scholarly journals Diffusion tensor regularization with metric double integrals

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Leon Frischauf ◽  
Melanie Melching ◽  
Otmar Scherzer
Keyword(s):  
Regularization Method ◽  
Diffusion Tensor ◽  
Real Data ◽  
Magnetic Resonance Images ◽  
Uniqueness Result ◽  
Stability And Convergence ◽  
Infinite Dimensional ◽  
Euclidean Setting ◽  
Variational Regularization ◽  
Double Integrals

Abstract In this paper, we propose a variational regularization method for denoising and inpainting of diffusion tensor magnetic resonance images. We consider these images as manifold-valued Sobolev functions, i.e. in an infinite dimensional setting, which are defined appropriately. The regularization functionals are defined as double integrals, which are equivalent to Sobolev semi-norms in the Euclidean setting. We extend the analysis of [14] concerning stability and convergence of the variational regularization methods by a uniqueness result, apply them to diffusion tensor processing, and validate our model in numerical examples with synthetic and real data.

On the equation ut = ∆uα + uβ

1993 ◽  
Vol 123 (2) ◽  
pp. 373-390 ◽  
Author(s):  
Yuan-Wei Qi
Keyword(s):  
Dynamical System ◽  
Blow Up ◽  
Uniqueness Result ◽  
Fast Diffusion ◽  
Negative Solution ◽  
Fast Diffusion Equation ◽  
Infinite Dimensional ◽  
The Cauchy Problem ◽  
Dimensional Dynamical System ◽  
Infinite Dimensional Dynamical System

SynopsisThe Cauchy problem of ut, = ∆uα + uβ, where 0 < α < l and α>1, is studied. It is proved that if 1< β<α + 2/n then every nontrivial non-negative solution is not global in time. But if β>α+ 2/n there exist both blow-up solutions and global positive solutions which decay to zero as t–1/(β–1) when t →∞. Thus the famous Fujita result on ut = ∆u + up is generalised to the present fast diffusion equation. Furthermore, regarding the equation as an infinite dimensional dynamical system on Sobolev space W1,s (W2.s) with S > 1, a non-uniqueness result is established which shows that there exists a positive solution u(x, t) with u(., t) → 0 in W1.s (W2.s) as t → 0.

Diffusion tensor imaging of cerebral white matter in patients with myotonic dystrophy

2005 ◽  
Vol 46 (1) ◽  
pp. 104-109 ◽  
Author(s):  
H. Fukuda ◽  
J. Horiguchi ◽  
C. Ono ◽  
T. Ohshita ◽  
J. Takaba ◽  
...  
Keyword(s):  
Diffusion Tensor Imaging ◽  
White Matter ◽  
Myotonic Dystrophy ◽  
Normal Control ◽  
Diffusion Tensor ◽  
Age At Onset ◽  
Mean Diffusivity ◽  
Magnetic Resonance Images ◽  
Microstructural Changes ◽  
Control Subjects

Purpose: To determine whether myotonic dystrophy (MyD) patients have diffusion tensor abnormalities suggestive of microstructural changes in normal‐appearing white matter (NAWM). Material and Methods: Conventional and diffusion tensor magnetic resonance images of the brain were obtained in 19 MyD patients and 19 age‐matched normal control subjects. Fractional anisotropy (FA) and mean diffusivity (MD) values were calculated in white matter lesions (WMLs) and NAWM in MyD patients and in the white matter of normal control subjects. Differences between WML and NAWM values and between MyD patient and control subject values were analyzed statistically. Results: Significantly lower FA and higher MD values were found in all regions of interest in the NAWM of MyD patients than in the white matter of control subjects ( P<0.01), as well as significantly lower FA and higher MD values in WMLs than in NAWM of MyD patients ( P<0.05). There was no significant correlation of mean FA or MD values in NAWM with patient age, age at onset, or duration of illness ( P>0.1). Conclusion: Diffusion tensor imaging analysis suggests the presence of diffuse microstructural changes in NAWM of MyD patients that may play an important role in the development of disability.

The SURE-LET Approach for MR Brain Image Denoising Using Different Shrinkage Rules

2012 ◽  
pp. 227-235 ◽  
Author(s):  
D. Selvathi ◽  
S. Thamarai Selvi ◽  
C. Loorthu Sahaya Malar
Keyword(s):  
Image Denoising ◽  
Mean Squared Error ◽  
Diffusion Tensor ◽  
Curvelet Transform ◽  
Research Area ◽  
Magnetic Resonance Images ◽  
Image Domain ◽  
Active Research ◽  
Unbiased Risk ◽  
Wavelet Image

SURE-LET Approach is used for reducing or removing noise in brain Magnetic Resonance Images (MRI). Removing or reducing noise is an active research area in image processing. Rician noise is the dominant noise in MRIs. Due to this type of noise, the abnormal tissue (cancerous tissue) may be misclassified as normal tissue and introduces bias into MRI measurements that can have significant impact on the shapes and orientations of tensors in diffusion tensor MRIs. SURE is a new approach to Orthonormal wavelet image denoising. It is an image-domain minimization of an estimate of the mean squared error—Stein’s unbiased risk estimates (SURE). Here, the denoising process can be expressed as a linear combination of elementary denoising processes-linear expansion of thresholds (LET). Different Shrinkage functions such as Soft and Hard and Shrinkage rules and Universal and BayesShrink are used to remove noise and the performance of these results are compared. The algorithm is applied on brain MRIs with different noisy conditions by varying standard deviation of noise. The performance of this approach is compared with performance of the Curvelet transform.

scholarly journals The Surgical White Matter Chassis: A Practical 3-Dimensional Atlas for Planning Subcortical Surgical Trajectories

2017 ◽  
Vol 14 (5) ◽  
pp. 469-482 ◽  
Author(s):  
Jonathan E Jennings ◽  
Amin B Kassam ◽  
Melanie B Fukui ◽  
Alejandro Monroy-Sosa ◽  
Srikant Chakravarthi ◽  
...  
Keyword(s):  
White Matter ◽  
Surgical Planning ◽  
Diffusion Tensor ◽  
Magnetic Resonance Images ◽  
Practical Implementation ◽  
Fiber Types ◽  
3.0 Tesla ◽  
3 Dimensional ◽  
White Matter Anatomy

AbstractBACKGROUNDThe imperative role of white matter preservation in improving surgical functional outcomes is now recognized. Understanding the fundamental white matter framework is essential for translating the anatomic and functional literature into practical strategies for surgical planning and neuronavigation.OBJECTIVETo present a 3-dimensional (3-D) atlas of the structural and functional scaffolding of human white matter—ie, a “Surgical White Matter Chassis (SWMC)”—that can be used as an organizational tool in designing precise and individualized trajectory-based neurosurgical corridors.METHODSPreoperative diffusion tensor imaging magnetic resonance images were obtained prior to each of our last 100 awake subcortical resections, using a clinically available 3.0 Tesla system. Tractography was generated using a semiautomated deterministic global seeding algorithm. Tract data were conceptualized as a 3-D modular chassis based on the 3 major fiber types, organized along median and paramedian planes, with special attention to limbic and neocortical association tracts and their interconnections.RESULTSWe discuss practical implementation of the SWMC concept, and highlight its use in planning select illustrative cases. Emphasis has been given to developing practical understanding of the arcuate fasciculus, uncinate fasciculus, and vertical rami of the superior longitudinal fasciculus, which are often-neglected fibers in surgical planning.CONCLUSIONA working knowledge of white matter anatomy, as embodied in the SWMC, is of paramount importance to the planning of parafascicular surgical trajectories, and can serve as a basis for developing reliable safe corridors, or modules, toward the goal of “zero-footprint” transsulcal access to the subcortical space.

Cluster Analysis of Diffusion Tensor Magnetic Resonance Images in Human Head Injury

Neurosurgery ◽  
2000 ◽  
Vol 47 (2) ◽  
pp. 306-314 ◽  
Author(s):  
Derek K. Jones ◽  
Ronan Dardis ◽  
Max Ervine ◽  
Mark A. Horsfield ◽  
Martin Jeffree ◽  
...  
Keyword(s):  
Cluster Analysis ◽  
Head Injury ◽  
Magnetic Resonance ◽  
Diffusion Tensor ◽  
Human Head ◽  
Magnetic Resonance Images

scholarly journals The infinitesimal generator of the stochastic Burgers equation

2020 ◽  
Vol 178 (3-4) ◽  
pp. 1067-1124
Author(s):  
Massimiliano Gubinelli ◽  
Nicolas Perkowski
Keyword(s):  
Burgers Equation ◽  
Martingale Problem ◽  
Uniqueness Result ◽  
Stochastic Pdes ◽  
Martingale Approach ◽  
Infinite Dimensional ◽  
Cylinder Test ◽  
Stochastic Burgers Equation ◽  
Trivial Intersection ◽  
Backward Equation

Abstract We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for “energy solutions” of the stochastic Burgers equation of Gubinelli and Perkowski (J Am Math Soc 31(2):427–471, 2018) to a wider class of equations. As applications of our approach we prove that the stochastic Burgers equation on the torus is exponentially $$L^2$$ L 2 -ergodic, and that the stochastic Burgers equation on the real line is ergodic.

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