Fourier
Analysis of the 2D Screened Poisson Equation 
Pravin
Bhat, Brian
Curless, Michael Cohen, C. Lawrence Zitnick 
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
welldefined sharpen filter that generalizes Laplacian sharpening, which
itself can be mapped to gradient domain filtering. Results using a DCTbased
screened Poisson solver are demonstrated on several applications including
image blending for panoramas, image sharpening, and deblocking of compressed
images. 
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