Gradient-based image recovery methods from incomplete Fourier measurements.

Area of Science:

• Medical Imaging
• Signal Processing
• Applied Mathematics

Background:

• Image reconstruction from limited Fourier samples is crucial for applications like MRI and SAR.
• Compressive sensing (CS) leverages image sparsity for accurate recovery using sub-Nyquist sampling.
• Traditional CS methods often use total-variation (TV) minimization, which can be computationally intensive.

Purpose of the Study:

• To develop a novel CS algorithm for improved image reconstruction from sparse Fourier samples.
• To exploit the inherent sparsity of image gradients for more efficient recovery.
• To compare the proposed method against existing leading reconstruction techniques.

Main Methods:

• Developed the GradientRec algorithm, which applies CS to recover horizontal and vertical image gradients.
• Estimated the original image from recovered gradients using two inverse problem-solving methods: least-squares optimization and a generalized Poisson solver.
• Conducted experiments to validate the effectiveness of GradientRec.

Main Results:

• The GradientRec algorithm demonstrates effectiveness in recovering image gradients, which are sparser than traditional TV representations.
• The proposed method shows competitive or superior performance compared to other leading CS-based image reconstruction techniques.
• Successful image reconstruction was achieved using significantly fewer Fourier samples than conventional methods.

Conclusions:

• GradientRec offers a promising alternative for image reconstruction in undersampled scenarios, particularly in medical imaging.
• Exploiting image gradient sparsity provides an effective strategy within the compressive sensing framework.
• The algorithm's ability to reconstruct images from minimal data has significant implications for reducing scan times and power consumption.