A regularized deep matrix factorized model of matrix completion for image restoration

It has been an important approach of using matrix completion to perform image restoration. Most previous works on matrix completion focus on the low-rank property by imposing explicit constraints on the recovered matrix, such as the constraint of the nuclear norm or limiting the dimension of the matrix factorization component. Recently, theoretical works suggest that deep linear neural network has an implicit bias towards low rank on matrix completion. However, low rank is not adequate to reflect the intrinsic characteristics of a natural image. Thus, algorithms with only the constraint of low rank are insufficient to perform image restoration well. In this work, we propose a Regularized Deep Matrix Factorized (RDMF) model for image restoration, which utilizes the implicit bias of the low rank of deep neural networks and the explicit bias of total variation. We demonstrate the effectiveness of our RDMF model with extensive experiments, in which our method surpasses the state of art models in common examples, especially for the restoration from very few observations. Our work sheds light on a more general framework for solving other inverse problems by combining the implicit bias of deep learning with explicit regularization

Noise2Inverse: Self-supervised deep convolutional denoising for tomography

Recovering a high-quality image from noisy indirect measurements is an important problem with many applications. For such inverse problems, supervised deep convolutional neural network (CNN)-based denoising methods have shown strong results, but the success of these supervised methods critically depends on the availability of a high-quality training dataset of similar measurements. For image denoising, methods are available that enable training without a separate training dataset by assuming that the noise in two different pixels is uncorrelated. However, this assumption does not hold for inverse problems, resulting in artifacts in the denoised images produced by existing methods. Here, we propose Noise2Inverse, a deep CNN-based denoising method for linear image reconstruction algorithms that does not require any additional clean or noisy data. Training a CNN-based denoiser is enabled by exploiting the noise model to compute multiple statistically independent reconstructions. We develop a theoretical framework which shows that such training indeed obtains a denoising CNN, assuming the measured noise is element-wise independent and zero-mean. On simulated CT datasets, Noise2Inverse demonstrates an improvement in peak signal-to-noise ratio and structural similarity index compared to state-of-the-art image denoising methods and conventional reconstruction methods, such as Total-Variation Minimization. We also demonstrate that the method is able to significantly reduce noise in challenging real-world experimental datasets.Comment: This paper appears in: IEEE Transactions on Computational Imaging On page(s): 1320-1335 Print ISSN: 2333-9403 Online ISSN: 2333-9403 Digital Object Identifier: 10.1109/TCI.2020.301964

Solving ill-posed inverse problems using iterative deep neural networks

We propose a partially learned approach for the solution of ill posed inverse problems with not necessarily linear forward operators. The method builds on ideas from classical regularization theory and recent advances in deep learning to perform learning while making use of prior information about the inverse problem encoded in the forward operator, noise model and a regularizing functional. The method results in a gradient-like iterative scheme, where the "gradient" component is learned using a convolutional network that includes the gradients of the data discrepancy and regularizer as input in each iteration. We present results of such a partially learned gradient scheme on a non-linear tomographic inversion problem with simulated data from both the Sheep-Logan phantom as well as a head CT. The outcome is compared against FBP and TV reconstruction and the proposed method provides a 5.4 dB PSNR improvement over the TV reconstruction while being significantly faster, giving reconstructions of 512 x 512 volumes in about 0.4 seconds using a single GPU