Convex Recovery of Continuous Domain Piecewise Constant Images From Nonuniform Fourier Samples

• Published in: IEEE transactions on signal processing Vol. 66; no. 1; pp. 236 - 250
• Main Authors: Ongie, Greg, Biswas, Sampurna, Jacob, Mathews
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.01.2018
• Subjects: Convolution; finite rate of innovation; Heuristic algorithms; Image edge detection; Indexes; Jacobian matrices; Off-the-grid image recovery; Signal processing algorithms; structured low-rank matrix completion; Technological innovation; Structured Low-Rank Matrix Completion; Finite Rate of Innovation; Off-the-Grid Image Recovery
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2017.2750111

We consider the recovery of a continuous domain piecewise constant image from its nonuniform Fourier samples using a convex matrix completion algorithm. We assume the discontinuities/edges of the image are localized to the zero level set of a bandlimited function. This assumption induces linear dependencies between the Fourier coefficients of the image, which results in a two-fold block Toeplitz matrix constructed from the Fourier coefficients being low rank. The proposed algorithm reformulates the recovery of the unknown Fourier coefficients as a structured low-rank matrix completion problem, where the nuclear norm of the matrix is minimized subject to structure and data constraints. We show that the exact recovery is possible with high probability when the edge set of the image satisfies an incoherency property. We also show that the incoherency property is dependent on the geometry of the edge set curve, implying higher sampling burden for smaller curves. This paper generalizes recent work on the super-resolution recovery of isolated Diracs or signals with finite rate of innovation to the recovery of piecewise constant images.