Reconstruction of large, irregularly sampled multidimensional images. A tensor-based approach

Area of Science:

• Image reconstruction
• Scientific computing
• Numerical analysis

Background:

• Image reconstruction from irregularly sampled data is crucial for many applications.
• Current spline-based methods face computational challenges (curse of dimensionality) in higher dimensions.
• Existing methods struggle with large-scale, high-dimensional interpolation problems.

Purpose of the Study:

• To develop a computationally efficient and memory-saving algorithm for image reconstruction from irregularly sampled data.
• To overcome the limitations of existing methods in handling high-dimensional (3-D, 3-D+time) interpolation problems.
• To enable feasible reconstruction of complex, high-dimensional datasets on standard hardware.

Main Methods:

• Revisiting the regularized least-squares formulation for image interpolation.
• Employing a uniform tensor-product B-spline basis for reconstruction.
• Utilizing tensor decomposition to exploit the sparsity of the multilinear system.
• Implementing a parallel, memory-efficient solver based on tensor properties.

Main Results:

• The proposed algorithm exhibits computational complexity that is essentially linear with the number of measurements.
• The dependency on the number of dimensions is significantly reduced compared to traditional sparse matrix methods.
• Demonstrated feasibility of 4-D reconstruction with millions of samples on desktop PCs.
• Successfully applied to reconstruct 3-D+time medical ultrasound images from irregularly sampled data.

Conclusions:

• The tensor-product B-spline approach offers a significant improvement in computational efficiency and memory usage for high-dimensional image reconstruction.
• This method overcomes the curse of dimensionality, enabling practical solutions for previously intractable problems.
• The algorithm's effectiveness is validated in 3-D/4-D and demonstrated in a real-world medical imaging scenario.