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IPI

Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability

In this work we wish to recover an unknown image from a blurry, or noisy-blurry version. We solve this inverse problem by energy
minimization and regularization. We seek a solution of the form $u + v$,
where $u$ is a function of bounded variation (cartoon component), while $v$ is
an oscillatory component (texture), modeled by a Sobolev function with
negative degree of differentiability. We give several results of existence and characterization of minimizers of the proposed optimization problem.
Experimental results show that this
cartoon + texture model better recovers textured details in natural images, by comparison with the more standard models where the unknown is restricted only to the space of functions of bounded variation.

keywords:
Sobolev spaces
,
oscillatory functions.
,
Image restoration
,
bounded variation
,
variational models

IPI

In this work, we wish to denoise HARDI (High Angular Resolution Diffusion Imaging) data arising in medical brain imaging. Diffusion imaging is a relatively new and powerful method to measure the three-dimensional profile of water diffusion at each point in the brain. These images can be used to reconstruct fiber directions and pathways in the living brain, providing detailed maps of fiber integrity and connectivity. HARDI data is a powerful new extension of diffusion imaging, which goes beyond the diffusion tensor imaging (DTI) model: mathematically, intensity data is given at every voxel and at any direction on the sphere. Unfortunately, HARDI data is usually highly contaminated with noise, depending on the

*b-value*which is a tuning parameter pre-selected to collect the data. Larger*b-values*help to collect more accurate information in terms of measuring diffusivity, but more noise is generated by many factors as well. So large*b-values*are preferred, if we can satisfactorily reduce the noise without losing the data structure. Here we propose two variational methods to denoise HARDI data. The first one directly denoises the collected data $S$, while the second one denoises the so-called sADC (spherical Apparent Diffusion Coefficient), a field of radial functions derived from the data. These two quantities are related by an equation of the form $S = S_0\exp(-b\cdot sADC)$ (in the noise-free case). By applying these two different models, we will be able to determine which quantity will most accurately preserve data structure after denoising. The theoretical analysis of the proposed models is presented, together with experimental results and comparisons for denoising synthetic and real HARDI data.## Year of publication

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