Relating brain tissue properties to diffusion tensor imaging (DTI) is limited when an image voxel contains partial volume of brain tissue with free water, such as cerebrospinal fluid or edema, rendering the DTI indices no longer useful for describing the underlying tissue properties. We propose here a method for separating diffusion properties of brain tissue from surrounding free water while mapping the free water volume. This is achieved by fitting a bi-tensor model for which a mathematical framework is introduced to stabilize the fitting. Applying the method on datasets from a healthy subject and a patient with edema yielded corrected DTI indices and a more complete tract reconstruction that passed next to the ventricles and through the edema. We were able to segment the edema into areas according to the condition of the underlying tissue. In addition, the volume of free water is suggested as a new quantitative contrast of diffusion MRI. The findings suggest that free water is not limited to the borders of the brain parenchyma; it therefore contributes to the architecture surrounding neuronal bundles and may indicate specific anatomical processes. The analysis requires a conventional DTI acquisition and can be easily merged with existing DTI pipelines. Key words: diffusion; brain; edema; partial volume; DTI; MRIDiffusion imaging is an MRI technique sensitive to the mean displacement of water molecules along a specified direction (1). Within typical experimental diffusion times (a few tens of milliseconds) water molecules in brain tissue are expected to have a mean displacement on the order of 5-10 m, reflecting the degree of hindrance by surrounding cellular structures. The most common method for inferring tissue macroscopic geometry from water displacement is diffusion tensor imaging (DTI) (2), which models displacements in multiple directions using a diffusion tensor. The geometry is inferred from DTI indices (3) such as: fractional anisotropy (FA), which provides good segmentation of white matter and an indication of white matter coherence; the principal eigenvector of the tensor, which provides the orientation of white matter bundles; and the mean diffusivity (MD) or apparent diffusion coefficient (ADC), which provides a contrast mechanism for identifying areas with increased bulk diffusivity that may represent an increase in tissue water content. DTI indices have proven to have significant value both in clinical evaluation and brain research (4), including the unique ability to delineate neuronal fibers via tractography (5).Free water is defined as water molecules that do not experience flow and are not restricted by their surroundings. In the human brain, free water is found as cerebrospinal fluid (CSF) confined to the ventricles and around the brain parenchyma. Free water may also accumulate in the form of vasogenic edema within the brain parenchyma in the extracellular space due to processes such as tumors, brain trauma, or hemorrhage that cause ruptures in the blood-brain barrier (6 -8). Free water ...

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The linear and nonlinear scale spaces, generated by the inherently real-valued diffusion equation, are generalized to complex diffusion processes, by incorporating the free Schrödinger equation. A fundamental solution for the linear case of the complex diffusion equation is developed. Analysis of its behavior shows that the generalized diffusion process combines properties of both forward and inverse diffusion. We prove that the imaginary part is a smoothed second derivative, scaled by time, when the complex diffusion coefficient approaches the real axis. Based on this observation, we develop two examples of nonlinear complex processes, useful in image processing: a regularized shock filter for image enhancement and a ramp preserving denoising process.

Abstract

Signal and image enhancement is considered in the context of a new type of diffusion process that simultaneously enhances, sharpens, and denoises images. The nonlinear diffusion coefficient is locally adjusted according to image features such as edges, textures, and moments. As such, it can switch the diffusion process from a forward to a backward (inverse) mode according to a given set of criteria. This results in a forward-and-backward (FAB) adaptive diffusion process that enhances features while locally denoising smoother segments of the signal or image. The proposed method, using the FAB process, is applied in a super-resolution scheme. The FAB method is further generalized for color processing via the Beltrami flow, by adaptively modifying the structure tensor that controls the nonlinear diffusion process. The proposed structure tensor is neither positive definite nor negative, and switches between these states according to image features. This results in a forward-and-backward diffusion flow where different regions of the image are either forward or backward diffused according to the local geometry within a neighborhood.

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An analysis of the BRST cohomology of the G G topological models is performed for the case of A (1) 1 . Invoking a special free field parametrization of the various currents, the cohomology on the corresponding Fock space is extracted. We employ the singular vector structure and fusion rules to translate the latter into the cohomology on the space of irreducible representations. Using the physical states we calculate the characters and partition function, and verify the index interpretation.We twist the energy-momentum tensor to establish an intriguing correspondence between the SL (2) SL(2) model with level k = p q − 2 and (p, q) models coupled to gravity.

Denoising algorithms based on gradient dependent regularizers, such as nonlinear diffusion processes and total variation denoising, modify images towards piecewise constant functions. Although edge sharpness and location is well preserved, important information, encoded in image features like textures or certain details, is often compromised in the process of denoising. We propose a mechanism that better preserves fine scale features in such denoising processes. A basic pyramidal structure-texture decomposition of images is presented and analyzed. A first level of this pyramid is used to isolate the noise and the relevant texture components in order to compute spatially varying constraints based on local variance measures. A variational formulation with a spatially varying fidelity term controls the extent of denoising over image regions. Our results show visual improvement as well as an increase in the signal-to-noise ratio over scalar fidelity term processes. This type of processing can be used for a variety of tasks in partial differential equation-based image processing and computer vision, and is stable and meaningful from a mathematical viewpoint.