Beyond Low Rank + Sparse: Multi-scale Low Rank Matrix Decomposition

We present a natural generalization of the recent low rank + sparse matrix decomposition and consider the decomposition of matrices into components of multiple scales. Such decomposition is well motivated in practice as data matrices often exhibit local correlations in multiple scales. Concretely, we propose a multi-scale low rank modeling that represents a data matrix as a sum of block-wise low rank matrices with increasing scales of block sizes. We then consider the inverse problem of decomposing the data matrix into its multi-scale low rank components and approach the problem via a convex formulation. Theoretically, we show that under various incoherence conditions, the convex program recovers the multi-scale low rank components \revised{either exactly or approximately}. Practically, we provide guidance on selecting the regularization parameters and incorporate cycle spinning to reduce blocking artifacts. Experimentally, we show that the multi-scale low rank decomposition provides a more intuitive decomposition than conventional low rank methods and demonstrate its effectiveness in four applications, including illumination normalization for face images, motion separation for surveillance videos, multi-scale modeling of the dynamic contrast enhanced magnetic resonance imaging and collaborative filtering exploiting age information

Parallel Magnetic Resonance Imaging as Approximation in a Reproducing Kernel Hilbert Space

In Magnetic Resonance Imaging (MRI) data samples are collected in the spatial frequency domain (k-space), typically by time-consuming line-by-line scanning on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of data using multiple receivers (parallel imaging), and by using more efficient non-Cartesian sampling schemes. As shown here, reconstruction from samples at arbitrary locations can be understood as approximation of vector-valued functions from the acquired samples and formulated using a Reproducing Kernel Hilbert Space (RKHS) with a matrix-valued kernel defined by the spatial sensitivities of the receive coils. This establishes a formal connection between approximation theory and parallel imaging. Theoretical tools from approximation theory can then be used to understand reconstruction in k-space and to extend the analysis of the effects of samples selection beyond the traditional g-factor noise analysis to both noise amplification and approximation errors. This is demonstrated with numerical examples.Comment: 28 pages, 7 figure

Deficiency and abelianized deficiency of some virtually free groups

Let $Q_m$ be the HNN extension of $\Z/m \times \Z/m$ where the stable letter conjugates the first factor to the second. We explore small presentations of the groups $\Gamma_{m,n}=Q_m \ast Q_n$. We show that for certain choices of (m,n), for example (2,3), the group $\Gamma_{m,n}$ has a relation gap unless it admits a presentation with at most 3 defining relations, and we establish restrictions on the possible form of such a presentation. We then associate to each (m,n) a 3-complex with 16 cells. This 3-complex is a counterexample to the D(2) conjecture if $\Gamma_{m,n}$ has a relation gap.Comment: 7 pages; no figures. Minor changes; now to appear in Math. Proc. Camb. Phil. So

Beat Pilot Tone: Versatile, Contact-Free Motion Sensing in MRI with Radio Frequency Intermodulation

Motion in Magnetic Resonance Imaging (MRI) scans results in image corruption and remains a barrier to clinical imaging. Motion correction algorithms require accurate sensing, but existing sensors are limited in sensitivity, comfort, or general usability. We propose Beat Pilot Tone (BPT), a radio frequency (RF) motion sensing system that is sensitive, comfortable, versatile, and scalable. BPT operates by a novel mechanism: two or more transmitted RF tones form standing wave patterns that are modulated by motion and sensed by the same receiver coil arrays used for MR imaging. By serendipity, the tones are mixed through nonlinear intermodulation in the receiver chain and digitized simultaneously with the MRI data. We demonstrate BPT's mechanism in simulations and experiments. Furthermore, we show in healthy volunteers that BPT can sense head, bulk, respiratory, and cardiac motion, including small vibrations such as displacement ballistocardiograms. BPT can distinguish between different motion types, achieve greater sensitivity than other methods, and operate as a multiple-input multiple-output (MIMO) system. Thus, BPT can enable motion-robust MRI scans at high spatiotemporal resolution in many applications

Printed Receive Coils with High Acoustic Transparency for Magnetic Resonance Guided Focused Ultrasound.

In magnetic resonance guided focused ultrasound (MRgFUS) therapy sound waves are focused through the body to selectively ablate difficult to access lesions and tissues. A magnetic resonance imaging (MRI) scanner non-invasively tracks the temperature increase throughout the tissue to guide the therapy. In clinical MRI, tightly fitted hardware comprised of multichannel coil arrays are required to capture high quality images at high spatiotemporal resolution. Ablating tissue requires a clear path for acoustic energy to travel but current array materials scatter and attenuate acoustic energy. As a result coil arrays are placed outside of the transducer, clear of the beam path, compromising imaging speed, resolution, and temperature accuracy of the scan. Here we show that when coil arrays are fabricated by additive manufacturing (i.e., printing), they exhibit acoustic transparency as high as 89.5%. This allows the coils to be placed in the beam path increasing the image signal to noise ratio (SNR) five-fold in phantoms and volunteers. We also characterize printed coil materials properties over time when submerged in the water required for acoustic coupling. These arrays offer high SNR and acceleration capabilities, which can address current challenges in treating head and abdominal tumors allowing MRgFUS to give patients better outcomes