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Quartic Box-Spline Reconstruction on the BCC Lattice.

IEEE Transactions on Visualization and Computer Graphics

|May 23, 2012

Summary

A new seven-direction quartic box-spline filter (M7) offers a computationally efficient alternative for body-centered cubic (BCC) lattices. M7 reduces aliasing in reconstruction, outperforming existing methods with lower polynomial degrees and smaller support.

Area of Science:

Applied Mathematics
Digital Signal Processing
Lattice Theory

Background:

Body-centered cubic (BCC) lattices are fundamental in various scientific domains.
Existing box-spline filters, like the eight-direction quintic box-spline (M8), face limitations in computational efficiency and support size.
Minimizing aliasing is crucial for accurate signal reconstruction in discrete systems.

Purpose of the Study:

To introduce and evaluate a novel seven-direction quartic box-spline filter (M7) for BCC lattices.
To compare the performance of M7 against the established M8 filter in terms of approximation order, polynomial degree, support size, and computational efficiency.
To assess the aliasing reduction capabilities of M7 in quasi-interpolation-based reconstruction.

Main Methods:

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Development of the seven-direction quartic box-spline (M7) filter.

Comparative analysis of M7 and M8 regarding approximation order, polynomial degree, and support size.

Application of M7 in quasi-interpolation prefilters for signal reconstruction.

Quantitative verification using integral filter metrics and frequency error kernels.

Visualization and analysis of distributional aliasing via spectral evaluation on planes and lines.

Main Results:

M7 achieves the same approximation order as M8 but with a lower polynomial degree and smaller support.
M7 demonstrates superior computational efficiency compared to M8.
Application of M7 in reconstruction shows a significant reduction in aliasing.
Quantitative metrics and frequency error kernels confirm M7's improved aliasing characteristics.

Conclusions:

The M7 filter presents a more efficient and effective alternative for BCC lattice applications.
M7 offers significant advantages in aliasing reduction for signal reconstruction tasks.
This work provides a valuable new tool for researchers working with BCC lattices and signal processing.