The work presented in this dissertation is motivated by classical problems in signal and image processing from the perspective of variational and PDE-based methods. Analytically encoding qualitative features of signals into variational energies in conjunction with modern methods in sparse optimization allows for well-founded and robust models, the optimization of which yields meaningful and cohesive signal decomposition.

Part I of this dissertation is based on joint work Variational Mode Decomposition [DZ14] with Dominique Zosso, in which the goal is to recursively decompose a signal into different modes of separate spectral bands, which are unknown beforehand. In the late nineties,

Huang [HSL98] introduced the Hilbert-Huang transform, also known as Empirical Mode Decomposition, in order to decompose a signal into so-called intrinsic mode functions (IMF) along with a trend, and obtain instantaneous frequency data. The HHT/EMD algorithm is widely used today, although there is no exact mathematical model corresponding to this algorithm, and, consequently, the exact properties and limits are widely unknown. We

propose an entirely non-recursive variational mode decomposition model, where the modes are extracted concurrently. The model looks for a number of modes and their respective center frequencies, such that the modes reproduce the input signal, while being smooth after demodulation into baseband. In Fourier domain, this corresponds to a narrow-band prior. Our model provides a solution to the decomposition problem that is theoretically well-founded, tractable, and motivated. The variational model is efficiently optimized using

an alternating direction method of multipliers approach. Preliminary results show excellent performance with respect to existing mode decomposition models.

Part II of this dissertation is the n-dimensional extension of the Variational Mode Decomposition. Decomposing multidimensional signals, such as images, into spatially compact, potentially overlapping modes of essentially wavelike nature makes these components accessible for further downstream analysis such as space-frequency analysis, demodulation, estimation of local orientation, edge and corner detection, texture analysis, denoising, inpainting,

and curvature estimation. The model decomposes the input signal into modes with narrow Fourier bandwidth; to cope with sharp region boundaries, incompatible with narrow bandwidth, we introduce binary support functions that act as masks on the narrow-band mode for image re-composition. L1 and TV-terms promote sparsity and spatial compactness. Constraining the support functions to partitions of the signal domain, we effectively get an image segmentation model based on spectral homogeneity. By coupling several submodes together with a single support function we are able to decompose an image into several crystal grains. Our efficient algorithm is based on variable splitting and alternate direction optimization; we employ Merriman-Bence-Osher-like [MBO92] threshold dynamics to handle eciently the motion by mean curvature of the support function boundaries under the sparsity promoting terms. The versatility and effectiveness of our proposed model

is demonstrated on a broad variety of example images from different modalities. These demonstrations include the decomposition of images into overlapping modes with smooth or sharp boundaries, segmentation of images of crystal grains, and inpainting of damaged image regions through artifact detection.

Part III of this dissertation is based on joint work with Igor Yanovsky of NASA Jet Propulsion Laboratory. We introduce a variational method for destriping data acquired by pushbroom type satellite imaging systems. The model leverages sparsity in signals and is based on current research in sparse optimization and compressed sensing. It is based on the basic

principles of regularization and data fidelity with certain constraints using modern methods in variational optimization - namely total variation (TV), both L1 and L2 fidelity, and the alternate direction method of multipliers. The main algorithm in Part III uses sparsity promoting energy functionals to achieve two important imaging effects. The TV term maintains boundary sharpness of content in the underlying clean image, while the L1 fidelity allows for the equitable removal of stripes without over- or under-penalization, providing a more accurate model of presumably independent sensors with unspecified and unrestricted bias distribution. A comparison is made between the TV-L1 and TV-L2 models to exemplify the qualitative efficacy of an L1 striping penalty. The model makes use of novel minimization splittings and proximal mapping operators, successfully yielding more realistic destriped images in very few iterations.