Affine-invariant Riemannian Metrics Research Articles

Time irreversibility analysis and abnormality detection based on Riemannian geometry for complex time series

Time irreversibility (TIR) is one of the basic properties of nonlinear systems, and it can be used as an effective way to explore the nonlinear complexity of dynamic systems. In this paper, we propose the TIR measure algorithm based on Riemannian geometry and apply this algorithm to abnormality detection tasks. The core idea is to calculate the positive definite covariance matrices of the forward sequence and its inverse sequence based on the theories of sliding window and phase space reconstruction, and compute the affine invariant Riemannian metric (AIRM) between them to characterize the TIR of the time series. The advantage of this algorithm is that the framework based on Riemannian geometry can more effectively use the spatial information of time series. In addition, compared with the traditional methods, the new method is suitable for nonstationary complex systems, which introduces a sliding window to make subsequences satisfy weak stationary conditions to measure local time irreversibility. The simulated and surrogate data are used to verify the effectiveness of the algorithm. Finally, we apply the algorithm to empirical analysis. The results show that, the algorithm detects the financial crisis period and stable period of the stock markets in the analysis of the financial system, and the loss degree of TIR of physiologic signals of different types of cardiac patients is detected in the study of physiologic signals. In the research of Parkinson’s gait, we verify that the walker is an effective tool to improve the gait of early Parkinson’s patients through statistical tests.

A Non-Iterative Method for the Difference of Means on the Lie Group of Symmetric Positive-Definite Matrices

A non-iterative method for the difference of means is presented to calculate the log-Euclidean distance between a symmetric positive-definite matrix and the mean matrix on the Lie group of symmetric positive-definite matrices. Although affine-invariant Riemannian metrics have a perfect theoretical framework and avoid the drawbacks of the Euclidean inner product, their complex formulas also lead to sophisticated and time-consuming algorithms. To make up for this limitation, log-Euclidean metrics with simpler formulas and faster calculations are employed in this manuscript. Our new approach is to transform a symmetric positive-definite matrix into a symmetric matrix via logarithmic maps, and then to transform the results back to the Lie group through exponential maps. Moreover, the present method does not need to compute the mean matrix and retains the usual Euclidean operations in the domain of matrix logarithms. In addition, for some randomly generated positive-definite matrices, the method is compared using experiments with that induced by the classical affine-invariant Riemannian metric. Finally, our proposed method is applied to denoise the point clouds with high density noise via the K-means clustering algorithm.

Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry.

Symmetric positive definite (SPD) data have become a hot topic in machine learning. Instead of a linear Euclidean space, SPD data generally lie on a nonlinear Riemannian manifold. To get over the problems caused by the high data dimensionality, dimensionality reduction (DR) is a key subject for SPD data, where bilinear transformation plays a vital role. Because linear operations are not supported in nonlinear spaces such as Riemannian manifolds, directly performing Euclidean DR methods on SPD matrices is inadequate and difficult in complex models and optimization. An SPD data DR method based on Riemannian manifold tangent spaces and global isometry (RMTSISOM-SPDDR) is proposed in this research. The main contributions are listed: (1) Any Riemannian manifold tangent space is a Hilbert space isomorphic to a Euclidean space. Particularly for SPD manifolds, tangent spaces consist of symmetric matrices, which can greatly preserve the form and attributes of original SPD data. For this reason, RMTSISOM-SPDDR transfers the bilinear transformation from manifolds to tangent spaces. (2) By log transformation, original SPD data are mapped to the tangent space at the identity matrix under the affine invariant Riemannian metric (AIRM). In this way, the geodesic distance between original data and the identity matrix is equal to the Euclidean distance between corresponding tangent vector and the origin. (3) The bilinear transformation is further determined by the isometric criterion guaranteeing the geodesic distance on high-dimensional SPD manifold as close as possible to the Euclidean distance in the tangent space of low-dimensional SPD manifold. Then, we use it for the DR of original SPD data. Experiments on five commonly used datasets show that RMTSISOM-SPDDR is superior to five advanced SPD data DR algorithms.

Sparse coding-based representation of LBP difference for 3D/4D facial expression recognition

This paper presents an effective method for automated 3D/4D facial expression recognition based on Mesh-Local Binary Pattern Difference (mesh-LBPD). In contrast to most of existing methods, the proposed mesh-LBPD is based on a unified set of geometric and appearance features of different facial regions. Indeed, multiple features are combined into a compact form using covariance matrices, namely Cov − 3D − LBP. Then, the Cov − 3D − LBP atoms are represented as sparse data combinations. To that end, a Riemannian optimization objective for dictionary learning and sparse coding is used, in order to reduce the complexity of the problem, and the representation loss is characterized via an affine invariant Riemannian metric. In order to prove the effectiveness of the proposed compact combination of geometric and appearance features, we conducted extensive experimental validations on real-world datasets. In fact, obtained results show the capability of the proposed method to significantly outperform, or achieve comparable performances with, the state-of-the-art methods.

Sparse Representation Over Learned Dictionaries on the Riemannian Manifold for Automated Grading of Nuclear Pleomorphism in Breast Cancer.

Breast cancer is found to be the most pervasive type of cancer among women. Computer aided detection and diagnosis of cancer at the initial stages can increase the chances of recovery and thus reduce the mortality rate through timely prognosis and adequate treatment planning. The nuclear atypia scoring or histopathological breast tumor grading remains to be a challenging problem due to the various artifacts and variabilities introduced during slide preparation and also because of the complexity in the structure of the underlying tissue patterns. Inspired by the success of symmetric positive definite (SPD) matrices in many of the challenging tasks in machine learning and computer vision, a sparse coding and dictionary learning on SPD matrices is proposed in this paper for the breast tumor grading. The proposed covariance-based SPD matrices form a Riemannian manifold and are represented as the sparse combination of Riemannian dictionary atoms. Non-linearity of the SPD manifold is tackled by embedding into the reproducing kernel Hilbert space using kernels derived from log-Euclidean metric, Jeffrey and Stein divergences and compared with the non-kernel-based affine invariant Riemannian metric. The novelty of the work lies in exploiting the kernel approach for the Hilbert space embedding of the Riemannian manifold, that can achieve a better discrimination of the breast cancer tissues, following a sparse representation over learned dictionaries and henceforth it outperforms many of the state-of-the-art algorithms in breast cancer grading in terms of quantitative and qualitative analysis.

Riemannian Regression and Classification Models of Brain Networks Applied to Autism.

Functional connectivity from resting-state functional MRI (rsfMRI) is typically represented as a symmetric positive definite (SPD) matrix. Analysis methods that exploit the Riemannian geometry of SPD matrices appropriately adhere to the positive definite constraint, unlike Euclidean methods. Recently proposed approaches for rsfMRI analysis have achieved high accuracy on public datasets, but are computationally intensive and difficult to interpret. In this paper, we show that we can get comparable results using connectivity matrices under the log-Euclidean and affine-invariant Riemannian metrics with relatively simple and interpretable models. On ABIDE Preprocessed dataset, our methods classify autism versus control subjects with 71.1% accuracy. We also show that Riemannian methods beat baseline in regressing connectome features to subject autism severity scores.

Which is the most appropriate strategy for conducting multivariate voxel-based group studies on diffusion tensors?

There is a real need in the neuroscience community for efficient tools to compare Diffusion Tensor Magnetic Resonance Imaging across cohorts of subjects. Most studies focus on the comparison of scalar images such as fractional anisotropy or mean diffusivity. Although different statistical frameworks have been proposed to compare the whole diffusion tensor information, they are still seldom used in neuroimaging studies. In this paper, we investigate on both simulated and real data whether there is a real added value of considering the whole tensor information for conducting voxel-based group studies. Then, we compare two statistical tests dedicated to tensor, namely the Cramér test and a tensor-based extension of the General Linear Model (GLM), the latter presenting the advantage to account for covariates. We also evaluate the impact of different metrics (Euclidean, Log-Euclidean and affine-invariant Riemannian metrics) for estimating the GLM parameters. Finally, we address the problem of interpreting the change detection maps obtained by tensor-based methods by proposing a way to characterize each of the detected clusters according to several scalar indices. Our study suggests that if there is no prior assumption about the nature of the expected changes, it is really preferable to use tensor-based rather than scalar-based statistical analysis. The Cramér test can advantageously be used when no confounding variable hampers the group comparison, otherwise the GLM should be considered. Finally, the different metrics show similar performance in the real scenario, with a significant computational overhead for the Riemannian framework.

Combined Tensor Fitting and TV Regularization in Diffusion Tensor Imaging Based on a Riemannian Manifold Approach

In this paper, we consider combined TV denoising and diffusion tensor fitting in DTI using the affine-invariant Riemannian metric on the space of diffusion tensors. Instead of first fitting the diffusion tensors, and then denoising them, we define a suitable TV type energy functional which incorporates the measured DWIs (using an inverse problem setup) and which measures the nearness of neighboring tensors in the manifold. To approach this functional, we propose generalized forward- backward splitting algorithms which combine an explicit and several implicit steps performed on a decomposition of the functional. We validate the performance of the derived algorithms on synthetic and real DTI data. In particular, we work on real 3D data. To our knowledge, the present paper describes the first approach to TV regularization in a combined manifold and inverse problem setup.

Log-Determinant Divergences Revisited: Alpha-Beta and Gamma Log-Det Divergences

This work reviews and extends a family of log-determinant (log-det) divergences for symmetric positive definite (SPD) matrices and discusses their fundamental properties. We show how to use parameterized Alpha-Beta (AB) and Gamma log-det divergences to generate many well-known divergences; in particular, we consider the Stein’s loss, the S-divergence, also called Jensen-Bregman LogDet (JBLD) divergence, Logdet Zero (Bhattacharyya) divergence, Affine Invariant Riemannian Metric (AIRM), and other divergences. Moreover, we establish links and correspondences between log-det divergences and visualise them on an alpha-beta plane for various sets of parameters. We use this unifying framework to interpret and extend existing similarity measures for semidefinite covariance matrices in finite-dimensional Reproducing Kernel Hilbert Spaces (RKHS). This paper also shows how the Alpha-Beta family of log-det divergences relates to the divergences of multivariate and multilinear normal distributions. Closed form formulas are derived for Gamma divergences of two multivariate Gaussian densities; the special cases of the Kullback-Leibler, Bhattacharyya, Rényi, and Cauchy-Schwartz divergences are discussed. Symmetrized versions of log-det divergences are also considered and briefly reviewed. Finally, a class of divergences is extended to multiway divergences for separable covariance (or precision) matrices.

Nonlocal denoising using anisotropic structure tensor for 3D MRI

Noise in magnetic resonance imaging (MRI) data is widely recognized to be harmful to image processing and subsequent quantitative analysis. To ameliorate the effects of image noise, the authors present a structure-tensor based nonlocal mean (NLM) denoising technique that can effectively reduce noise in MRI data and improve tissue characterization. The proposed 3D NLM algorithm uses a structure tensor to characterize information around tissue boundaries. The similarity weight of a pixel (or patch), which determines its contribution to the denoising process, is determined by the intensity and structure tensor simultaneously. Meanwhile, similarity of structure tensors is computed using an affine-invariant Riemannian metrics, which compares tensor properties more comprehensively and avoids orientation inaccuracy of structure subsequently. The proposed method is further extended for denoising high dimensional MRI data such as diffusion weighted MRI. It is also extended to handle Rician noise corruption so that denoising effects are further enhanced. The proposed method was implemented in both simulated datasets and multiply modalities of real 3D MRI datasets. Comparisons with related state-of-the-art algorithms demonstrated that this method improves denoising performance qualitatively and quantitatively. In this paper, high order structure information of 3D MRI was characterized by 3D structure tensor and compared for NLM denoising in a Riemannian space. Experiments with simulated and real human MRI data demonstrate a great potential of the proposed technique for routine clinical use.

Log‐Euclidean metrics for fast and simple calculus on diffusion tensors

Diffusion tensor imaging (DT-MRI or DTI) is an emerging imaging modality whose importance has been growing considerably. However, the processing of this type of data (i.e., symmetric positive-definite matrices), called "tensors" here, has proved difficult in recent years. Usual Euclidean operations on matrices suffer from many defects on tensors, which have led to the use of many ad hoc methods. Recently, affine-invariant Riemannian metrics have been proposed as a rigorous and general framework in which these defects are corrected. These metrics have excellent theoretical properties and provide powerful processing tools, but also lead in practice to complex and slow algorithms. To remedy this limitation, a new family of Riemannian metrics called Log-Euclidean is proposed in this article. They also have excellent theoretical properties and yield similar results in practice, but with much simpler and faster computations. This new approach is based on a novel vector space structure for tensors. In this framework, Riemannian computations can be converted into Euclidean ones once tensors have been transformed into their matrix logarithms. Theoretical aspects are presented and the Euclidean, affine-invariant, and Log-Euclidean frameworks are compared experimentally. The comparison is carried out on interpolation and regularization tasks on synthetic and clinical 3D DTI data.

A Riemannian Framework for Tensor Computing

Tensors are nowadays a common source of geometric information. In this paper, we propose to endow the tensor space with an affine-invariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries (null eigenvalues are at the infinity), the geodesic between two tensors and the mean of a set of tensors are uniquely defined, etc. We have previously shown that the Riemannian metric provides a powerful framework for generalizing statistics to manifolds. In this paper, we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation, filtering, diffusion and restoration of missing data. For instance, most interpolation and Gaussian filtering schemes can be tackled efficiently through a weighted mean computation. Linear and anisotropic diffusion schemes can be adapted to our Riemannian framework, through partial differential evolution equations, provided that the metric of the tensor space is taken into account. For that purpose, we provide intrinsic numerical schemes to compute the gradient and Laplace-Beltrami operators. Finally, to enforce the fidelity to the data (either sparsely distributed tensors or complete tensors fields) we propose least-squares criteria based on our invariant Riemannian distance which are particularly simple and efficient to solve.