Stable image reconstruction using total variation minimization

2012/02/29 by Deanna Needell, Rachel Ward, Needell, Deanna +1
Engineering · Mathematics · Medicine · #41A46 #68Q25 #68W20 #90C27 #Computer Vision and Pattern Recognition (cs.CV) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Medical Imaging Techniques and Applications #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Sparse and Compressive Sensing Techniques

paper · pdf · doi:10.48550/arxiv.1202.6429

openalex publication_date 2012/02/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

This article presents near-optimal guarantees for accurate and robust image recovery from under-sampled noisy measurements using total variation minimization. In particular, we show that from O(slog(N)) nonadaptive linear measurements, an image can be reconstructed to within the best s-term approximation of its gradient up to a logarithmic factor, and this factor can be removed by taking slightly more measurements. Along the way, we prove a strengthened Sobolev inequality for functions lying in the null space of suitably incoherent matrices.

Citations

• Atomic Decomposition by Basis Pursuit
• Sparse Approximate Solutions to Linear Systems
• Compressed Sensing MRI
• Single-pixel imaging via compressive sampling
• Stable signal recovery from incomplete and inaccurate measurements
• Iterative hard thresholding for compressed sensing

Related

• Exact recovery of the support of piecewise constant images via total variation regularization
• Improved recovery guarantees and sampling strategies for TV minimization in compressive imaging
• Guarantees of Total Variation Minimization for Signal Recovery
• Approximation Theory of Total Variation Minimization for Data Completion
• Exact reconstruction and reconstruction from noisy data with anisotropic total variation