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

Diffusion Tensor Imaging (DTI) is a noise-sensitive method, where a low Signal-to-Noise Ratio (SNR) results in significant errors in the estimated tensor field. Post-reconstruction tensor field smoothing is a simple and effective solution for alleviating this problem. Diffusion tensors can be represented by Symmetric Positive-Definite (SPD) matrices which can be viewed as a Riemannian manifold after defining a suitable metric on the space of SPD matrices. The Log-Cholesky metric is a recently developed concept with several advantages over previously defined metrics, e.g., Frobenius, Log-Euclidean, and affine-invariant metrics. In this work, we implemented a smoothing method based on the Log-Cholesky metric and show its effectiveness as a simple solution to filtering noisy diffusion tensor fields.

The symmetric positive definite (SPD) matrices have been widely used in image and vision problems. Recently there are growing interests in studying sparse representation (SR) of SPD matrices, motivated by the great success of SR for vector data. Though the space of SPD matrices is well-known to form a Lie group that is a Riemannian manifold, existing work fails to take full advantage of its geometric structure. This paper attempts to tackle this problem by proposing a kernel based method for SR and dictionary learning (DL) of SPD matrices. We disclose that the space of SPD matrices, with the operations of logarithmic multiplication and scalar logarithmic multiplication defined in the Log-Euclidean framework, is a complete inner product space. We can thus develop a broad family of kernels that satisfies Mercer's condition. These kernels characterize the geodesic distance and can be computed efficiently. We also consider the geometric structure in the DL process by updating atom matrices in the Riemannian space instead of in the Euclidean space. The proposed method is evaluated with various vision problems and shows notable performance gains over state-of-the-arts.

Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle of balanced metrics that relate the affine-invariant metric with the Euclidean and inverse-Euclidean metric, or the Bogoliubov-Kubo-Mori metric with the Euclidean and log-Euclidean metrics. We introduce two new families of balanced metrics, the mixed-power-Euclidean and the mixed-power-affine metrics and we discuss the relation between this new principle of balanced metrics and the concept of dual connections in information geometry.

Diffusion tensor imaging (DTI) is the only noninvasive technique of analyzing and qualifying water molecule's diffusion anisotropy in brain tissues. This paper presented a novel algorithm to analyze DTI and brain white matter tractography based on Riemannian manifold. Firstly, a 3×3 symmetric positive definite covariant tensor was constructed for each voxel using DTI, so brain white matte can be represented as a tensor field. Secondly, the tensor field was regarded as Riemannian manifold, and the fluid motion in the tensor field was represented by Navier-Stoke equation, so the problem of brain white matter tractography between two voxels can be transformed into the computation of smallest distance between two points in Riemannian manifold. Finally, distances between two points in Riemannian manifold can be represented by geodesic, and the numerical solution was based on Level-Set method, which was the brain white matter tractography. In experiment, this paper compared our method and the traditional algorithm based on a digital DTI phantom. The experiment result showed that our method could accurately retrieve the DTI tractography, and was more robust than traditional algorithm.

Diffusion tensor (DT) imaging is an emerging magnetic resonance (MR) imaging technique for evaluating the microstructure of well-organized biologic tissues such as muscles and nerves. DT imaging provides information about tissue microstructure by producing three-dimensional maps of water molecule movements. The two main parameters of measurement at DT imaging, fractional anisotropy and the apparent diffusion coefficient, allow quantitation of architectural changes occurring in tissue. These parameters are modified in the presence of cervical spondylotic myelopathy, cervical spine trauma, carpal tunnel syndrome, lumbar nerve compression, peripheral nerve tumors, and muscle ischemia. Their alteration may be observed at DT imaging even when no abnormality is seen at conventional MR imaging, a fact that suggests that DT imaging allows the detection of abnormalities at an earlier stage of injury. Experimental studies in animals have shown that DT imaging consistently allows identification of pathophysiologic alterations in tissue that correlate with histologic findings. Tractographic images accurately depict both normal and abnormal diffusion in anatomic structures such as the thigh and pelvic muscles, cervical spine, and lumbar nerves. Patients with chronic diseases also may benefit from follow-up evaluation with DT imaging, although DT imaging sequences must be further adapted to improve the evaluation of specific anatomic regions by reducing artifacts, optimizing spatial resolution, and minimizing acquisition time. Given its proven potential for use in identifying abnormalities that are otherwise identifiable only with electrophysiologic and histopathologic studies, and with future technical improvements, DT imaging could soon become a standard method for early diagnosis, management, and follow-up of disease in the spine, muscles, and peripheral nerves.

The geometry of mixed-Euclidean metrics on symmetric positive definite matrices

Symmetric positive definite (SPD) matrices used as feature descriptors in image recognition are usually high dimensional. Traditional manifold learning is only applicable for reducing the dimension of high-dimensional vector-form data.For high-dimensional SPD matrices, directly using manifold learning algorithms to reduce the dimension of matrix-form data is impossible. The SPD matrix must first be transformed into a long vector, and then the dimension of this vector must be reduced. However, this approach breaks the spatial structure of the SPD matrix space. To overcome this limitation, we propose a new dimension reduction algorithm on SPD matrix space to transform high-dimensional SPD matrices into low-dimensional SPD matrices. Our work is based on the fact that the set of all SPD matrices with the same size has a Lie group structure, and we aim to transform the manifold learning to the SPD matrix Lie group. We use the basic idea of the manifold learning algorithm called locality preserving projection (LPP) to construct the corresponding Laplacian matrix on the SPD matrix Lie group. Thus, we call our approach Lie-LPP to emphasize its Lie group character. We present a detailed algorithm analysis and show through experiments that Lie-LPP achieves effective results on human action recognition and human face recognition.

We propose an unbiased group-wise diffeomorphic registration technique to normalize a group of diffusion tensor (DT) images. Our method uses an implicit reference group-wise registration framework to avoid bias caused by reference selection. Log-Euclidean metrics on diffusion tensors are used for the tensor interpolation and computation of the similarity cost functions. The overall energy function is constructed by a diffeomorphic demons approach. The tensor reorientation is performed and implicitly optimized during the registration procedure. The performance of the proposed method is compared with reference-based diffusion tensor imaging (DTI) registration methods. The registered DTI images have smaller shape differences in terms of reduced variance of the fractional anisotropy maps and more consistent tensor orientations. We demonstrate that fiber tract atlas construction can benefit from the group-wise registration by producing fiber bundles with higher overlaps.

To explore the utility of diffusion tensor imaging (DTI) and diffusion tensor metrics (DTM) in characterizing the structural pathology of epidermoid cysts. DTI gives information about the tissue structure; a high fractional anisotropy (FA) indicates a highly structured orientation of the tissue, fibers, or white matter tracts. Based on the tensor rank, a set of three metrics has been described that can be used to measure the directional dependence of diffusion: linear anisotropy (CL), planar anisotropy (CP), and spherical anisotropy (CS). DTM takes into account the shape of diffusion anisotropy and hence may provide better insight into the orientation of structures than FA. DTI was performed in three patients with epidermoid cysts. FA, directionally-averaged mean diffusivity (Dav), exponential apparent diffusion coefficient (eADC), and DTM, such as CL, CP, and CS, were measured from the tumor core as well as from the normal-appearing white matter. Histopathological correlation was obtained. Epidermoid cysts showed high FA with Dav values similar to that of normal white matter. eADC maps did not show any restriction of diffusion. FA values were high, but not as high as that for the white matter. CP values were higher and CL values were lower than those obtained for the white matter in various regions. High CP values suggest preferential diffusion of water molecules along a two-dimensional geometry, which could be attributed to the well-structured orientation of keratin filaments and flakes within the tumor as demonstrated by histopathology. Advanced imaging modalities like DTI with DTM can provide information regarding the microstructural anatomy of the epidermoid cysts.

This thesis aimed to investigate the feasibility of two diffusion imaging techniques, Diffusion Tensor Imaging (DTI) and Diffusion Kurtosis Imaging (DKI), on detecting subtle physiological or pathological microstructural changes in normal-appearing neural tissues of human central nervous system. 
\n At first, ten patients with neuromyelitis optica (NMO) and twelve age- and gender-matched healthy subjects were recruited. DTI-derived indices including fractional anisotropy (FA), mean diffusivity (MD), axial and radial diffusivities were quantified in the lateral and dorsal columns of cervical spinal cord. Based on the regions of interest (ROIs) measurement, NMO patients showed reduced FA, increased MD and radial diffusivity compared to control subjects, while axial diffusivity did not show any significant difference. The three former DTI metrics also showed significant correlations with disability scores, and especially FA was found to be sensitive to mild NMO. Our results show that DTI-derived indices can quantitatively assess the white matter (WM) abnormalities with seemingly normal appearance in conventional MRI, and are associated with the level of clinical disability, suggesting that DTI may have great potential as a useful diagnostic tool in the clinical setting.
\n DKI is an extension of conventional DTI to probe the non-Gaussian diffusion property in biological tissues. Besides the four conventional DTI-derived metrics, DKI also provide three additional kurtosis metrics (mean kurtosis (MK), axial and radial kurtosis). In the second study, ROI-based analysis was used to characterize age-related microstructural changes in WM, cortical and subcortical gray matter (GM) of 27 healthy adults (21~59 yrs). Though the volumes of GM and WM were still preserved, DTI-derived metrics can detect the subtle changes in WM and GM. Meanwhile, MK and radial kurtosis significantly increased in both caudate nucleus and putamen while Thalamus showed little aging effect in the diffusivity and kurtosis metrics but significantly decreased only in FA. Our results demonstrated that DKI is sensitive to detect the age-related alterations in neural microstructures at the stage of early aging.
\n In addition, DKI has been applied to detect the pathological changes in the normal-appearing neural tissues of 18 patients with multiple sclerosis (MS), compared to 22 healthy controls. Diffuse WM abnormalities have been observed extensively in the brain, revealed by DKI-derived metrics. Though the volumetric and voxel-wise analysis revealed no significant changes in the volume of cortical GM, decreased FA and kurtoses with increased diffusivities in MS group were sensitive to disclose the subtle alterations in global and regional cortical GM tissues. Significant correlations have been found between FA in the global, frontal and temporal cortical GM in relapsing-remitting MS patients and their disability scores, suggesting FA as an important biomarker to monitor the disease progress in cortical GM. Moreover, elevated kurtosis indices in MS patients did not correlate with diffusivities in caudate nucleus, putamen and thalamus, suggesting these metrics may be vulnerable to different pathologic aspects of the disease.
\n In conclusion, DKI is sensitive to neural alterations during normal aging and in MS pathologies, and can provide complementary information to conventional MRI and DTI.

Improved segmentation of white matter tracts with adaptive Riemannian metrics.

To correct eddy-current artifacts in diffusion tensor (DT) images without the need to obtain auxiliary scans for the sole purpose of correction. DT images are susceptible to distortions caused by eddy currents induced by large diffusion gradients. We propose a new postacquisition correction algorithm that does not require any auxiliary reference scans. It also avoids the problematic procedure of cross-correlating images with significantly different contrasts. A linear model is used to describe the dependence of distortion parameters (translation, scaling, and shear) on the diffusion gradients. The model is solved numerically to provide an individual correction for every diffusion-weighted (DW) image. The assumptions of the linear model were successfully verified in a series of experiments on a silicon oil phantom. The correction obtained for this phantom was compared with correction obtained by a previously published method. The algorithm was then shown to markedly reduce eddy-current distortions in DT images from human subjects. The proposed algorithm can accurately correct eddy-current artifacts in DT images. Its principal advantages are that only images with comparable signals and contrasts are cross-correlated, and no additional scans are required.

Introduction We present the first study in literature that documents the changes in diffusion tensor imaging (DTI) indices and tractography after surgical decompression in patients with cervical myelopathy. DTI, an established tool for tracking nerve fibers in brain, is now widely evaluated for its usefulness in spine pathologies. Although there are many studies directly comparing DTI indices of control and cervical myelopathic patients, there are no studies that have evaluated the prognostic applicability of DTI indices. The present study analyzes the pre- and postdecompression DTI indices and tractography in cervical myelopathic patients and its prognostic value. Materials and Methods Total 56 subjects including 16 cervical myelopathy patients who required surgical decompression (mean age: 53.5 ± 12.3 years, males: 15, females: (1) and forty normal controls (mean age: 38.18 ± 13.74 years, males: 20, females: 20) were included in the study. DTI of cervical spine was obtained using an axial single shot echo-planar parallel diffusion-weighted imaging sequence in a 1.5 tesla magnetic resonance imaging (MRI). Apparent diffusion coefficient (ADC), fractional anisotropy (FA), and Eigen values (E1, E2, and E3) were obtained at each region of interest (cervical disk level) from C1-C2 to C7-T1. The DTI indices of myelopathic patients were compared with normal volunteers after matching was done based on sex, cord level, and cord size. Postoperatively, the patients were followed up after 6 months and DTI scan was performed again. The clinical improvement and the DTI indices observed at the follow-up were compared with the preoperative values. Results Analysis showed significant differences between the controls and patients in all the indices (p<0.05) at the site of compression. At the compressed levels, FA decreased while all the other indices increased as compared to the controls. The mean DTI indices of myelopathic patients before surgery at the sites of spinal cord compression were as follows: FA (0.49 ± 0.092), ADC (1.8 ± 0.32 µm2/ms), E1 (2.80 ± 0.44 µm2/ms), E2 (1.63 ± 0.45 µm2/ms), E3 (0.95 ± 0.23 µm2/ms). The average DTI indices of matched controls were as follows: FA (0.55 ± 0.033), ADC (1.47 ± 0.066 µm2/ms), E1 (2.49 ± 0.150 µm2/ms), E2 (1.16 ± 0.050 µm2/ms), E3 (0.77 ± 0.058 µm2/ms. The Nurick grading distribution in patients before surgery was as follows: Grade 2 = 2, Gr 3 = 8, Gr 4 = 2, Gr 5 = 4. Postoperatively, all the patients had improvement of neurology by at least one Nurick grade except for one patient who had no improvement. The Nurick grading distribution in patients during 6 months follow-up after surgery was as follows: Grade 0 = 3, Gr 1 = 6, Gr 2 = 2, Gr 3 = 3, Gr 4 = 2. Mean postoperative values of ADC, E1, and E2 showed improvement towards normal values especially in patients with preoperative Nurick grade 2 and 3 (Table 1). Patients with Nurick grade 4 and 5 did not show any significant change. The mean FA and E3 values worsened when compared to preoperative DTI indices. The postoperative DTI indices for the patients who had neurological improvement are as follows: FA (0.45 ± 0.087), ADC (1.73 ± 0.31 µm2/ms), E1 (2.59 ± 0.28 µm2/ms), E2 (1.56 ± 0.38 µm2/ms), E3 (1.07 ± 0.31 µm2/ms. Tractography also showed interruption of nerve fibre tract continuity at the level of severe compression in patients with myelopathy. At levels of moderate compression over the spinal cord, the changes in direction of nerve fibres due to bulging disc-osteophytes were evident as changes in color coding of the tracts. After surgical decompression, the restitution of fibre tract pathways was clearly demonstrated in DT tractography. Conclusion For the first time, the prognostic applicability of DTI by comparing pre- and postoperative indices of cervical myelopathy patients has been described. All preoperative DTI indices at compressed levels showed significant differences from controls indicating abnormality in neuronal integrity. All but one of the patients showed clinical improvement at 6 months. The postoperative DTI values (ADC, E1 and E2) of patients with Nurick grading 2 and 3 were observed to approach normal values indicating restoration of linear diffusion in neuronal tracts. DTI indices in patients with Nurick grade 4 and 5 remained same or worsened more. This could probably indicate persistent axonal damage. At 6 months postsurgery, DTI indices show evidence of improvement towards normal values, even though the values did not match the true extent of clinical improvement. Further DTI evaluation at 12 months and 2 years is required to analyze the extent of usefulness of DTI indices in evaluating cervical myelopathy. The results suggest that DTI indices, especially ADC and Eigen vectors can be promising indicators in the evaluation and prognostic assessment of myelopathy patients. I confirm having declared any potential conflict of interest for all authors listed on this abstract Yes Disclosure of Interest None declared

Objective To analyze the characteristics of magnetic resonance diffusion tensor imaging(DTI)and diffusion tensor tractography (DTIT) in patients with cerebral infarction, and explore the diagnosis values and prognosis of diffusion weighted imaging (DWI) and diffusion tensor imaging (DTI) in patients with cerebral infarction in different stages. Methods 58 patients with cerebral infarction in different stages and 25 healthy volunteers were examined by magnetic resonance imaging (MRI), including conventional T1 and T2 weighted imaging, DWI and DTI. Fractional anisotropy (FA) images were reconstructed. The values of FA and apparent diffusion coefficient (ADC) were measured in the infarcted regions, corresponding contralateral normal regions and corresponding normal regions in normal control group. Results ①DWI and DTI showed size of infarction focus was more accurate and clearer than that of conventional MRI; ②The FA and ADC values of the infarcted regions during superacute stage, acute stage, subacute stage and chronic stage were (0.24±0.02, 0.31 ±0.11), (0.20±0.02, 0.32±0.12), (0.18±0.02, 0.34±0.11) and (0.16±0.02, 0.37±0.13), respectively, lower than those in the contralateral corresponding regions which were (0.40±0.03, 0.70±0.21), (0.37±0.03, 0.71±0.21), (0.39±0.03, 0.72±0.22) and (0.40:±0.03, 0.72±0.23), respectively. The differences were statistically significant (P＜0.05). The FA and ADC values had no significant differences between the uninjured sides in patients with cerebral infarction and the corresponding regions in the normal control group (P＞0.05); ③The FA and ADC values in brain tissues changed regularly with the time of infarction after cerebral infarction. The FA values in the affected sides had no consistent changes as compared with the contralateral sides in the superacute stage. They increased or decreased slightly, then (during acute stage, subacute stage and chronic stage) decreased irreversibly. The ADC values in the affected sides changed with time regularly, they decreased significantly, gradually returned to normal, and then increased again. Conclusion DTI and DTT examination contribute to the diagnosis of cerebral infarction. The combination of the FA and ADC values may more accurately conduct clinical staging and evaluate the time of the occurrence of cerebral infarction. Key words: Magnetic resonance; Cerebral infarction; Diffusion tensor imaging; Diffusion tensor tractography

Magnetic resonance diffusion tensor imaging (DTI) and functional MRI (fMRI) are two active research areas in neuroimaging. DTI is sensitive to the anisotropic diffusion of water exerted by its macromolecular environment and has been shown useful in characterizing structures of ordered tissues such as the brain white matter, myocardium, and cartilage. The diffusion tensor provides two new types of information of water diffusion: the magnitude and the spatial orientation of water diffusivity inside the tissue. This information has been used for white matter fiber tracking to review physical neuronal pathways inside the brain. Functional MRI measures brain activations using the hemodynamic response. The statistically derived activation map corresponds to human brain functional activities caused by neuronal activities. The combination of these two methods provides a new way to understand human brain from the anatomical neuronal fiber connectivity to functional activities between different brain regions. In this study, virtual reality (VR) based MR DTI and fMRI visualization with high resolution anatomical image segmentation and registration, ROI definition and neuronal white matter fiber tractography visualization and fMRI activation map integration is proposed. Rationale and methods for producing and distributing stereoscopic videos are also discussed.