Complex-valued Deep Learning with Applications to Magnetic Resonance Image Synthesis

Magnetic resonance imaging (MRI) has the ability to produce a series of images that each have different visual contrast between tissues, allowing clinicians to qualitatively assess pathologies that may be visible in one contrast-weighted image but not others. Unfortunately, these standard contrast-weighted images do not contain quantitative values, producing challenges for post-processing, assessment, and longitudinal studies. MR fingerprinting is a recent technique that produces quantitative tissue maps from a single pseudorandom acquisition, but it relies on computationally heavy nearest neighbor algorithms to solve the associated nonlinear inverse problem. In this dissertation, we present our deep learning methods to speed up quantitative MR fingerprinting and synthesize the standard contrast-weighted images directly from the same MR fingerprinting scan.Adapting deep learning methodologies to MR image synthesis presents two specific challenges: 1) complex-valued data and 2) the presence of noise while undersampling.MRI signals are inherently complex-valued, as they are measurements of rotating magnetization within the body. However, modern neural networks are not designed to support complex values. As an example, the pervasive ReLU activation function is undefined for complex numbers. This limitation curtails the impact of deep learning for complex data applications, such as MRI, radio frequency modulation identification, and target recognition in synthetic-aperture radar images. In this dissertation, we discuss the motivation for complex-valued networks, the changes that we have made to implement complex backpropagation, and our new complex cardioid activation function that made it possible to outperform real-valued networks for MR fingerprinting image synthesis.In Fourier-based medical imaging, undersampling results in an underdetermined system, in which a linear reconstruction will exhibit artifacts. Another consequence is lower signal-to-noise ratio (SNR) because of fewer acquired measurements. The coupled effects of low SNR and underdetermined system during reconstruction makes it difficult to model the signal and analyze image reconstruction algorithms. We demonstrate that neural networks trained only with a Gaussian noise model fail to process in vivo MR fingerprinting data, while our proposed empirical noise model allows neural networks to successfully synthesize quantitative images. Additionally, to better understand the impact of noise on undersampled imaging systems, we present an image quality prediction process that reconstructs fully sampled, fully determined data with noise added to simulate the SNR loss induced by a given undersampling pattern. The resulting prediction image empirically shows the effects of noise in undersampled image reconstruction without any effect from an underdetermined system, allowing MR pulse sequence and reconstruction developers to determine if low SNR, rather than the underdetermined system, is the limiting factor for a successful reconstruction.