Abstract. We analyze the problem of reconstructing a 2D function that approximates a set of desired gradients and a data term. The combined data and gradient terms enable operations like modifying the gradients of an image while staying close to the original image. Starting with a variational formulation, we arrive at the "screened Poisson equation" known in physics. Analysis of this equation in the Fourier domain leads to a direct, exact, and efficient solution to the problem. Further analysis reveals the structure of the spatial filters that solve the 2D screened Poisson equation and shows gradient scaling to be a well-defined sharpen filter that generalizes Laplacian sharpening, which itself can be mapped to gradient domain filtering. Results using a DCT-based screened Poisson solver are demonstrated on several applications including image blending for panoramas, image sharpening, and de-blocking of compressed images.
In 2008 REF they proposed a variational formulation that combines a data term with a gradient term, the combination of these two kinds of terms allowed them to derive several kinds of filters such as Sharpening, Flattening and de-blocking.
Fourier Analysis of the 2D Screened Poisson Equation for Gradient Domain Problems
ECCV
Mathematics Computer Science