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).

Famous quotes containing the words model, signal and/or image:

“She represents the unavowed aspiration of the male human being, his potential infidelity—and infidelity of a very special kind, which would lead him to the opposite of his wife, to the “woman of wax” whom he could model at will, make and unmake in any way he wished, even unto death.”
—Marguerite Duras (b. 1914)

“Mistakes, scandals, and failures no longer signal catastrophe. The crucial thing is that they be made credible, and that the public be made aware of the efforts being expended in that direction. The “marketing” immunity of governments is similar to that of the major brands of washing powder.”
—Jean Baudrillard (b. 1929)

“The exile is a singular, whereas refugees tend to be thought of in the mass. Armenian refugees, Jewish refugees, refugees from Franco Spain. But a political leader or artistic figure is an exile. Thomas Mann yesterday, Theodorakis today. Exile is the noble and dignified term, while a refugee is more hapless.... What is implied in these nuances of social standing is the respect we pay to choice. The exile appears to have made a decision, while the refugee is the very image of helplessness.”
—Mary McCarthy (1912–1989)