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Joe Kileel1, Nicholas F Marshall2, Oscar Mickelin3

  • 1Department of Mathematics, University of Texas at Austin, Austin, TX 78712 USA.

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|March 12, 2026

View abstract on PubMed

Summary
This summary is machine-generated.

We developed fast algorithms to convert 3D functions between voxel grids and ball harmonic expansions. These methods significantly improve computational efficiency for scientific data analysis.

Keywords:
33C1033C5542-0465D1865R10Laplacian eigenfunctionsfast transformsspherical Besselspherical harmonics

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Area of Science:

  • Mathematics
  • Numerical Analysis
  • Scientific Computing

Background:

  • Representing 3D functions in Cartesian voxel grids is common in scientific applications.
  • Ball harmonics provide an alternative, efficient basis for representing functions within a unit ball.
  • Transforming between these representations is computationally intensive with naive methods.

Purpose of the Study:

  • To devise fast and provably accurate algorithms for transforming 3D functions between Cartesian voxel and ball harmonic representations.
  • To significantly reduce the computational complexity compared to existing direct methods.

Main Methods:

  • Developed novel algorithms for transforming N x N x N Cartesian voxel data to ball harmonic expansions.
  • Algorithms achieve provable accuracy (epsilon) with improved time complexity.
  • Leveraged properties of the Dirichlet Laplacian eigenbasis on the unit ball.

    • Main Results:

    • Achieved relative L1-L-infinity accuracy of epsilon in time complexity O(N^3(log N)^2 + N^3|log epsilon|^2).
    • This is a substantial improvement over the naive O(N^6) complexity.
    • Demonstrated the effectiveness of the algorithms through numerical examples.

    Conclusions:

    • The developed algorithms offer a computationally efficient solution for 3D function representation transformations.
    • These methods have potential applications in various scientific fields requiring 3D data analysis and manipulation.
    • The provable accuracy and improved speed make these algorithms valuable for complex simulations and modeling.