We introduce a new chamfering paradigm, locally connecting pixels to produce path distances that approximate Euclidean space by building a small network (a replacement product) inside each pixel. These " RE -grid graphs" maintain near-Euclidean polygonal distance contours even in noisy data sets, making them useful tools for approximation when exact numerical solutions are unobtainable or impractical. The RE -grid graph creates a modular global architecture with lower pixel-to-pixel valency and simplified topology at the cost of increased computational complexity due to its internal structure. We present an introduction to chamfering replacement products with a number of case study examples to demonstrate the potential of these graphs for path-finding in high frequency and low resolution image spaces which motivate further study. Possible future applications include morphology, watershed segmentation, halftoning, neural network design, anisotropic image processing, image skeletonization, dendritic shaping, and cellular automata.

Magnetic resonance imaging (MRI) plays a critical role in medicine today. Over the last few years, there has been an interest in techniques that use multiple coils on one side of the imaging system such as parallel excitation or parallel imaging to improve performance in high-field MRI. In this paper, we explore the use of multiple coils at both the transmit and receive sides for enhanced imaging. We term this a multiple-input multiple-output (MIMO) MRI system. We use tools from the well-developed MIMO communication models and propose a novel radio-frequency (RF) pulse design method for multiple coils. We show that our MIMO MRI techniques can provide an improved signal-to-noise ratio (SNR) in the reconstructed image.

The theory and techniques of compressed sensing (CS) have shown their potential as a breakthrough in accelerating k-space data acquisition for parallel magnetic resonance imaging (pMRI). However, the performance of CS reconstruction models in pMRI has not been fully maximized, and CS recovery guarantees for pMRI are largely absent. To improve reconstruction accuracy from parsimonious amounts of k-space data while maintaining flexibility, a new CS SENSitivity Encoding (SENSE) pMRI reconstruction framework promoting joint sparsity (JS) across channels (JS CS SENSE) is proposed in this paper. The recovery guarantee derived for the proposed JS CS SENSE model is demonstrated to be better than that of the conventional CS SENSE model and similar to that of the coil-by-coil CS model. The flexibility of the new model is better than the coil-by-coil CS model and the same as that of CS SENSE. For fast image reconstruction and fair comparisons, all the introduced CS-based constrained optimization problems are solved with split Bregman, variable splitting, and combined-variable splitting techniques. For the JS CS SENSE model in particular, these techniques lead to an efficient algorithm. Numerical experiments show that the reconstruction accuracy is significantly improved by JS CS SENSE compared with the conventional CS SENSE. In addition, an accurate residual-JS regularized sensitivity estimation model is also proposed and extended to calibration-less (CaL) JS CS SENSE. Numerical results show that CaL JS CS SENSE outperforms other state-of-the-art CS-based calibration-less methods in particular for reconstructing non-piecewise constant images.