# First order algorithms in variational image processing

arXiv: https://arxiv.org/abs/1412.4237 · HTML: https://mlsanity.com/paper/1412.4237/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Dec 2014.

## Built on by (most cited first, 4 of 4)

- [Denoising: A Powerful Building-Block for Imaging, Inverse Problems, and Machine Learning](https://mlsanity.com/paper/2409.06219/) (2024)
- [An Operator-Splitting Method for the Gaussian Curvature Regularization Model with Applications to Surface Smoothing and Imaging](https://mlsanity.com/paper/2108.01914/) (2021)
- [Lagrangian Motion Magnification with Double Sparse Optical Flow Decomposition](https://mlsanity.com/paper/2204.07636/) (2022)
- [Geometric Generative Models based on Morphological Equivariant PDEs and GANs](https://mlsanity.com/paper/2403.14897/) (2024)

## Abstract (arXiv)

Variational methods in imaging are nowadays developing towards a quite universal and flexible tool, allowing for highly successful approaches on tasks like denoising, deblurring, inpainting, segmentation, super-resolution, disparity, and optical flow estimation. The overall structure of such approaches is of the form cal D(Ku) + α cal R (u) rightarrow minu ; where the functional cal D is a data fidelity term also depending on some input data f and measuring the deviation of Ku from such and cal R is a regularization functional. Moreover K is a (often linear) forward operator modeling the dependence of data on an underlying image, and α is a positive regularization parameter. While cal D is often smooth and (strictly) convex, the current practice almost exclusively uses nonsmooth regularization functionals. The majority of successful techniques is using nonsmooth and convex functionals like the total variation and generalizations thereof or ℓ1-norms of coefficients arising from scalar products with some frame system. The efficient solution of such variational problems in imaging demands for appropriate algorithms. Taking into account the specific structure as a sum of two very different terms to be minimized, splitting algorithms are a quite canonical choice. Consequently this field has revived the interest in techniques like operator splittings or augmented Lagrangians. Here we shall provide an overview of methods currently developed and recent results as well as some computational studies providing a comparison of different methods and also illustrating their success in applications.