ADMM-TGV image restoration for scientific applications with unbiased parameter choice

<jats:title>Abstract</jats:title><jats:p>Image restoration via alternating direction method of multipliers (ADMM) has gained large interest within the last decade. Solving standard problems of Gaussian and Poisson noise, the set of “Total Variation” (TV)-based regularizers proved to be efficient and versatile. In the last few years, the “Total Generalized Variation” (TGV) approach combined TV regularizers of different orders adaptively to better suit local regions in the image. This improved the technique significantly. The approach solved the staircase problem inherent of the first-order TV while keeping the beneficial edge preservation. The iterative minimization for the augmented Lagrangian of TGV problems requires four important parameters: two penalty parameters <jats:inline-formula><jats:alternatives><jats:tex-math>$${\rho }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$${\eta }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> and two regularization parameters <jats:inline-formula><jats:alternatives><jats:tex-math>$${\lambda _{0}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$${\lambda _{1}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula>. The choice of penalty parameters decides on the convergence speed, and the regularization parameters decide on the impact of the respective regularizer and are determined by the noise level in the image. For scientific applications of such algorithms, an automated and thus objective method to determine these parameters is essential to receive unbiased results independent of the user. Obviously, both sets of parameters are to be well chosen to achieve optimal results, too. In this paper, a method is proposed to adaptively choose optimal <jats:inline-formula><jats:alternatives><jats:tex-math>$${\rho }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$${\eta }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> values for the iteration to converge faster, based on the primal and dual residuals arising from the optimality conditions of the augmented Lagrangian. Further, we show how to choose <jats:inline-formula><jats:alternatives><jats:tex-math>$${\lambda _{0}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$${\lambda _{1}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> based on the inherent noise in the image.</jats:p>