Incorporating structural constraints is critical for exploiting prior information in estimation problems. In this dissertation, we consider estimation problems in two areas with complex priors: magnetic resonance imaging (MRI) and millimeter wave (mmWave) channel estimation.

In MRI estimation, we present a novel low-rank model-based parametric matrix estimation method for joint MR image reconstruction and estimation of intrinsic tissue parameter suitable for highly accelerated (i.e. highly undersampled) scans. The MR image typically has a low-rank structure. This is a reasonable expectation if there are a limited number of tissue types in the field of view, and pixels from the same tissue type have the same parameter value. Also, we add an additional penalty term to the optimization that enforces that the temporal evolution of each pixel in the MR image is approximately described by some parametric structure. Unfortunately, due to the parametric term, the optimization is non-convex. However, we show that a simple variant of the commonly-used Iterative Shrinkage and Thresholding Algorithm (ISTA) can provide convergence to a local minimum of the optimization. Specifically, for cardiac T2 (transverse spin relaxation time) mapping, we assume a monoexponential model to the time response of the image as a function of the pixel's T2 value.

For mmWave channel estimation, we present a novel method for estimation of the receive-side spatial covariance matrix of a channel from a sequence of power measurements made in different angular directions. In mmWave cellular system, the channels will likely have a low-rank structure relative to the number of antenna elements. Extensive measurements at 28 and 73 GHz in New York City (NYC)---a dense, urban environment similar to likely initial deployments for mmWave systems---have shown that the mmWave channel energy is often concentrated in a small number of relatively narrow-beam clusters. Analysis of this data has revealed that the channel is often well approximated by a rank three or four channel, typically much smaller than the antenna dimension. This low-rank property implies that the spatial covariance matrix can be characterized by a relatively small number of parameters for the purpose of channel estimation. Of course, the use of low-rank spatial structure is widely used in array processing and underlies many classic channel estimations for wireless systems. The contribution in this work is to consider low-rank channel estimation from analog measurements. Furthermore, we exploit that the spatial covariance matrix is nonnegative. The non-negative nature of the covariance matrix reduces the number of measurements required when the matrix is low-rank. We show that maximum likelihood (ML) reconstruction of the channel covariance matrix from a collection of such measurements made at random angles is similar to a non-negative matrix completion problem that has been used widely in machine learning and image processing.

Previous mmWave statistical channel models generally only describe the channel at a single location. Here, we follow a standard model where the channel at each point can be described by a set of ''path clusters''. We model number of Line of sight (LOS) and number of Non-Line of Sight (NLOS) path clusters as a spatial random process whose variations are due to randomness in the environment. While a complete description of the random process would require describing the joint density of any finite set of points, here, we focus on developing statistical models for the pairwise joint densities for neighboring points. By marginalizing these pairwise joint densities, we obtain the point-wise distribution. Also, by further assumption that the process is Markov, we can generate a random trajectory for any given route.