make up an orthonormal basis for L2(R). In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x1 cos θ + x2 sin θ). We derive a formula giving the ridgelet coefficients of a ridge function in terms of the 1-D wavelet coefficients of the ridge profile r(t), and we study the properties of the linear approximation operator which ‘kills’ coefficients at high angular scale or high ridge scale. We also show that partial orthonormal ridgelet expansions can give efficient nonlinear approximations to pure ridge functions. In effect, the rearranged weighted ridgelet coefficients of a ridge function decay at essentially the same rate as the rearranged weighted 1-D wavelet coefficients of the 1-D ridge profile r(t). This shows that simple thresholding in the ridgelet basis is, for certain purposes, equally as good as ideal nonlinear ridge approximation.