Fully smoothed ℓ1-TV models: Bounds for the minimizers and parameter choice

F. Baus (1) , Mila Nikolova (2) , Gabriele Steidl (1)

1 University of Kaiserslautern [Kaiserslautern]
2 CMLA - Centre de Mathématiques et de Leurs Applications

We consider a class of convex functionals that can be seen as C1 smooth approximations of the ℓ1-TV model. The minimizers of such functionals were shown to exhibit a qualitatively different behavior compared to the nonsmooth ℓ1-TV model [12]. Here we focus on the way the parameters involved in these functionals determine the features of the minimizers u*. We give explicit relationships between the minimizers and these parameters. Given an input digital image f, we prove that the error ∥u*−f∥_infty obeys b−ε ≤ ∥u*−f∥_infty ≤b where b is a constant independent of the input image. Further we can set the parameters so that ε > 0 is arbitrarily close to zero. More precisely, we exhibit explicit formulae relating the model parameters, the input image f and the values b and ε. Conversely, we can fix the parameter values so that the error ∥u*−f∥_infty satisfy some prescribed b, ε. All theoretical results are confirmed using numerical tests on natural digital images of different sizes with disparate content and quality.

Mots clés: Parameter estimation for smoothed ℓ1−TV model; Histogram specification; Quantisation noise; ℓ∞ error; Convex optimization; image processing

Domaines: Analyse numérique [math.NA]

F. Baus, Mila Nikolova, Gabriele Steidl. Fully smoothed ℓ1-TV models: Bounds for the minimizers and parameter choice. Journal of Mathematical Imaging and Vision, 2013, ⟨10.1007/s10851-013-0420-0⟩. ⟨hal-00722743v3⟩