Tensor Estimation Research Articles

Tensor Estimation With Structured Priors

We consider rank-one symmetric tensor estimation when the tensor is corrupted by gaussian noise and the spike forming the tensor is a structured signal coming from a generalized linear model. The latter is a mathematically tractable model of a non-trivial hidden lower-dimensional latent structure in a signal. We work in a large dimensional regime with fixed ratio of signal-to-latent space dimensions. Remarkably, in this asymptotic regime, the mutual information between the spike and the observations can be expressed as a finite-dimensional variational problem, and it is possible to deduce the minimum-mean-square-error from its solution. We discuss, on examples, properties of the phase transitions as a function of the signal-to-noise ratio. Typically, the critical signal-to-noise ratio decreases with increasing signal-to-latent space dimensions. We discuss the limit of vanishing ratio of signal-to-latent space dimensions and determine the limiting tensor estimation problem. We also point out similarities and differences with the case of matrices.

Quality estimation of magnetotelluric impedance tensors using neural networks

While much of the workflow for geophysical data processing and analysis is automated, final quality assurance typically requires the decisions of skilled human analysts and interpreters. The quality and reliability of geophysical inverse models depends directly on the effectiveness of input data. The influence of spurious data on derived data products used in the inversion has been reduced by this effectiveness. Failure to identify and mitigate bias in data products can lead to costly errors. By applying supervised machine learning (ML), a neural network can be trained to recognize features in data that a human domain expert would identify as characteristic of poor data quality. In this study, we use magnetotelluric (MT) data as an example of a geophysical data set appropriate for such a training exercise. While MT data are used to estimate the resistivity structure of the subsurface, the concepts we discuss are universal to seismic, potential fields, and other geophysical data sets. We train a neural network, pyMAGIQ (Python-based magnetotelluric impedance qualifier), with multiple hidden layers and demonstrate that it successfully generates the nonlinear mapping function required to assess the quality of MT data. The training set is a large database of frequency-domain MT impedance tensors from the National Science Foundation-funded EarthScope MT project. A human-assigned quality index is associated with each impedance. We apply pyMAGIQ to unrated MT data from the United States and Canada and confirm that the ML-assigned quality factors are consistent with those assigned by trained human operators. We also apply sensitivity analysis to the trained neural network. This reveals that the human- and ML-assigned data quality index depends on the magnitude of the confidence limits on (1) the phases and (2) the continuity of the apparent resistivities and phases with respect to frequency.

Investigation of intravoxel incoherent motion tensor imaging for the characterization of the in vivo human heart

To investigate intravoxel incoherent motion (IVIM) tensor imaging of the in vivo human heart and elucidate whether the estimation of IVIM tensors is affected by the complexity of pseudo-diffusion components in myocardium. The cardiac IVIM data of 10 healthy subjects were acquired using a diffusion weighted spin-echo echo-planar imaging sequence along 6 gradient directions with 10 b values (0~400 s/mm2 ). The IVIM data of left ventricle myocardium were fitted to the IVIM tensor model. The complexity of myocardial pseudo-diffusion components was reduced through exclusion of low b values (0 and 5 s/mm2 ) from the IVIM curve-fitting analysis. The fractional anisotropy, mean fraction/mean diffusivity, and Westin measurements of pseudo-diffusion tensors (fp and D*) and self-diffusion tensor (D), as well as the angle between the main eigenvector of fp (or D*) and that of D, were computed and compared before and after excluding low b values. The fractional anisotropy values of fp and D* without low b value participation were significantly higher (P < .001) than those with low b value participation, but an opposite trend was found for the mean fraction/diffusivity values. Besides, after removing low b values, the angle between the main eigenvector of fp (or D*) and that of D became small, and both fp and D* tensors presented significant decrease of spherical components and significant increase of linear components. The presence of multiple pseudo-diffusion components in myocardium indeed influences the estimation of IVIM tensors. The IVIM tensor model needs to be further improved to account for the complexity of myocardial microcirculatory network and blood flow.

Complexity Estimation of Cubical Tensor Represented through 3D Frequency-Ordered Hierarchical KLT

In this work is introduced one new hierarchical decomposition for cubical tensor of size 2n, based on the well-known orthogonal transforms Principal Component Analysis and Karhunen–Loeve Transform. The decomposition is called 3D Frequency-Ordered Hierarchical KLT (3D-FOHKLT). It is separable, and its calculation is based on the one-dimensional Frequency-Ordered Hierarchical KLT (1D-FOHKLT) applied on a sequence of matrices. The transform matrix is the product of n sparse matrices, symmetrical at the point of their main diagonal. In particular, for the case in which the angles which define the transform coefficients for the couples of matrices in each hierarchical level of 1D-FOHKLT are equal to π/4, the transform coincides with this of the frequency-ordered 1D Walsh–Hadamard. Compared to the hierarchical decompositions of Tucker (H-Tucker) and the Tensor-Train (TT), the offered approach does not ensure full decorrelation between its components, but is close to the maximum. On the other hand, the evaluation of the computational complexity (CC) of the new decomposition proves that it is lower than that of the above-mentioned similar approaches. In correspondence with the comparison results for H-Tucker and TT, the CC decreases fast together with the increase of the hierarchical levels’ number, n. An additional advantage of 3D-FOHKLT is that it is based on the use of operations of low complexity, while the similar famous decompositions need large numbers of iterations to achieve the coveted accuracy.

Three-Dimensional Ballistic-Diffusive Heat Transport in Silicon: Transient Response and Thermal Conductivity

Abstract Phonons are the main contributors to thermal energy transfer in thin films. The size dependence of the thermal transport characteristics alters the film properties such as thermal conductivity. Hence, in the present study, three-dimensional, transient phonon transport in dielectric material is studied through the Equation of Phonon Radiative Transport (EPRT) to assess the size dependence of thermal conductivity. The numerical scheme is introduced solving the EPRT in three dimensions and the governing algorithm is described in detail. A parametric study is carried out examining the effect of the Kn \mathrm{Kn} number on the thermal energy transport characteristics in three-dimensional thermally excited film. The formulation and estimation of the effective thermal conductivity tensor is presented and discussed, thereby extending, to some extent, the one-dimensional results obtained earlier. We demonstrate that thermal conductivity changes in all directions, depending on the size effect. In addition, the directions of the temperature gradient and heat flux vectors differ as the Kn \mathrm{Kn} number approaches unity.

Integrated fusion framework based on semicoupled sparse tensor factorization for spatio-temporal–spectral fusion of remote sensing images

Remote sensing image fusion is considered a cost effective method for handling the tradeoff between the spatial, temporal and spectral resolutions of current satellite systems. However, most current fusion methods concentrate on fusing images in two domains among the spatial, temporal and spectral domains, and a few efforts have been made to comprehensively explore the relationships of spatio-temporal–spectral features. In this study, we propose a novel integrated spatio-temporal–spectral fusion framework based on semicoupled sparse tensor factorization to generate synthesized frequent high-spectral and high-spatial resolution images by blending multisource observations. Specifically, the proposed method regards the desired high spatio-temporal–spectral resolution images as a four-dimensional tensor and formulates the integrated fusion problem as the estimation of the core tensor and the dictionary along each mode. The high-spectral correlation across the spectral domain and the high self-similarity (redundancy) features in the spatial and temporal domains are jointly exploited using the low dimensional and sparse core tensors. In addition, assuming that the sparse coefficients in the core tensors across the observed and desired image spaces are not strictly the same, we formulate the estimation of the core tensor and the dictionaries as a semicoupled sparse tensor factorization of available heterogeneous spatial, spectral and temporal remote sensing observations. Finally, the proposed method can exploit the multicomplementary spatial, temporal and spectral information of any combination of remote sensing data based on this single unified model. Experiments on multiple data types, including spatio-spectral, spatio-temporal, and spatio-temporal–spectral data fusion, demonstrate the effectiveness and efficiency of the proposed method.

Probabilistic Moment Tensor Inversion for Hydrocarbon-Induced Seismicity in the Groningen Gas Field, The Netherlands, Part 1: Testing

ABSTRACTSince 1991, induced earthquakes have been observed and linked to gas production in the Groningen field. Recorded waveforms are complex, resulting partly from a Zechstein salt layer overlying the reservoir and partly from free-surface reverberations, internal multiples, interface conversions, guided waves, and waves diving below the reservoir. Therefore, picking of polarities or amplitudes for use in moment tensor inversion is problematic, whereas phase identification may be circumvented employing full waveform techniques. Although moment tensors have become a basic tool to analyze earthquake sources, their uncertainties are rarely reported. We introduce a method for probabilistic moment tensor estimation and demonstrate its use on the basis of a single event within the Groningen field, concentrating on detailed tests of input data and inversion parameters to derive rules of good practice for moment tensor estimation of events recorded in the Groningen field. In addition to the moment tensor, event locations are provided. Hypocenters estimated simultaneously with moment tensors are often less sensitive to uncertainties in crustal structure, which is pertinent for the application to the Groningen field, because the task of relating earthquakes to specific faults hitherto suffers from a limited resolution of earthquake locations. Because of the probabilistic approach, parameter trade-offs, uncertainties, and ambiguities are mapped. In addition, the implemented bootstrap method implicitly accounts for modeling errors affecting every station and phase differently. A local 1D velocity model extracted from a full 3D velocity model yields more consistent results than other models applied previously. For all velocity models and combinations of input data tested, a shift in location of 1 km to the south is observed for the test event compared to the public catalog. A full moment tensor computed employing the local 1D velocity model features negative isotropic components and may be interpreted as normal fault and collapse at reservoir level.

τ-SIMPLE Algorithm for the closure problem in homogenization of stokes flows

Estimation of the permeability tensor is crucial for many natural and industrial porous media applications. Homogenization provides a rigorous framework to calculate effective parameters from pore-scale images of a representative unit cell of the porous medium by solving a boundary value problem, also known as closure problem, subject to global constraints. However, the latter are hard to satisfy for arbitrarily complex geometries. We have developed a novel computational algorithm to calculate rigorously the permeability tensor. The approach, here referred to as τ-SIMPLE, is based on introducing an artificial time scale τ to satisfy the global constraint within the SIMPLE iteration. We show that the proposed algorithm has high accuracy for both two-dimensional and three-dimensional periodically arranged geometries.

Key-Frame Detection and Super-Resolution of Hyperspectral Video via Sparse-Based Cumulative Tensor Factorization

Thanks to the rapid development of hyperspectral sensors, hyperspectral videos (HSV) can now be collected with high temporal and spectral resolutions and utilized to handle invisible dynamic monitoring missions, such as chemical gas plume tracking. However, using such sequential large-scale data effectively is challenged, because the direct process of these data requires huge demands in terms of computational loads and memory. This paper presents a key-frame and target-detecting algorithm based on cumulative tensor CANDECOMP/PARAFAC (CP) factorization (CTCF) to select the frames where the target shows up, and a novel super-resolution (SR) method using sparse-based tensor Tucker factorization (STTF) is used to improve the spatial resolution. In the CTCF method, the HSV sequence is seen as cumulative tensors and the correlation of adjacent frames is exploited by applying CP tensor approximation. In the proposed STTF-based SR method, we consider the HSV frame as a third-order tensor; then, HSV frame super-resolution problem is transformed into estimations of the dictionaries along three dimensions and estimation of the core tensor. In order to promote sparse core tensors, a regularizer is incorporated to model the high spatial-spectral correlations. The estimations of the core tensor and the dictionaries along three dimensions are formulated as sparse-based Tucker factorizations of each HSV frame. Experimental results on real HSV data set demonstrate the superiority of the proposed CTCF and STTF algorithms over the comparative state-of-the-art target detection and SR approaches.

Use of multi-flip angle measurements to account for transmit inhomogeneity and non-Gaussian diffusion in DW-SSFP

Diffusion-weighted steady-state free precession (DW-SSFP) is an SNR-efficient diffusion imaging method. The improved SNR and resolution available at ultra-high field has motivated its use at 7T. However, these data tend to have severe B1 inhomogeneity, leading not only to spatially varying SNR, but also to spatially varying diffusivity estimates, confounding comparisons both between and within datasets. This study proposes the acquisition of DW-SSFP data at two-flip angles in combination with explicit modelling of non-Gaussian diffusion to address B1 inhomogeneity at 7T. Data were acquired from five fixed whole human post-mortem brains with a pair of flip angles that jointly optimize the diffusion contrast-to-noise (CNR) across the brain. We compared one- and two-flip angle DW-SSFP data using a tensor model that incorporates the full DW-SSFP Buxton signal, in addition to tractography performed over the cingulum bundle and pre-frontal cortex using a ball & sticks model. The two-flip angle DW-SSFP data produced angular uncertainty and tractography estimates close to the CNR optimal regions in the single-flip angle datasets. The two-flip angle tensor estimates were subsequently fitted using a modified DW-SSFP signal model that incorporates a gamma distribution of diffusivities. This allowed us to generate tensor maps at a single effective b-value yielding more consistent SNR across tissue, in addition to eliminating the B1 dependence on diffusion coefficients and orientation maps. Our proposed approach will allow the use of DW-SSFP at 7T to derive diffusivity estimates that have greater interpretability, both within a single dataset and between experiments.

Inverting magnetotelluric data with distortion correction—stability, uniqueness and trade-off with model structure

SUMMARY Galvanic distortion of magnetotelluric (MT) data is a common effect that can impede the reliable imaging of subsurface structures. Recently, we presented an inversion approach that includes a mathematical description of the effect of galvanic distortion as inversion parameters and demonstrated its efficiency with real data. We now systematically investigate the stability of this inversion approach with respect to different inversion strategies, starting models and model parametrizations. We utilize a data set of 310 MT sites that were acquired for geothermal exploration. In addition to impedance tensor estimates over a broad frequency range, the data set also comprises transient electromagnetic measurements to determine near surface conductivity and estimates of distortion at each site. We therefore can compare our inversion approach to these distortion estimates and the resulting inversion models. Our experiments show that inversion with distortion correction produces stable results for various inversion strategies and for different starting models. Compared to inversions without distortion correction, we can reproduce the observed data better and reduce subsurface artefacts. In contrast, shifting the impedance curves at high frequencies to match the transient electromagnetic measurements reduces the misfit of the starting model, but does not have a strong impact on the final results. Thus our results suggest that including a description of distortion in the inversion is more efficient and should become a standard approach for MT inversion.

Hallucinating schizophrenia patients have longer left arcuate fasciculus fiber tracks: a DTI tractography study

The arcuate fasciculus (AF) has been implicated in the pathology behind schizophrenia and auditory verbal hallucinations (AVHs). White matter tracts forming the arcuate fasciculus can be quantified and visualized using diffusion tensor imaging (DTI) tractography. Although there have been a number of studies on this topic, the results have been conflicting. Studying the underlying white matter structure of the AF could shed light on the constrains for interaction between temporal and frontal language areas in AVHs. The participants were 66 patients with a schizophrenia diagnosis, where AVHs were defined from the Positive and Negative Syndrome Scale (PANSS), and compared with a healthy control group. DTI was performed on a 3T MR scanner, and tensor estimation was done using deterministic streamline tractography. Statistical analysis of the data showed significantly longer reconstructed tracks along the AF in patients with severe and frequent AVHs, as well as an overall significant asymmetry with longer tracks in the left compared to the right side. In addition, there were significant positive correlations between PANSS scores and track length, track volume, and number of track streamlines for the posterior AF segment on the left side. It is concluded that the present DTI results may have implications for interpretations of functional imaging results.

Unconventional flash technique for the identification of multilayer thermal diffusivity tensors

This paper presents a novel non-intrusive technique for the thermal characterization of opaque multilayer materials that consists in the simultaneous estimation of the thermal diffusivities tensors of each constituting layers that may be isotropic or orthotropic. The present identification method is based on the resolution of an inverse heat conduction problem consisting in the minimization of the least square objective function constructed with the temperature measurements and the prediction of a pseudo-analytical model. The pseudo-analytical model relies on the quadrupoles formalism that proves its capability to treat multi-layered materials and to mimic the experimental method investigated in this work. This latter, inspired from the flash method, relies on a unique and non-intrusive experiment, in terms of excitation and measurements. One of the sample surfaces is locally and shortly heated by means of a CO2 laser, while the resulting transient temperature field is continuously recorded during the cooling period by an IR camera. The minimization procedure invokes a PSO algorithm that has been proved to be convenient for such complex problems that deal with a non-linearity and a large number of unknown parameters, including those related to the excitation. The overall identification method is validated using an isotropic opaque monolayer material of known properties. The identification method is then performed on an actual two layers sample. The accuracy and robustness is discussed depending on the face of excitation and measurement leading to two distinct experimental configurations. A strong emphasis is placed on the sensitivity analysis in order to test the feasibility of the estimation for both configurations.

New estimation of triaxial three-layered Earth's inertia tensor and solutions of Earth rotation normal modes

Until now, the calculation of the principal inertia moment of the triaxial three-layered Earth mainly adopts the scaling method. This method assumes that the corresponding principal inertia axes of the layers coincide each other, but this is not the case. In this paper, a rigorous tensor transformation rule is adopted to calculate the principal inertia moments (PIMs) of different layers. Appling the new estimated PIMs to the triaxial three-layered Earth rotation theory with considering various couplings, the numerical calculations show that the periods of the Chandler Wobble (CW), Free Core Nutation (FCN), Free Inner Core Nutation (FICN) and Inner Core Wobble (ICW) are respectively 433.0, 430.8, 943.9 and 2735.9 mean solar days, which are well comparable with the corresponding values accepted at present in geoscience community. Better estimates of the PIMs of different layers may provide better constrains on relevant physical parameters of the Earth's interior.

Sparse and Low-rank Tensor Estimation via Cubic Sketchings.

In this paper, we propose a general framework for sparse and low-rank tensor estimation from cubic sketchings. A two-stage non-convex implementation is developed based on sparse tensor decomposition and thresholded gradient descent, which ensures exact recovery in the noiseless case and stable recovery in the noisy case with high probability. The non-asymptotic analysis sheds light on an interplay between optimization error and statistical error. The proposed procedure is shown to be rate-optimal under certain conditions. As a technical by-product, novel high-order concentration inequalities are derived for studying high-moment sub-Gaussian tensors. An interesting tensor formulation illustrates the potential application to high-order interaction pursuit in high-dimensional linear regression.

An online and generalized non-negativity constrained model for large-scale sparse tensor estimation on multi-GPU

Non-negative Tensor Factorization (NTF) models are effective and efficient in extracting useful knowledge from various types of probabilistic distribution with multi-way information. Current NTF models are mostly designed for problems in computer vision which involve the whole Matricized Tensor TimesKhatri−RaoProduct (MTTKRP). Meanwhile, a Sparse NTF (SNTF) proposed to solve the problem of sparse Tensor Factorization (TF) can result in large-scale intermediate data. A Single-thread-based SNTF (SSNTF) model is proposed to solve the problem of non-linear computing and memory overhead caused by large-scale intermediate data. However, the SSNTF is not a generalized model. Furthermore, the above methods cannot describe the stream-like data from industrial applications in mainstream processors, e.g, Graphics Processing Unit (GPU) and multi-GPU in an online way. To address these two issues, a Generalized SSNTF (GSSNTF) is proposed, which extends the works of SSNTF to the Euclidean distance, KullbackLeibler (KL)-divergence, and ItakuraSaito (IS)-divergence. The GSSNTF only involves the feature elements instead of the entire factor matrices during its update process, which can avoid the formation of large-scale intermediate matrices with convergence and accuracy promises. Furthermore, GSSNTF can merge the new data into the state-of-the-art built tree dataset for sparse tensor, and then online learning has the promise of the correct data format. At last, a model of Compute Unified Device Architecture (CUDA) parallelizing GSSNTF (CUGSSNTF) is proposed on GPU and Multi-GPU (MCUGSSNTF). Thus, CUGSSNTF has linear computing complexity and space requirement, and linear communication overhead on multi-GPU. CUGSSNTF and MCUGSSNTF are implemented on 8 P100 GPUs in this work, and the experimental results from real-world industrial data sets indicate the linear scalability and 40X speedup performances of CUGSSNTF than the state-of-the-art parallelized approachs.

Comparative Study of Planar Coil EMI Sensors for Inversion-Based Detection of Buried Objects

In a context of buried objects detection using electromagnetic induction (EMI), dipole inversion refers to the estimation of object's location and magnetic polarizability tensor from EMI and sensor's positional data. In case of planar coil sensors, dipole inversion may become a surprisingly difficult and potentially ill-posed problem due to non-uniform distribution of directional sensitivities, nonlinear nature of target localization, as well as strong correlations between tensor elements and target's depth. In this paper, we evaluate inversion performances of two categories of planar coil sensors; single-receiver sensors used in conventional metal detection (MD), and multi-receiver sensors aimed at metal characterization (MC). We use three different inversion methods; nonlinear least squares (NLS), HAP method featuring novel auxiliary source model for improved object localization, and differential evolution (DE). Comparative study is performed using synthetic EMI and sensor's positional data under realistic scenarios involving different targets, depths and signal-to-noise ratios (SNRs). Our results suggest that relatively simple planar MC sensors clearly outperform conventional MD sensors, especially at greater depths. On average, DE method notably improves the invertibility of MD sensors, while a denser scan pattern may help to tackle lower SNR at greater depths.

Spatio-Temporal Patterns of the Development of Strong Seismic Activations (1999–2007) in the Northern Baikal Area

Abstract —The development of strong seismic activations in the northern Baikal area in 1999–2007 is considered. Based on maps of earthquake epicenter density, it has been shown that each activation is a separate group of seismic shocks (a cluster), whose scale and spatio-temporal pattern depend strongly on the stress-strain state of the crust. Estimates of seismic-moment tensors for strong earthquakes within the clusters demonstrate that the most numerous groups of shocks form in the rift stress field. With moving away to the southeast from the conventional axis of the Baikal rift, this field changes under amplification of the compressional stresses in Transbaikalia. Simultaneously, we observe a decrease in the number of seismic events and in their energy level. Totally, the considered seismic activations prove the small-scale block structure of the crust in the northern Baikal area and reflect the main features of the modern geotectonic development of this area related to the adjacent morphostructural zone, where most of fragments are involved in rifting and the other are affected by the activation of positive-sign block movements. The obtained results should be taken into account in the assessment of the seismic hazard of the studied territory.

Tensor Regression Using Low-Rank and Sparse Tucker Decompositions

This paper studies a tensor-structured linear regression model with a scalar response variable and tensor-structured predictors, such that the regression parameters form a tensor of order $d$ (i.e., a $d$-fold multiway array) in $\mathbb{R}^{n_1 \times n_2 \times \cdots \times n_d}$. It focuses on the task of estimating the regression tensor from $m$ realizations of the response variable and the predictors where $m\ll n = \prod \nolimits_{i} n_i$. Despite the seeming ill-posedness of this problem, it can still be solved if the parameter tensor belongs to the space of sparse, low Tucker-rank tensors. Accordingly, the estimation procedure is posed as a non-convex optimization program over the space of sparse, low Tucker-rank tensors, and a tensor variant of projected gradient descent is proposed to solve the resulting non-convex problem. In addition, mathematical guarantees are provided that establish the proposed method linearly converges to an appropriate solution under a certain set of conditions. Further, an upper bound on sample complexity of tensor parameter estimation for the model under consideration is characterized for the special case when the individual (scalar) predictors independently draw values from a sub-Gaussian distribution. The sample complexity bound is shown to have a polylogarithmic dependence on $\bar{n} = \max \big\{n_i: i\in \{1,2,\ldots,d \} \big\}$ and, orderwise, it matches the bound one can obtain from a heuristic parameter counting argument. Finally, numerical experiments demonstrate the efficacy of the proposed tensor model and estimation method on a synthetic dataset and a collection of neuroimaging datasets pertaining to attention deficit hyperactivity disorder. Specifically, the proposed method exhibits better sample complexities on both synthetic and real datasets, demonstrating the usefulness of the model and the method in settings where $n \gg m$.

Research progress of diffusion sensitive gradient field encoding schemes in magnetic resonance diffusion tensor imaging

Magnetic resonance imaging is a medical imaging technique in which the phenomenon of nuclear magnetic resonance is used. This technique is widely used in clinical and scientific research at present. The diffusion of water molecules is isotropic in a homogeneous medium, while it is anisotropic in the structure of human tissue. Magnetic resonance diffusion tensor imaging (DTI) is for studying the microscopic structure inside body by using the water molecules’ diffusion effect which will reduce the signal intensity of magnetic resonance. Besides, it can quantitatively study the anisotropy of water molecules in three-dimensional space, and thus acquiring important pathological and physiological information without invading in vivo. In order to obtain the accurate result of the anisotropic diffusion of water molecules, according to a certain scheme, it is necessary to sequentially use diffusion sensitive gradient (DSG) magnetic fields in different diffusion orientations to measure the diffusion coefficient of water molecules for estimating the diffusion tensor. The precision of estimating diffusion tensor is affected by the applied DSG encoding scheme, and it is usually necessary to use a large number of linearly independent and evenly spatial distributions of DSG magnetic fields in order to make the tensor measurement result more accurate. Diffusion spectroscopy imaging technique and high angular resolution diffusion imaging (HARDI) technique are proposed for more complex fiber bundles crossing in human tissue, one of which, HARDI, has higher requirement for the number and the direction distribution uniformity of DSGs. In this paper, the basic principle of DTI and the DSG encoding schemes are reviewed, which includes completely random scheme, heuristic scheme, regular polyhedral scheme, numerically optimized scheme, etc. For the above various schemes their respective advantages and limitations are analyzed. At present, the Golden Ratio method is to be used in a new spherical DSG encoding scheme which meets the requirements for HARDI and can offer more accurate tensor estimation results in face of the corruption of data sets encountered in clinical practice.