Box spline based 3D tomographic reconstruction of diffusion propagators from MRI data

Abstract3D box splines are defined by convolving a 1D box function with itself along different directions. In volume visualization, box splines are mainly used as reconstruction kernels that are easy to adapt to various sampling lattices, such as the Cartesian Cubic (CC), Body‐Centered Cubic (BCC), and Face‐Centered Cubic (FCC) lattices. The usual way of tailoring a box spline to a specific lattice is to span the box spline by exactly those principal directions that span the lattice itself. However, in this case, the preferred directions of the box spline and the lattice are the same, amplifying the anisotropic effects of each other. This leads to an anisotropic volume representation with strongly preferred directions. Therefore, in this paper, we retailor box splines to lattices such that the sets of vectors that span the box spline and the lattice are disjoint sets. As the preferred directions of the box spline and the lattice compensate each other, a more isotropic volume representation can be achieved. We demonstrate this by comparing different combinations of box splines and lattices concerning their anisotropic behavior in tomographic reconstruction and volume visualization.

Splines over regular triangulations in numerical simulation

On convolutions of B-splines

This paper presents a versatile hyper-ellipsoidal basis function for function approximation in a given high dimensional space. This hyper-ellipsoidal basis function can be translated and rotated to cover the data based upon the distribution of data in a given high dimensional space. Based on this function, we propose a one-pass hyper-ellipsoidal learning algorithm for which any new incoming data can be fed for learning without involving the previously learned one. This learning algorithm is used to adjust the parameters of the versatile hyper-ellipsoidal basis function. In addition, we propose the hyper-ellipsoidal basis function (HEBF) neural network that uses the one-pass hyper-ellipsoidal neural learning algorithm. The structure of this neural network is similar to the radial basis function (RBF) neural networks. The hidden neurons in the HEBF neural network can be increased or decreased during learning process. The number of the hidden neurons in the network can be grown based on geometric growth criterion and can be reduced by merging the two hidden neurons into a new hidden neuron based on merging criterion during learning process. The merging process can be done independently without considering the learned data set.KeywordsRadial Basis FunctionHide NeuronRadial Basis Function Neural NetworkHigh Dimensional SpaceInitial WidthThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present a method to stabilize bases with local supports by means of extension. It generalizes the known approach for tensor product B-splines to a much broader class of functions, which includes hierarchical and weighted variants of polynomial, trigonometric, and exponential splines, but also box splines, T-splines, and other function spaces of interest with a local basis. Extension removes elements that cause instabilities from a given basis by linking them with the remaining ones by means of a specific linear combination. The two guiding principles for this process are locality and persistence. Locality aims at coupling basis functions whose supports are close together, while persistence guarantees that a given set of globally supported functions, like certain monomials in the case of polynomial splines, remain in the span of the basis after extension. Furthermore, we study how extension influences the approximation power and the condition of Gramian matrices associated with the basis, and present a series of examples illustrating the potential of the method.

Wavelets and pre-wavelets in low dimensions

Über die kreiseleigenschaften des atomkerns

Rain gauges, weather radars, satellite sensors and modelled data from weather centres are used operationally for estimating the spatial-temporal variability of rainfall. However, the associated uncertainties can be very high, especially in poorly equipped regions of the world. Very recently, an innovative method, named SM2RAIN, that uses soil moisture observations to infer rainfall, has been proposed by Brocca et al. (2013) with very promising results when applied with in situ and satellite-derived data. However, a thorough analysis of the physical consistency of the SM2RAIN algorithm has not been carried out yet. In this study, synthetic soil moisture data generated from a physically-based soil water balance model are employed to check the reliability of the assumptions made in the SM2RAIN algorithm. Next, high quality and multiyear in situ soil moisture observations, at different depths (5-30 cm), and rainfall for ten sites across Europe are used for testing the performance of the algorithm, its limitations and applicability range. SM2RAIN shows very high accuracy in the synthetic experiments with a correlation coefficient, R, between synthetically generated and simulated data, at daily time step, higher than 0.940 and an average Bias lower than 4%. When real datasets are used, the agreement between observed and simulated daily rainfall is slightly lower with average R-values equal to 0.87 and 0.85 in the calibration and validation periods, respectively. Overall, the performance is found to be better in humid temperate climates and for sensors installed vertically. Interestingly, algorithms of different complexity in the reproduction of the underlying hydrological processes provide similar results. The average contribution of surface runoff and evapotranspiration components amounts to less than 4% of the total rainfall, while the soil moisture variations (63%) and subsurface drainage (30%) terms provide a much higher contribution. Overall, the SM2RAIN algorithm is found to perform well both in the synthetic and real data experiments, thus offering a new and independent source of data for improving rainfall estimation, and consequently enhancing hydrological, meteorological and climatic studies.

For many medical procedures 3D bone models are built from computed tomography (CT) or magnetic resonance imaging (MRI) data, both of which are expensive and time consuming, and unavailable in most operating rooms. We propose 3D tomographic reconstruction from fluoroscopy as an alternative to CT and MRI. Although there are fluoroscopy machines that can perform 3D reconstruction, they must be instrumented and constrained to gather images along a pre-defined path. We present a methodology for 3D reconstruction from imagery collected along an arbitrary path on standard fluoroscopy machines. Using metal beads on the scanned object as fiducial markers, we can recover the path of the X-ray from the image data and apply cone-beam tomographic reconstruction methods. In this paper we describe our cone-beam tomographic reconstruction algorithm and demonstrate reconstruction of a phantom and a cadaver knee from data collected on a standard fluoroscopy unit.

Satellite soil moisture estimates have received increasing attention over the past decade. This paper examines the applicability of estimating soil moisture states and soil hydraulic parameters through two particle filter (PF) methods: The PF with commonly used sampling importance resampling (PF-SIR) and the PF with recently developed Markov chain Monte Carlo sampling (PF-MCMC) methods. In a synthetic experiment, the potential of assimilating remotely sensed near-surface soil moisture measurements into a 1-D mechanistic soil water model (HYDRUS-1D) using both the PF-SIR and PF-MCMC algorithms is analyzed. The effects of satellite temporal resolution and accuracy, soil type, and ensemble size on the assimilation of soil moisture are analyzed. In a real data experiment, we first validate the Advanced Microwave Scanning Radiometer--Earth Observing System (AMSR-E) soil moisture products in the Oklahoma Little Washita Watershed. Aside from rescaling the remotely sensed soil moisture, a bias correction algorithm is implemented to correct the deep soil moisture estimate. Both the ascending and descending AMSR-E soil moisture data are assimilated into the HYDRUS-1D model. The synthetic assimilation results indicated that, whereas both updating schemes showed the ability to correct the soil moisture state and estimate hydraulic parameters, the PF-MCMC scheme is consistently more accurate than PR-SIR. For real data case, the quality of remotely sensed soil moisture impacts the benefits of their assimilation into the model. The PF-MCMC scheme brought marginal gains than the open-loop simulation in RMSE at both surface and root-zone soil layer, whereas the PF-SIR scheme degraded the open-loop simulation.

In hydrological modeling, the identification of model mechanisms best suited for representing individual hydrological (physical) processes is of major scientific and operational interest. We present a statistical hypothesis‐testing perspective on this model identification challenge and contribute a mechanism identification framework that combines: (i) Bayesian estimation of posterior probabilities of individual mechanisms from a given ensemble of model structures; (ii) a test statistic that defines a “dominant” mechanism as a mechanism more probable than all its alternatives given observed data; and (iii) a flexible modeling framework to generate model structures using combinations of available mechanisms. The uncertainty in the test statistic is approximated using bootstrap sampling from the model ensemble. Synthetic experiments (with varying error magnitude and multiple replicates) and real data experiments are conducted using the hydrological modeling system FUSE (7 processes and 2–4 mechanisms per process yielding 624 feasible model structures) and data from the Leizarán catchment in northern Spain. The mechanism identification method is reliable: it identifies the correct mechanism as dominant in all synthetic trials where an identification is made. As data/model errors increase, statistical power (identifiability) decreases, manifesting as trials where no mechanism is identified as dominant. The real data case study results are broadly consistent with the synthetic analysis, with dominant mechanisms identified for 4 of 7 processes. Insights on which processes are most/least identifiable are also reported. The mechanism identification method is expected to contribute to broader community efforts on improving model identification and process representation in hydrology.

Form invariant stress-strain relationships are presented for anisotropic media. The case of transversely isotropic media is then considered and the five elastic constants needed to describe such a medium are obtained in terms of the elements of the stiffness tensor. The dispersion equation is solved and approximate basis functions are constructed to expand the displacement fields inside such an anisotropic inclusion. The scattering of elastic waves by spherical or nonspherical inclusions is studied as a function of frequency and geometry and compared with the isotropic case.

In this paper, a robust L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> filtering method is proposed for a class of switched linear parameter varying (LPV) systems in which the state-space matrices and the time delay rely on real-time measured parameter. By using multiple delay-dependent Lyapunov-Krasovskii function, the delay-dependent L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> performance criterion for the switched LPV systems is first established. As there exists coupling between Lyapunov function and system parameter matrices, we utilized a slack matrix variable to decouple it. Because of the dependence on parameter, approximate basis function and gridding technique are utilized, and the solution for the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> filters can be transformed into finite dimensional convex optimization problem. Based on the switching logic with the minimum average dwell time (ADT) and parameter linear matrix inequalities (PLMIs) technique, the resulting filters guarantee the filtering error systems to be exponentially stable with a prespecified L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> disturbance attenuation level. A numerical example is presented to verify the effectiveness of the proposed method.

Gaussian processes are powerful non-parametric probabilistic models for stochastic functions. However, the direct implementation entails a complexity that is computationally intractable when the number of observations is large, especially when estimated with fully Bayesian methods such as Markov chain Monte Carlo. In this paper, we focus on a low-rank approximate Bayesian Gaussian processes, based on a basis function approximation via Laplace eigenfunctions for stationary covariance functions. The main contribution of this paper is a detailed analysis of the performance, and practical recommendations for how to select the number of basis functions and the boundary factor. Intuitive visualizations and recommendations, make it easier for users to improve approximation accuracy and computational performance. We also propose diagnostics for checking that the number of basis functions and the boundary factor are adequate given the data. The approach is simple and exhibits an attractive computational complexity due to its linear structure, and it is easy to implement in probabilistic programming frameworks. Several illustrative examples of the performance and applicability of the method in the probabilistic programming language Stan are presented together with the underlying Stan model code.

The separable nonnegative matrix factorization (NMF) model has been adopted in many applications (e.g., topic modeling and community detection), as it ensures identifiability of the latent factors with polynomial time algorithms under reasonable assumptions. Separable NMF is often formulated as a self-dictionary (SD) sparse regression problem, and algorithms under this formulation can be categorized into two classes, namely, greedy methods and all-at-once convex optimization. The latter has been shown to be more resilient to noise, but has serious scalability issues. Particularly, existing convex optimization algorithms for the SD sparse regression problem cost a memory that grows quadratically with respect to the data size-making it hardly useful in practice. This work puts forth a Frank-Wolfe algorithm to tackle this challenge. The proposed method has a memory cost that grows linearly in the data size-with provable guarantees. To our best knowledge, this is the first linear memory algorithm for this long-existing problem. Synthetic and real data experiments are used to showcase the memory efficiency of the proposed algorithm.