Considering the context of functional data analysis, we developed and applied a new Bayesian approach via the Gibbs sampler to select basis functions for a finite representation of functional data. The proposed methodology uses Bernoulli latent variables to assign zero to some of the basis function coefficients with a positive probability. This procedure allows for an adaptive basis selection since it can determine the number of bases and which ones should be selected to represent functional data. Moreover, the proposed procedure measures the uncertainty of the selection process and can be applied to multiple curves simultaneously. The methodology developed can deal with observed curves that may differ due to experimental error and random individual differences between subjects, which one can observe in a real dataset application involving daily numbers of COVID-19 cases in Brazil. Simulation studies show the main properties of the proposed method, such as its accuracy in estimating the coefficients and the strength of the procedure to find the true set of basis functions. Despite having been developed in the context of functional data analysis, we also compared the proposed model via simulation with the well-established LASSO and Bayesian LASSO, which are methods developed for non-functional data.

Raman spectroscopy is widely used in discriminative tasks. It provides a wide-range physio-chemical fingerprint in a rapid and non-invasive way. The Raman spectrometry uses a sensor array to convert photon signals into digital spectroscopic data. This analog-to-digital process can benefit from the compressed sensing (CS) technique. The major benefits include less memory usage, shorter acquisition time, and more cost-efficient sensor. Traditional compressed sensing and reconstruction is a series of mathematical operations performed on the signal. Meanwhile, for discriminative tasks, both the signal and the categorical information are involved. For such scenarios, this paper proposes a method that uses both domain signal and categorical information to optimize CS hyper-parameters, including 1) the sampling ratio or the sensing matrix, 2) the basis matrix for the sparse transform, and 3) the regularization rate or shrinkage factor for L1-norm minimization. A case study of formula milk brand identification proves the proposed method can generate effective compressed sensing while preserving enough discriminative power in the reconstructed signal. Under the optimized hyper-parameters, a 100% classification accuracy is retained by only sampling 20% of the original signal.

OBJECTIVE: A new subspace-based iterative reconstruction method, termed Self-supporting Tailored k-space Estimation for Parallel imaging reconstruction (STEP), is presented and evaluated in comparison to the existing autocalibrating method SPIRiT and calibrationless method SAKE.
METHODS: In STEP, two tailored schemes including k-space partition and basis selection are proposed to promote spatially variant signal subspace and incorporated into a self-supporting structured low rank model to enforce properties of locality, sparsity, and rank deficiency, which can be formulated into a constrained optimization problem and solved by an iterative algorithm. Simulated and in vivo datasets were used to investigate the performance of STEP in terms of overall image quality and detail structure preservation.
RESULTS: The advantage of STEP on image quality is demonstrated by retrospectively undersampled multichannel Cartesian data with various patterns. Compared with SPIRiT and SAKE, STEP can provide more accurate reconstruction images with less residual aliasing artifacts and reduced noise amplification in simulation and in vivo experiments. In addition, STEP has the capability of combining compressed sensing with arbitrary sampling trajectory.
CONCLUSIONS: Using k-space partition and basis selection can further improve the performance of parallel imaging reconstruction with or without calibration signals.