In one dimension, the Laguerre polynomials are orthogonal under exponential weighting: $ \int_0^\infty L_n(x) L_m(x) e^{-x} \, dx = 0, n \ne m $ Does anyone know what the corresponding basis functions would be in 2 dimensions? $ \int_{-\infty}^\infty \int_{-\infty}^\infty F_n(x,y) F_m(x,y) e^{-r} \, dx \, dy = 0, r= \sqrt{x^2+y^2}, n \ne m $ The Zernike polynomials are orthogonal, but with uniform weight and over the unit disk.
The underlying problem is to compute an estimator for a missing pixel. A series of orthogonal functions are helpful since you can then incrementally compute a 1-st order estimator, then a second order, then a third order, and so on. The exponential arises since (in natural scenes anyway) presumably pixels further away have less influence.