Representing a function of two variables as a superposition over angles of ridge functions with a given shape via linear programming

A ridge function with shape function g in the horizontal direction is a function of the form g(x)h(y,0). Along each horizontal line it has the shape g(x), multiplied by a function h(y,0) which depends on the y-value of the horizontal line. Similarly a ridge function with shape function g in the vertical direction has the form g(y)h(x,π/2). For a given shape function g it may or may not be possible to represent an arbitrary function f(x,y) as a superposition over all angles of a ridge function with shape g in each direction, where h=hf=hf,g depends on the functions f and g and also on the direction, θ:h=hf,g(·,θ). We show that if g is Gaussian centered at zero then this is always possible and we give the function hf,g for a given f(x,y). For highpass or for odd shapes g, we show it is impossible to represent an arbitrary f(x,y), i.e. in general there is no hf,g. Note that our problem is similar to tomography, where the problem is to invert the Radon transform, except that the use of the word inversion is here somewhat “inverted”: in tomography f(x,y) is unknown and we find it by inverting the projections of f; here, f(x,y) is known, g(z) is known, and hf(·,θ)=hf,g(·,θ) is the unknown.