Bi-Linear Modeling of Data Manifolds for Dynamic-MRI Recovery.

This paper puts forth a novel bi-linear modeling framework for data recovery via manifold-learning and sparse-approximation arguments and considers its application to dynamic magnetic-resonance imaging (dMRI). Each temporal-domain MR image is viewed as a point that lies onto or close to a smooth manifold, and landmark points are identified to describe the point cloud concisely. To facilitate computations, a dimensionality reduction module generates low-dimensional/compressed renditions of the landmark points. Recovery of high-fidelity MRI data is realized by solving a non-convex minimization task for the linear decompression operator and affine combinations of landmark points which locally approximate the latent manifold geometry. An algorithm with guaranteed convergence to stationary solutions of the non-convex minimization task is also provided. The aforementioned framework exploits the underlying spatio-temporal patterns and geometry of the acquired data without any prior training on external data or information. Extensive numerical results on simulated as well as real cardiac-cine MRI data illustrate noteworthy improvements of the advocated machine-learning framework over state-of-the-art reconstruction techniques.

MLS: Joint Manifold-Learning and Sparsity-Aware Framework for Highly Accelerated Dynamic Magnetic Resonance Imaging.

Manifold-based models have been recently exploited for accelerating dynamic magnetic resonance imaging (dMRI). While manifold-based models have shown to be more efficient than conventional low-rank approaches, joint low-rank and sparsity-aware modeling still appears to be very efficient due to the inherent sparsity within dMR images. In this paper, we propose a joint manifold-learning and sparsity-aware framework for dMRI. The proposed method establishes a link between the recently developed manifold models and conventional sparsity-aware models. Dynamic MR images are modeled as points located on or close to a smooth manifold, and a novel data-driven manifold-learning approach, which preserves affine relation among images, is used to learn the low-dimensional embedding of the dynamic images. The temporal basis learnt from such an approach efficiently captures the inherent periodicity of dynamic images and hence sparsity along temporal direction is enforced during reconstruction. The proposed framework is validated by extensive numerical tests on phantom and in-vivo data sets.

M-MRI: A Manifold-based Framework to Highly Accelerated Dynamic Magnetic Resonance Imaging.

High-dimensional signals, including dynamic magnetic resonance (dMR) images, often lie on low dimensional manifold. While many current dynamic magnetic resonance imaging (dMRI) reconstruction methods rely on priors which promote low-rank and sparsity, this paper proposes a novel manifold-based framework, we term M-MRI, for dMRI reconstruction from highly undersampled k-space data. Images in dMRI are modeled as points on or close to a smooth manifold, and the underlying manifold geometry is learned through training data, called "navigator" signals. Moreover, low-dimensional embeddings which preserve the learned manifold geometry and effect concise data representations are computed. Capitalizing on the learned manifold geometry, two regularization loss functions are proposed to reconstruct dMR images from highly undersampled k-space data. The advocated framework is validated using extensive numerical tests on phantom and in-vivo data sets.

Bi-Linear Modeling of Manifold-Data Geometry for Dynamic-MRI Recovery.

This paper establishes a modeling framework for data located onto or close to (unknown) smooth manifolds, embedded in Euclidean spaces, and considers its application to dynamic magnetic resonance imaging (dMRI). The framework comprises several modules: First, a set of landmark points is identified to describe concisely a data cloud formed by highly under-sampled dMRI data, and second, low-dimensional renditions of the landmark points are computed. Searching for the linear operator that decompresses low-dimensional data to high-dimensional ones, and for those combinations of landmark points which approximate the manifold data by affine patches, leads to a bi-linear model of the dMRI data, cognizant of the intrinsic data geometry. Preliminary numerical tests on synthetically generated dMRI phantoms, and comparisons with state-of-the-art reconstruction techniques, underline the rich potential of the proposed method for the recovery of highly under-sampled dMRI data.

ACCELERATING DYNAMIC MAGNETIC RESONANCE IMAGING BY NONLINEAR SPARSE CODING.

Although being high-dimensional, dynamic magnetic resonance images usually lie on low-dimensional manifolds. Nonlinear models have been shown to capture well that latent low-dimensional nature of data, and can thus lead to improvements in the quality of constrained recovery algorithms. This paper advocates a novel reconstruction algorithm for dynamic magnetic resonance imaging (dMRI) based on nonlinear dictionary learned from low-spatial but high-temporal resolution images. The nonlinear dictionary is initially learned using kernel dictionary learning, and the proposed algorithm subsequently alternates between sparsity enforcement in the feature space and the data-consistency constraint in the original input space. Extensive numerical tests demonstrate that the proposed scheme is superior to popular methods that use linear dictionaries learned from the same set of training data.

Adaptive multiregression in reproducing kernel Hilbert spaces: the multiaccess MIMO channel case.

This paper introduces a wide framework for online, i.e., time-adaptive, supervised multiregression tasks. The problem is formulated in a general infinite-dimensional reproducing kernel Hilbert space (RKHS). In this context, a fairly large number of nonlinear multiregression models fall as special cases, including the linear case. Any convex, continuous, and not necessarily differentiable function can be used as a loss function in order to quantify the disagreement between the output of the system and the desired response. The only requirement is the subgradient of the adopted loss function to be available in an analytic form. To this end, we demonstrate a way to calculate the subgradients of robust loss functions, suitable for the multiregression task. As it is by now well documented, when dealing with online schemes in RKHS, the memory keeps increasing with each iteration step. To attack this problem, a simple sparsification strategy is utilized, which leads to an algorithmic scheme of linear complexity with respect to the number of unknown parameters. A convergence analysis of the technique, based on arguments of convex analysis, is also provided. To demonstrate the capacity of the proposed method, the multiregressor is applied to the multiaccess multiple-input multiple-output channel equalization task for a setting with poor resources and nonavailable channel information. Numerical results verify the potential of the method, when its performance is compared with those of the state-of-the-art linear techniques, which, in contrast, use space-time coding, more antenna elements, as well as full channel information.

Adaptive learning in complex reproducing kernel Hilbert spaces employing Wirtinger's subgradients.

This paper presents a wide framework for non-linear online supervised learning tasks in the context of complex valued signal processing. The (complex) input data are mapped into a complex reproducing kernel Hilbert space (RKHS), where the learning phase is taking place. Both pure complex kernels and real kernels (via the complexification trick) can be employed. Moreover, any convex, continuous and not necessarily differentiable function can be used to measure the loss between the output of the specific system and the desired response. The only requirement is the subgradient of the adopted loss function to be available in an analytic form. In order to derive analytically the subgradients, the principles of the (recently developed) Wirtinger's calculus in complex RKHS are exploited. Furthermore, both linear and widely linear (in RKHS) estimation filters are considered. To cope with the problem of increasing memory requirements, which is present in almost all online schemes in RKHS, the sparsification scheme, based on projection onto closed balls, has been adopted. We demonstrate the effectiveness of the proposed framework in a non-linear channel identification task, a non-linear channel equalization problem and a quadrature phase shift keying equalization scheme, using both circular and non circular synthetic signal sources.

Adaptive kernel-based image denoising employing semi-parametric regularization.

The main contribution of this paper is the development of a novel approach, based on the theory of Reproducing Kernel Hilbert Spaces (RKHS), for the problem of noise removal in the spatial domain. The proposed methodology has the advantage that it is able to remove any kind of additive noise (impulse, gaussian, uniform, etc.) from any digital image, in contrast to the most commonly used denoising techniques, which are noise dependent. The problem is cast as an optimization task in a RKHS, by taking advantage of the celebrated Representer Theorem in its semi-parametric formulation. The semi-parametric formulation, although known in theory, has so far found limited, to our knowledge, application. However, in the image denoising problem, its use is dictated by the nature of the problem itself. The need for edge preservation naturally leads to such a modeling. Examples verify that in the presence of gaussian noise the proposed methodology performs well compared to wavelet based technics and outperforms them significantly in the presence of impulse or mixed noise.