Code accompanying the paper "Mixed Noise and Posterior Estimation with Conditional DeepGEM". Here we learn the noise parameters $(a,b)$ of a Bayesian inverse problem $Y = f(X) + \xi,$ where $\xi \sim \mathcal{N}\big(0, a^2 + b^2 \text{diag} (f(x)^2) \big)$ is a mixture of additive and multiplicative Gaussian noise. This approach allows to incorporate information from several measurements within one model and therefore for more accurate reconstruction of the noise parameters, when more than one measurement is available. Here the distance to the true a and b decreases as more information is given.

We build upon the deepGEM framework [2], and combine it with conditional normalizing flows [3] to solve scatterometric inverse problems [4,5]. If you use the forward models from the low-dimensional photo mask please cite the corresponding papers [4,5]. if you use the forward model from the oxide layer please cite the zenodo page [6] and our paper [1].

The code is split up into reverse and forward KL for the high-dimensional version. For the lower dimensional scatterometry example, we also included baselines, where we chose a and b on a grid with comparable run time.

[1] Mixed Noise and Posterior Estimation with Conditional DeepGEM, Hagemann et al, arXiv:2402.02964

[2] DeepGEM: Generalized Expectation-Maximization for Blind Inversion, Gao et al, NeurIPS 2021

[3] Guided Image Generation with Conditional Invertible Neural Networks, Ardizzone et al, arXiv 1907.02392

[4] Bayesian approach to the statistical inverse problem of scatterometry: Comparison of three surrogate models, Heidenreich et al, International Journal for Uncertainty Quantification, 5(6), 2015

[5] Bayesian approach to determine critical dimensions from scatterometric measurements, Heidenreich et al, Metrologia, 55(6):S201, 2018

[6] Zenodo link

@article{Hagemann_2024,
doi = {10.1088/2632-2153/ad5926},
url = {https://dx.doi.org/10.1088/2632-2153/ad5926},
year = {2024},
month = {jul},
publisher = {IOP Publishing},
volume = {5},
number = {3},
pages = {035001},
author = {Paul Hagemann and Johannes Hertrich and Maren Casfor and Sebastian Heidenreich and Gabriele Steidl},
title = {Mixed noise and posterior estimation with conditional deepGEM},
journal = {Machine Learning: Science and Technology}
}