Aq-Series Approach to Deblurring the Discrete Gaussian

Abstract

A method is presented for deblurring an image blurred by the discrete Gaussian. The method, based on classical theorems of Jacobi and Ramanujan, not only provides exact formulas for the deblurring, but also condition numbers and error bounds estimating the agreement between the original and reconstructed image. The use of the Jacobi Triple Product Theorem provides a convenient factorization of the formulas used in the inversion process into a product of three infinite series. These three series correspond to a constant together with a Toeplitz operator and its transpose. In the finite setting this factorization corresponds to the factorization of a matrix into the product of Toeplitz matrices, where the entries can be computed using simple recursion formulas. For selected choices of σ, condition numbers are calculated for these operators. The results are similar to a method developed by Kimia and Zucker.