Fourier Analysis of the 2D Screened Poisson Equation for Gradient Domain Problems

Authors:
Pravin Bhat;Brian Curless;Michael Cohen;C. Lawrence Zitnick
Affiliations:
University of Washington, ;University of Washington, ;University of Washington, and Microsoft Research, ;University of Washington,
Venue:
ECCV '08 Proceedings of the 10th European Conference on Computer Vision: Part II
Year:
2008

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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.