Potts Model - The Potts Model in Signal and Image Processing

The Potts Model in Signal and Image Processing

The Potts model has applications in signal reconstruction. Assume that we are given noisy observation of a piecewise constant signal To recover from the noisy observation vector, on seeks a minimizer of the corresponding inverse problem, the -Potts functional which is defined by

$P_\gamma(u) = \gamma \| \nabla u \|_0 + \| u-f\|_p^p = \gamma \# \{ i : u_i \neq u_{i+1} \} + \sum_{i=1}^n |u_i - f_i|^p$

The jump penalty forces piecewise constant solutions and the data term couples the minimizing candidate to the data The parameter controls the tradeoff between regularity and data fidelity. There are fast algorithms for the exact minimization of the L^1 and the L^2-Potts functional (Friedrich, Kempe, Liebscher, Winkler, 2008).

In image processing, the Potts functional is related to the segmentation problem. However, in two dimensions the problem is NP-hard (Boykov, Veksler, Zabih, 2001).

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