American Institute of Mathematical Sciences

Advanced
May  2010, 4(2): 273-310. doi: 10.3934/ipi.2010.4.273

HARDI data denoising using vectorial total variation and logarithmic barrier

Yunho Kim 1, , Paul M. Thompson 2, and  Luminita A. Vese 3,

1. 

Department of Mathematics, University of California, Irvine, Irvine, CA 92697-3875, United States
2. 

Laboratory of Neuro Imaging and Department of Neurology, UCLA School of Medicine, Los Angeles, CA 90095-7334, United States
3. 

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095-1555, United States

Received  March 2009 Revised  November 2009 Published  May 2010

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.
Mathematics Subject Classification: 35A15, 35J50, 35K10, 35Q92, 35Q93, 35R30, 49J40, 49J45, 49N45, 65K10, 65M06, 65M32, 68U1.
Citation: Yunho Kim, Paul M. Thompson, Luminita A. Vese. HARDI data denoising using vectorial total variation and logarithmic barrier. Inverse Problems & Imaging, 2010, 4 (2) : 273-310. doi: 10.3934/ipi.2010.4.273
[1]

Nicolas Lermé, François Malgouyres, Dominique Hamoir, Emmanuelle Thouin. Bayesian image restoration for mosaic active imaging. Inverse Problems & Imaging, 2014, 8 (3) : 733-760. doi: 10.3934/ipi.2014.8.733
[2]

Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013
[3]

Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007
[4]

Chunlin Wu, Juyong Zhang, Xue-Cheng Tai. Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Problems & Imaging, 2011, 5 (1) : 237-261. doi: 10.3934/ipi.2011.5.237
[5]

Bartomeu Coll, Joan Duran, Catalina Sbert. Half-linear regularization for nonconvex image restoration models. Inverse Problems & Imaging, 2015, 9 (2) : 337-370. doi: 10.3934/ipi.2015.9.337
[6]

Yunho Kim, Luminita A. Vese. Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability. Inverse Problems & Imaging, 2009, 3 (1) : 43-68. doi: 10.3934/ipi.2009.3.43
[7]

Jing Xu, Xue-Cheng Tai, Li-Lian Wang. A two-level domain decomposition method for image restoration. Inverse Problems & Imaging, 2010, 4 (3) : 523-545. doi: 10.3934/ipi.2010.4.523
[8]

Ke Chen, Yiqiu Dong, Michael Hintermüller. A nonlinear multigrid solver with line Gauss-Seidel-semismooth-Newton smoother for the Fenchel pre-dual in total variation based image restoration. Inverse Problems & Imaging, 2011, 5 (2) : 323-339. doi: 10.3934/ipi.2011.5.323
[9]

Petteri Harjulehto, Peter Hästö, Juha Tiirola. Point-wise behavior of the Geman--McClure and the Hebert--Leahy image restoration models. Inverse Problems & Imaging, 2015, 9 (3) : 835-851. doi: 10.3934/ipi.2015.9.835
[10]

Jia Li, Zuowei Shen, Rujie Yin, Xiaoqun Zhang. A reweighted $l^2$ method for image restoration with Poisson and mixed Poisson-Gaussian noise. Inverse Problems & Imaging, 2015, 9 (3) : 875-894. doi: 10.3934/ipi.2015.9.875
[11]

Abdulkarim Hassan Ibrahim, Jitsupa Deepho, Auwal Bala Abubakar, Kazeem Olalekan Aremu. A modified Liu-Storey-Conjugate descent hybrid projection method for convex constrained nonlinear equations and image restoration. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021022
[12]

Alexey Penenko. Convergence analysis of the adjoint ensemble method in inverse source problems for advection-diffusion-reaction models with image-type measurements. Inverse Problems & Imaging, 2020, 14 (5) : 757-782. doi: 10.3934/ipi.2020035
[13]

Amir Averbuch, Pekka Neittaanmäki, Valery Zheludev. Periodic spline-based frames for image restoration. Inverse Problems & Imaging, 2015, 9 (3) : 661-707. doi: 10.3934/ipi.2015.9.661
[14]

Ying Zhang, Xuhua Ren, Bryan Alexander Clifford, Qian Wang, Xiaoqun Zhang. Image fusion network for dual-modal restoration. Inverse Problems & Imaging, 2021, 15 (6) : 1409-1419. doi: 10.3934/ipi.2021067
[15]

Ruiqiang He, Xiangchu Feng, Xiaolong Zhu, Hua Huang, Bingzhe Wei. RWRM: Residual Wasserstein regularization model for image restoration. Inverse Problems & Imaging, 2021, 15 (6) : 1307-1332. doi: 10.3934/ipi.2020069
[16]

Josselin Garnier, George Papanicolaou. Resolution enhancement from scattering in passive sensor imaging with cross correlations. Inverse Problems & Imaging, 2014, 8 (3) : 645-683. doi: 10.3934/ipi.2014.8.645
[17]

Lingbing He, Yulong Zhou. High order approximation for the Boltzmann equation without angular cutoff. Kinetic & Related Models, 2018, 11 (3) : 547-596. doi: 10.3934/krm.2018024
[18]

Anis Theljani, Ke Chen. An augmented lagrangian method for solving a new variational model based on gradients similarity measures and high order regulariation for multimodality registration. Inverse Problems & Imaging, 2019, 13 (2) : 309-335. doi: 10.3934/ipi.2019016
[19]

Ennio Fedrizzi. High frequency analysis of imaging with noise blending. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 979-998. doi: 10.3934/dcdsb.2014.19.979
[20]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy