Log-Euclidean Metrics Research Articles

Log-Cholesky filtering of diffusion tensor fields: Impact on noise reduction

Diffusion tensor imaging (DTI) is a powerful neuroimaging technique that provides valuable insights into the microstructure and connectivity of the brain. By measuring the diffusion of water molecules along neuronal fibers, DTI allows the visualization and study of intricate networks of neural pathways.DTI is a noise-sensitive method, where a low signal-to-noise ratio (SNR) results in significant errors in the estimated tensor field. Tensor field regularization is an effective solution for noise reduction.Diffusion tensors are represented by symmetric positive-definite (SPD) matrices. The space of SPD matrices may be viewed as a Riemannian manifold after defining a suitable metric on its tangent bundle. The Log-Cholesky metric is a recently developed concept with advantages over previously defined Riemannian metrics, such as the affine-invariant and Log-Euclidean metrics. The utility of the Log-Cholesky metric for tensor field regularization and noise reduction has not been investigated in detail.This manuscript provides a quantitative investigation of the impact of Log-Cholesky filtering on noise reduction in DTI. It also provides sufficient details of the linear algebra and abstract differential geometry concepts necessary to implement this technique as a simple and effective solution to filtering diffusion tensor fields.

Adaptive Log-Euclidean Metrics for SPD Matrix Learning.

Symmetric Positive Definite (SPD) matrices have received wide attention in machine learning due to their intrinsic capacity to encode underlying structural correlation in data. Many successful Riemannian metrics have been proposed to reflect the non-Euclidean geometry of SPD manifolds. However, most existing metric tensors are fixed, which might lead to sub-optimal performance for SPD matrix learning, especially for deep SPD neural networks. To remedy this limitation, we leverage the commonly encountered pullback techniques and propose Adaptive Log-Euclidean Metrics (ALEMs), which extend the widely used Log-Euclidean Metric (LEM). Compared with the previous Riemannian metrics, our metrics contain learnable parameters, which can better adapt to the complex dynamics of Riemannian neural networks with minor extra computations. We also present a complete theoretical analysis to support our ALEMs, including algebraic and Riemannian properties. The experimental and theoretical results demonstrate the merit of the proposed metrics in improving the performance of SPD neural networks. The efficacy of our metrics is further showcased on a set of recently developed Riemannian building blocks, including Riemannian batch normalization, Riemannian Residual blocks, and Riemannian classifiers.

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.

Smooth Gradation of Anisotropic Meshes Using Log–Euclidean Metrics

The anisotropic mesh-size function represented by metric tensors includes two features: mesh sizes and mesh orientation. Such metrics are widely used in scientific computing for adaptation, but the solution metrics are not smooth, which adversely affects adaptation. A novel algorithm is proposed in this paper to smooth the metric as a whole and to improve anisotropic mesh-size gradation. First, the concept of Log–Euclidean metrics is used to convert the metric tensors from the Riemannian space into a Euclidean space. Then, the variations of metric tensors are limited by constraining the gradients of metric tensors in this space over the region of each background mesh element. Finally, a convex nonlinear optimization problem is formulated to smooth the metric tensors over the entire meshing domain. Theoretical analysis reveals the existence of a globally optimal smoothed sizing function. Numerical experiments on anisotropic meshes in both analytical and real laminar and turbulent flow cases are presented to demonstrate the effectiveness of the proposed method.

Batch Mode Active Learning on the Riemannian Manifold for Automated Scoring of Nuclear Pleomorphism in Breast Cancer.

Breast cancer is the most prevalent invasive type of cancer among women. The mortality rate of the disease can be reduced considerably through timely prognosis and felicitous treatment planning, by utilizing the computer aided detection and diagnosis techniques. With the advent of whole slide image (WSI) scanners for digitizing the histopathological tissue samples, there is a drastic increase in the availability of digital histopathological images. However, these samples are often unlabeled and hence they need labeling to be done through manual annotations by domain experts and experienced pathologists. But this annotation process required for acquiring high quality large labeled training set for nuclear atypia scoring is a tedious, expensive and time consuming job. Active learning techniques have achieved widespread acceptance in reducing this human effort in annotating the data samples. In this paper, we explore the possibilities of active learning on nuclear pleomorphism scoring over a non-Euclidean framework, the Riemannian manifold. Active learning technique adopted for the cancer grading is in the batch-mode framework, that adaptively identifies the apt batch size along with the batch of instances to be queried, following a submodular optimization framework. Samples for annotation are selected considering the diversity and redundancy between the pair of samples, based on the kernelized Riemannian distance measures such as log-Euclidean metrics and the two Bregman divergences - Stein and Jeffrey divergences. Results of the adaptive Batch Mode Active Learning on the Riemannian metric show a superior performance when compared with the state-of-the-art techniques for breast cancer nuclear pleomorphism scoring, as it makes use of the information from the unlabeled samples.

Log-Euclidean Metrics for Contrast Preserving Decolorization.

This paper presents a novel Log-Euclidean metric inspired color-to-gray conversion model for faithfully preserving the contrast details of color image, which differs from the traditional Euclidean metric approaches. In the proposed model, motivated by the fact that Log-Euclidean metric has promising invariance properties such as inversion invariant and similarity invariant, we present a Log-Euclidean metric-based maximum function to model the decolorization procedure. The Gaussian-like penalty function consisting of the Log-Euclidean metric between gradients of the input color image and transformed grayscale image is incorporated to better reflect the degree of preserving feature discriminability and color ordering in color-to-gray conversion. A discrete searching algorithm is employed to solve the proposed model with linear parametric and non-negative constraints. Extensive evaluation experiments show that the proposed method outperforms the state-of-the-art methods both quantitatively and qualitatively.

On the averaging of cardiac diffusion tensor MRI data: the effect of distance function selection

Diffusion tensor magnetic resonance imaging (DT-MRI) allows a unique insight into the microstructure of highly-directional tissues. The selection of the most proper distance function for the space of diffusion tensors is crucial in enhancing the clinical application of this imaging modality. Both linear and nonlinear metrics have been proposed in the literature over the years. The debate on the most appropriate DT-MRI distance function is still ongoing. In this paper, we presented a framework to compare the Euclidean, affine-invariant Riemannian and log-Euclidean metrics using actual high-resolution DT-MRI rat heart data. We employed temporal averaging at the diffusion tensor level of three consecutive and identically-acquired DT-MRI datasets from each of five rat hearts as a means to rectify the background noise-induced loss of myocyte directional regularity. This procedure is applied here for the first time in the context of tensor distance function selection. When compared with previous studies that used a different concrete application to juxtapose the various DT-MRI distance functions, this work is unique in that it combined the following: (i) metrics were judged by quantitative—rather than qualitative—criteria, (ii) the comparison tools were non-biased, (iii) a longitudinal comparison operation was used on a same-voxel basis. The statistical analyses of the comparison showed that the three DT-MRI distance functions tend to provide equivalent results. Hence, we came to the conclusion that the tensor manifold for cardiac DT-MRI studies is a curved space of almost zero curvature. The signal to noise ratio dependence of the operations was investigated through simulations. Finally, the ‘swelling effect’ occurrence following Euclidean averaging was found to be too unimportant to be worth consideration.

Cluster Analysis of Diffusion Tensor Fields with Application to the Segmentation of the Corpus Callosum

Accurate segmentation of the Corpus Callosum (CC) is an important aspect of clinical medicine and is used in the diagnosis of various brain disorders. In this paper, we propose an automated method for two and three dimensional segmentation of the CC using diffusion tensor imaging. It has been demonstrated that Hartigan's K-means is more efficient than the traditional Lloyd algorithm for clustering. We adapt Hartigan's K-means to be applicable for use with the metrics that have a f -mean (e.g. Cholesky, root Euclidean and log Euclidean). Then we applied the adapted Hartigan's K-means, using Euclidean, Cholesky, root Euclidean and log Euclidean metrics along with Procrustes and Riemannian metrics (which need numerical solutions for mean computation), to diffusion tensor images of the brain to provide a segmentation of the CC. The log Euclidean and Riemannian metrics provide more accurate segmentations of the CC than the other metrics as they present the least variation of the shape and size of the tensors in the CC for 2D segmentation. They also yield a full shape of the splenium for the 3D segmentation.

A reduced-order matrices fitting scheme with Log-Euclidean metrics for fast approximation of dynamic response of parametric structural systems

Structural engineering practice often involves tasks, such as design, optimization or statistical analysis, where iterative solutions of dynamical systems are required varying some parameter which affect the system matrices in a generally non-linear way. The approach of interpolating among matrices of reduced-order models (ROMs) is not new in the literature and it is very promising in order to speed up the calculations. In particular, the method shows a great potential since it is applicable to any kind of FE model and of geometry, with no restrictions regarding the element type constituting the full-order model or the range of parameter variability. Yet, dealing with symmetric positive definite (SPD) full-order matrices, the SPD nature of the reduced matrices must be preserved. To this end, a special mathematical framework must be used, which allows to linearize the curved space of SPD matrices. In the literature of ROMs interpolation, the common choice is to rely on the principles of Riemannian geometry through the concept of the tangent plane to the SPD manifold. Although mathematically robust, this approach brings a marked distortion in the space of SPD matrices. Also, it shows numerical instabilities when dealing with strongly anisotropic matrices. In this work, a polynomial fitting of the reduced matrices is proposed with Log-Euclidean metrics. These metrics were obtained by giving the SPD manifold the structure of a Lie group. While maintaining the already proposed approach of interpolating between a number of sampling ROMs, the advantages of using Log-Euclidean metrics are illustrated in details. In particular, a great improvement is shown in case of a unique reduction matrix whose columns span the solution of the parameter space of interest in an accurate enough way, and in case of substructuring reduction techniques. The reported examples of applications show excellent accuracy, while maintaining the dramatic decrease in the computational time of the interpolatory scheme.

Fully automatic stitching of diffusion tensor images in spinal cord

Diffusion tensor imaging (DTI) has become an important tool for studying the spinal cord pathologies. To enable high resolution imaging for modern studies, the DTI technique utilizes a small field of view (FOV) to capture partial human spinal cords. However, normal aging and many other diseases which affect the entire spinal cord increase the desire of acquiring the continuous full-view of the spinal cord. To overcome this problem, this paper presents a novel pipeline for automatic stitching of three-dimensional (3D) DTI of different portions of the spinal cord. The proposed technique consists of two operations, e.g. feature-based registration and adaptive composition to stitch every source image together to create a panoramic image. In the feature-based registration process, feature points are detected from the apparent diffusion coefficient map, and then a novel feature descriptor is designed to characterize feature points directly from a tensor neighborhood. 3D affine transforms are achieved by determining the correspondence matching. In the adaptive composition process, an effective feathering approach is presented to compute the tensors in the overlap region by the Log-Euclidean metrics. We evaluate the algorithm on real datasets from one healthy subject and one adolescent idiopathic scoliosis (AIS) patient. The colored FA maps and fiber tracking results show the effectiveness and accuracy of the proposed stitching framework.

Mueller matrix interpolation in polarization optics

The question of the physical significance of the Mueller matrix average is addressed by means of an analysis of interpolation processes. We draw a comparison between two interpolation processes. The first one is related to the classical Euclidean metrics and the second one is based on the log-Euclidean metrics. Both the associated interpolation procedures are depicted with their underlying physical models. Addressing the question of the physical meaning of the log-Euclidean process of interpolation is founded on a very similar approach to the layered-medium interpretation proposed by Jones [J. Opt. Soc. Am.38, 671 (1948)] in the seventh paper of his series. Based on the analysis of their respective properties, we eventually show that the choice between both these interpolation processes may depend on what statistical situation is considered or what underlying physical model is assumed.

Clinical DT-MRI Estimation, Smoothing, and Fiber Tracking With Log-Euclidean Metrics

Diffusion tensor magnetic resonance imaging (DT-MRI or DTI) is an imaging modality that is gaining importance in clinical applications. However, in a clinical environment, data have to be acquired rapidly, often at the expense of the image quality. This often results in DTI datasets that are not suitable for complex postprocessing like fiber tracking. We propose a new variational framework to improve the estimation of DT-MRI in this clinical context. Most of the existing estimation methods rely on a log-Gaussian noise (Gaussian noise on the image logarithms), or a Gaussian noise, that do not reflect the Rician nature of the noise in MR images with a low signal-to-noise ratio (SNR). With these methods, the Rician noise induces a shrinking effect: the tensor volume is underestimated when other noise models are used for the estimation. In this paper, we propose a maximum likelihood strategy that fully exploits the assumption of a Rician noise. To further reduce the influence of the noise, we optimally exploit the spatial correlation by coupling the estimation with an anisotropic prior previously proposed on the spatial regularity of the tensor field itself, which results in a maximum a posteriori estimation. Optimizing such a nonlinear criterion requires adapted tools for tensor computing. We show that Riemannian metrics for tensors, and more specifically the log-Euclidean metrics, are a good candidate and that this criterion can be efficiently optimized. Experiments on synthetic data show that our method correctly handles the shrinking effect even with very low SNR, and that the positive definiteness of tensors is always ensured. Results on real clinical data demonstrate the truthfulness of the proposed approach and show promising improvements of fiber tracking in the brain and the spinal cord.