# CNN-based Euler's Elastica Inpainting with Deep Energy and Deep Image Prior

arXiv: https://arxiv.org/abs/2207.07921 · HTML: https://mlsanity.com/paper/2207.07921/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 30 references that are in the map.
- First posted on arXiv 16 Jul 2022.

## Builds on (most cited first, 9 of 9)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Context Encoders: Feature Learning by Inpainting](https://mlsanity.com/paper/1604.07379/) (2016)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Free-Form Image Inpainting with Gated Convolution](https://mlsanity.com/paper/1806.03589/) (2018)
- [Early Stopping for Deep Image Prior](https://mlsanity.com/paper/2112.06074/) (2021)
- [Regularization by architecture: A deep prior approach for inverse problems](https://mlsanity.com/paper/1812.03889/) (2018)
- [Learning Euler's Elastica Model for Medical Image Segmentation](https://mlsanity.com/paper/2011.00526/) (2020)
- [Variational image regularization with Euler's elastica using a discrete gradient scheme](https://mlsanity.com/paper/1712.07386/) (2017)

## Built on by (most cited first, 2 of 2)

- [Makeup-Guided Facial Privacy Protection via Untrained Neural Network Priors](https://mlsanity.com/paper/2408.12387/) (2024)
- [Curvature regularization for Non-line-of-sight Imaging from Under-sampled Data](https://mlsanity.com/paper/2301.00406/) (2023)

## Abstract (arXiv)

Euler's elastica constitute an appealing variational image inpainting model. It minimises an energy that involves the total variation as well as the level line curvature. These components are transparent and make it attractive for shape completion tasks. However, its gradient flow is a singular, anisotropic, and nonlinear PDE of fourth order, which is numerically challenging: It is difficult to find efficient algorithms that offer sharp edges and good rotation invariance. As a remedy, we design the first neural algorithm that simulates inpainting with Euler's Elastica. We use the deep energy concept which employs the variational energy as neural network loss. Furthermore, we pair it with a deep image prior where the network architecture itself acts as a prior. This yields better inpaintings by steering the optimisation trajectory closer to the desired solution. Our results are qualitatively on par with state-of-the-art algorithms on elastica-based shape completion. They combine good rotation invariance with sharp edges. Moreover, we benefit from the high efficiency and effortless parallelisation within a neural framework. Our neural elastica approach only requires 3x3 central difference stencils. It is thus much simpler than other well-performing algorithms for elastica inpainting. Last but not least, it is unsupervised as it requires no ground truth training data.