Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability
Yunho Kim Luminita A. Vese
Inverse Problems & Imaging 2009, 3(1): 43-68 doi: 10.3934/ipi.2009.3.43
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
HARDI data denoising using vectorial total variation and logarithmic barrier
Yunho Kim Paul M. Thompson Luminita A. Vese
Inverse Problems & Imaging 2010, 4(2): 273-310 doi: 10.3934/ipi.2010.4.273
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.
keywords: bounded variation log barrier method. High angular resolution diffusion imaging variational models image restoration

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