imageProcessing/at-u0-v1

Computes a piecewise smooth approximation of a grey-level or color image, by optimizing the Ambrosio-Tortorelli functional (with u a 0-form and v a 1-form).

Author(s) of this documentation:

Marion Foare, Jacques-Olivier Lachaud

Usage: at-u0-v1 -i [input.pgm]

(for grey-level image restoration)

Usage: at-u0-v1 -i [input.ppm]

(for color image restoration)

The Ambrosio-Tortorelli functional is a classical relaxation of the Mumford-Shah functional.

Given an input grayscale image, defined in an open bounded domain \( \Omega \), we represent its gray levels by a function \( g \in L^{\infty}(\Omega) \). In the Ambrosio-Tortorelli functional [1], one wants to find a function \( u \in SBV(\Omega) \) which is a smooth approximation of the input image \( g \). The Ambrosio-Tortorelli functional [1] is defined by

\[ \displaystyle AT_{\varepsilon}(u,v) = \int_\Omega \alpha |u-g|^2 + v^2 |\nabla u|^2 + \lambda \varepsilon |\nabla v|^2 + \frac{\lambda}{4 \varepsilon} |1-v|^2 dx, \]

for functions \( u,v \in W^{1,2}(\Omega)\) with \( 0 \leq v \leq 1 \).

In AT functional, function \( v \) is a smooth approximation of the set of discontinuities, and takes value close to 0 in this set, while being close to 1 outside discontinuities. A remarkable property of this functional is that it \( \Gamma \)-converges to (a relaxation of) MS functional as \( \varepsilon \) tends to 0 (see [1]). The intuition is that a large \( \varepsilon \) induces a solution with a fuzzy set of discontinuities, which is then progressively narrowed to the crisp 1-dimensional set of discontinuites as \( \varepsilon \) goes to 0.

We discretize AT with discrete calculus and define \( u \) and \( g \) on the vertices and \( v \) on the edges. We denote this formulation AT10. Gray levels are seen as point mass on the center of pixels, so that functions \( u \) and \( g \) are both 0-forms, while \( v \) is a dual 1-form in between \( u \). It follows:

\[ \displaystyle AT10(u,v) = \Sigma_{i=1}^n \alpha \langle u_i - g_i , u_i - g_i \rangle_0 + \langle v , \mathbf{d_0} u_i \rangle_1 \langle v , \mathbf{d_0} u_i \rangle_1 \\ + \lambda \varepsilon \langle (\mathbf{d_1} + \bar{\mathbf{\star}} \bar{\mathbf{d_1}} \mathbf{\star}) v , (\mathbf{d_1} + \bar{\mathbf{\star}} \bar{\mathbf{d_1}} \mathbf{\star}) v \rangle_1 + \frac{\lambda}{4\varepsilon} \langle 1 - v , 1 - v \rangle_1. \]

For more details, see Image restoration and inpainting with Ambrosio-Tortorelli functional

Allowed options are:

-h [ --help ]                         display this message
-i [ --input ] arg                    the input image PPM filename.
-m [ --inpainting-mask ] arg          the input inpainting mask filename.
-o [ --output ] arg (=AT)             the output image basename.
-l [ --lambda ] arg                   the parameter lambda.
-1 [ --lambda-1 ] arg (=0.3125)       the initial parameter lambda (l1).
-2 [ --lambda-2 ] arg (=0.0005)       the final parameter lambda (l2).
-q [ --lambda-ratio ] arg (=1.414213) the division ratio for lambda from l1 
                                      to l2.
-a [ --alpha ] arg (=1)               the parameter alpha.
-e [ --epsilon ] arg                  the initial and final parameter epsilon
                                      of AT functional at the same time.
--epsilon-1 arg (=2)                  the initial parameter epsilon.
--epsilon-2 arg (=0.25)               the final parameter epsilon.
--epsilon-r arg (=2)                  sets the ratio between two consecutive 
                                      epsilon values of AT functional.
-n [ --nbiter ] arg (=10)             the maximum number of iterations.
--image-snr arg                       the input image without deterioration 
                                      if you wish to compute the SNR.
-p [ --pixel-size ] arg (=1)          the pixel size for outputing images 
                                      (useful when one wants to see the 
                                      discontinuities v on top of u).
-c [ --color-v ] arg (=0xff0000)      the color chosen for displaying the 
                                      singularities v (e.g. red is 0xff0000).
-v [ --verbose ] arg (=0)             the verbose level (0: silent, 1: less 
                                      silent, etc).

example:

./imageProcessing/at-u0-v1 -i ../imageProcessing/Images/degrade-b04.pgm --image-snr ../imageProcessing/Images/degrade.pgm -a 0.05 --epsilon-1 4 --epsilon-2 0.25 -l 0.005 -p 2 -c 0xff0000 -o degrade

Input image g	Reconstructed image u	Perfect image
Input image (noise = 0.4)	AT01 alpha=0.05 lambda=0.005	Perfect image
SNR of g = 21.9183	SNR of u = 34.4426	Perfect image

Note

Other restoration examples, parameter analysis, and image inpainting examples may be found in Image restoration and inpainting with Ambrosio-Tortorelli functional.