A Mathematical Framework for Deep Learning in Elastic Source Imaging

Abstract

An inverse elastic source problem with sparse measurements is of concern. A generic mathematical framework is proposed which extends a low-dimensional manifold regularization in the conventional source reconstruction algorithms thereby enhancing their performance with sparse data-sets. It is rigorously established that the proposed framework is equivalent to the so-called deep convolutional framelet expansion in machine learning literature for inverse problems. Apposite numerical examples are furnished to substantiate the efficacy of the proposed framework. (DCNN) for low-dose computed tomography (CT), winning the second place in 2016 American Association of Physicists in Medicine (AAPM) X-ray CT Low-dose Grand Challenge [44]. Jin et al. [26] and Han et al. [23] independently showed that the global streaking artifacts from the sparse-view CT can be removed efficiently with the deep network. In MRI, Wang et al. [57] applied deep learning to provide a soft initialization for compressed sensing MRI (CS-MRI). In photo-acoustic tomography, Antholzer et al.[9] proposed a U-Net architecture [48] to effectively remove streaking artifacts from inverse spherical Radon transform based reconstructed images. The power of machine learning for inverse problems has been also demonstrated in material discovery and designs, in which the goal is to find the material compositions and structures to satisfy the design goals under assorted design constraints [42,43,52].In spite of such intriguing performance improvement by deep learning approaches, the origin of the success for inverse problems was poorly understood. To address this, we recently proposed so-called deep convolutional framelets as a powerful mathematical framework to understand deep learning approaches for inverse problems [62]. The novelty of the deep convolutional framelets was the discovery that an encoder-decoder network structure emerges as the signal space manifestation from Hankel matrix decomposition in the higher dimensional space [62]. In addition, by controlling the number of filter channels, a neural network is trained to learn the optimal local bases so that it gives the best low-rank shrinkage [62]. This discovery demonstrates an important link between the deep learning and the compressed sensing approachs [17] through a Hankel structure matrix decomposition [25,27,63].Thus, the aim of this paper is to provide a deep learning reconstruction formula for elastic source imaging from sparse measurements. Specifically, a generic framework is provided that incorporates a low-dimensional manifold regularization in the

Deep Residual Learning for Accelerated MRI Using Magnitude and Phase Networks

“…See Appendix B and [29] for detailed descriptions regarding why and how to employ dual frame U-Net. In addition to the application of dual frame U-Net for sparse CT reconstruction problems [29], dual frame U-Net has been successfully used for elastic source imaging problems [33], a complicated inverse scattering problem, confirming the advantages of dual frame U-Net over U-Net. Therefore, we propose dual frame U-Net for MR reconstruction.…”

Photorealistic Style Transfer via Wavelet Transforms

“…To achieve better reconstruction, we adopted concatenation instead of summation for unpooling, similar to U-Net structure [13,28,36]. This enables the network to learn the weighted sum of components at the expense of interpretability and theoretical correctness.…”

Elastography mapped by deep convolutional neural networks

“…In other words, machine learning may inversely reconstruct elastography by regression analysis of the first-round data. The advent of machine learning, especially convolutional neural networks (CNNs) in deep learning, powers and boosts the processing of big data [31][32][33][34][35]. It has been demonstrated that deep CNNs have outstandingly succeeded in image classification and processing [31,35].…”

“…CNN regression allows multiple outputs, relying on convolutional operations to train the corresponding weights. However, limited efforts of ML have been made in elastography [33,34,40], only applied for image processing and enhancement without rigorous reconstruction involvement based on measured data. The reason is that the cost of data acquisition is unaffordable and even unreachable for having enough data to train partial differential equations governed problems [41].…”