Type
Bachelor thesis / Master thesis
Prerequisites
- Functional analysis, in particular operator theory
- Deep learning for image reconstruction
- Basic knowledge of PyTorch or Tensorflow
Description
The goal of inverse problems is to recover model parameters from a set of indirect measurements, for example, recovering images from data. In most experimental sciences, inverse problems play an important role since the physical observations are always mediated by measuring devices. An inverse problem is mathematically formulated as an operator inversion problem where the operator inverts the model's measurements in the absence of noise. The operator is referred to as "forward operator and is typically not invertible, making the associated inverse problem "ill-posed". Solving the ill-posed inverse problem by simply minimizing the miss-fit against the data leads to overfitting. To overcome this challenge, one needs to introduce apriori information to obtain solutions that are consistent with the physics of the measured signals. This can be, for example, the fact that the measured signals are sparse. Apriori information can be either modeled by first principles or learned from data; combining these approaches has yielded the best performance.
References
- Classical sparse coding algorithms and their mathematical foundations for image processing
Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (https://link.springer.com/book/10.1007/978-1-4419-7011-4) - A survey on theoretical developments of deep learning in inverse problems
- Theoretical Perspectives on Deep Learning Methods in Inverse Problems (https://ieeexplore.ieee.org/document/10035380)
- In addition to sparsity and inverse problems, the review paper provides an up-to-date introduction on improving optimization methods via machine learning, including plug-and-play and algorithm unrolling
Learning to Optimize: A Primer and A Benchmark (https://jmlr.org/papers/v23/21-0308.html) - Analysis on the relationship between deep learning and inverse problems
i) Convolutional Neural Networks Analyzed via Convolutional Sparse Coding (https://arxiv.org/abs/1607.08194)
ii) Theoretical Linear Convergence of Unfolded ISTA and its Practical Weights and Thresholds (https://arxiv.org/abs/1808.10038) - Important deep learning model, U-Net and visual transformer, for computer vision
i) U-Net: Convolutional Networks for Biomedical Image Segmentation (https://arxiv.org/abs/1505.04597)
ii) A Survey on Vision Transformer (https://arxiv.org/abs/2012.12556)