Hankel Transforms

DSF uses Hankel transforms to compute various quantities throughout the pipeline. The code provides a module, dsf.hankel, with a unified public interface for performing these transforms. The public layer is accessed through the dsf.hankel.hankel.HankelTransform class.

There are three backend implementations available:

1. fftlog: Uses the FFTLog algorithm to compute the transform. It supports projected and spherical 1D Hankel transforms. This is the fastest option for most 1D applications.

2. matrix_zeros: Uses identities of Bessel zeros to compute Hankel transform matrix operators. It supports projected 1D, 2D, and 3D Hankel transforms. Can exhibit ringing at small scales.

3. matrix_direct: Uses a user-defined k- and r-grid to compute Hankel transform matrix operators. It supports projected 1D and 2D Hankel transforms. Requires fine-tuning of the k- and r-sampling upon generation. Assuming the grid is sampled well, the method is robust against ringing at small scales, but can exhibit minor ringing at large scales.

Radial bin weights

Both matrix backends accept radial_weight_method when constructed:

ht = HankelTransform(
    backend="matrix_zeros", radial_weight_method="gradient"
)
r_bin, covariance_bin = ht.bin_radial_matrix(r, covariance, r_bins)

The available methods are:

  • gradient (default): the original np.gradient spacing on the union of the radial grid and bin edges, with each point assigned to its left-inclusive, right-exclusive bin.

  • voronoi: exact annular measures of midpoint cells clipped to each bin. A cell can contribute to multiple bins; sample values are assumed constant over each cell.

  • polynomial: the same union grid and bin assignment as gradient, using DerivKit cubic least-squares fits over up to nine nearby points to estimate spacing. Windows shift at endpoints. This reproduces the derivkit_gradient branch implementation.

These options control radial averaging only. Polynomial fitting can overshoot on irregular grids and does not guarantee positive weights or exact annular normalization. For covariance calculations, pass the selection through DeltaSigmaCovarianceBuilder(hankel_kwargs={ "radial_weight_method": "voronoi"}). The FFTLog backend does not support radial matrix binning.

As a rule of thumb, use fftlog to perform any 1D Hankel transforms. For any 2D transforms, try matrix_zeros if you don’t need to sample at particularly small scales; otherwise try matrix_direct. However, we recommend trying both backends to explicitly check for ringing behavior and speed.

Common Issue: When performing a matrix Hankel transform, the the input k-grid for the power spectrum must cover the full range of the k-values used to generate the Hankel transform matrix. The matrix_zeros backend can dynamically expend its k-grid during generation, depending on the other initialization parameters, meaning that the required k-grid for the power spectrum may be larger than you expect!