Fast Expansion into Harmonics on the Disk: A Steerable Basis with Fast Radial Convolutions
Abstract.
We present a fast and numerically accurate method for expanding digitized \(L \times L\) images representing functions on \([-1,1]^2\) supported on the disk \(\{x \in \mathbb{R}^2 : |x| \lt 1\}\) in the harmonics (Dirichlet Laplacian eigenfunctions) on the disk. Our method, which we refer to as the Fast Disk Harmonics Transform (FDHT), runs in \(\mathcal{O}(L^2 \log L)\) operations. This basis is also known as the Fourier–Bessel basis, and it has several computational advantages: it is orthogonal, ordered by frequency, and steerable in the sense that images expanded in the basis can be rotated by applying a diagonal transform to the coefficients. Moreover, we show that convolution with radial functions can also be efficiently computed by applying a diagonal transform to the coefficients.
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Acknowledgments.
The authors would like to thank Joakim Andén, Yunpeng Shi, and Gregory Chirikjian for their helpful comments on a draft of this paper. We also thank the two anonymous reviewers for their comments, which improved the exposition of the manuscript.
References
1.
A. H. Barnett, Aliasing error of the \(\exp (\beta \sqrt{1-z^2})\) kernel in the nonuniform fast Fourier transform, Appl. Comput. Harmon. Anal., 51 (2021), pp. 1–16.
2.
A. H. Barnett, J. Magland, and L. af Klinteberg, A parallel nonuniform fast Fourier transform library based on an “Exponential of semicircle” kernel, SIAM J. Sci. Comput., 41 (2019), pp. C479–C504, https://doi.org/10.1137/18M120885X.
3.
J.-P. Berrut and L. N. Trefethen, Barycentric Lagrange interpolation, SIAM Rev., 46 (2004), pp. 501–517, https://doi.org/10.1137/S0036144502417715.
4.
T. Bhamre, T. Zhang, and A. Singer, Denoising and covariance estimation of single particle cryo-EM images, J. Struct. Biol., 195 (2016), pp. 72–81.
5.
X. Cheng, Q. Qiu, R. Calderbank, and G. Sapiro, RotDCF: Decomposition of Convolutional Filters for Rotation-Equivariant Deep Networks, preprint, arXiv:1805.06846, 2018.
6.
Y. Cheng, N. Grigorieff, P. A. Penczek, and T. Walz, A primer to single-particle cryo-electron microscopy, Cell, 161 (2015), pp. 438–449.
7.
T. S. Cohen and M. Welling, Steerable CNNs, preprint, arXiv:1612.08498, 2016.
8.
Y. Colin de Verdière, On the remainder in the Weyl formula for the Euclidean disk, Séminaire de Théorie Spectrale et Géométrie, 29 (2010), pp. 1–13.
9.
F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds., NIST Digital Library of Mathematical Functions, Release 1.1.5 of 2022-03-15, http://dlmf.nist.gov/.
10.
A. Dutt, M. Gu, and V. Rokhlin, Fast algorithms for polynomial interpolation, integration, and differentiation, SIAM J. Numer. Anal., 33 (1996), pp. 1689–1711, https://doi.org/10.1137/0733082.
11.
A. Dutt and V. Rokhlin, Fast Fourier transforms for nonequispaced data, SIAM J. Sci. Comput., 14 (1993), pp. 1368–1393, https://doi.org/10.1137/0914081.
12.
A. Elbert, Some recent results on the zeros of Bessel functions and orthogonal polynomials, J. Comput. Appl. Math., 133 (2001), pp. 65–83.
13.
J. Frank, Three-Dimensional Electron Microscopy of Macromolecular Assemblies: Visualization of Biological Molecules in Their Native State, Oxford University Press, Oxford, UK, 2006.
14.
C. F. Gauss, De nexu inter multitudinem classium, in Quas formae binariae secundi gradus distribuuntur, earumque determinantem, Cambridge Library Collection–Mathematics 2, Cambridge University Press, Cambridge, UK, 2011, pp. 269–291, https://doi.org/10.1017/CBO9781139058230.012.
15.
A. Ghaani Farashahi and G. S. Chirikjian, Fourier-Zernike series of convolutions on disks, Mathematics, 6 (2018), 290.
16.
A. Ghaani Farashahi and G. S. Chirikjian, Discrete Spectra of Convolutions on Disks Using Sturm-Liouville Theory, preprint, arXiv:1901.05001, 2019.
17.
A. Ghaani Farashahi and G. S. Chirikjian, Fourier-Zernike series of compactly supported convolutions on SE(2), J. Approx. Theory, 271 (2021), 105621.
18.
A. Ghaani Farashahi and G. S. Chirikjian, Fourier-Bessel series of compactly supported convolutions on disks, Anal. Appl., 20 (2022), pp. 171–192.
19.
Z. Gimbutas, N. F. Marshall, and V. Rokhlin, A fast simple algorithm for computing the potential of charges on a line, Appl. Comput. Harmon. Anal., 49 (2020), pp. 815–830.
20.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 2014.
21.
L. Greengard and J.-Y. Lee, Accelerating the nonuniform fast Fourier transform, SIAM Rev., 46 (2004), pp. 443–454, https://doi.org/10.1137/S003614450343200X.
22.
H. Guo, Y. Gao, T. Li, T. Li, Y. Lu, L. Zheng, Y. Liu, T. Yang, F. Luo, S. Song, et al., Structures of Omicron spike complexes and implications for neutralizing antibody development, Cell Rep., 39 (2022), 110770.
23.
G. H. Hardy and E. Landau, The lattice points of a circle, Proc. R. Soc. Lond. Ser. A, Contain. Pap. Math. Phys. Character, 105 (1924), pp. 244–258.
24.
A. Kurganov and J. Rauch, The order of accuracy of quadrature formulae for periodic functions, in Advances in Phase Space Analysis of Partial Differential Equations, Springer, New York, 2009, pp. 155–159.
25.
B. Landa and Y. Shkolnisky, Approximation scheme for essentially bandlimited and space-concentrated functions on a disk, Appl. Comput. Harmon. Anal., 43 (2017), pp. 381–403.
26.
B. Landa and Y. Shkolnisky, Steerable principal components for space-frequency localized images, SIAM J. Imaging Sci., 10 (2017), pp. 508–534, https://doi.org/10.1137/16M1085334.
27.
C. L. Lawson, A. Patwardhan, M. L. Baker, C. Hryc, E. S. Garcia, B. P. Hudson, I. Lagerstedt, S. J. Ludtke, G. Pintilie, R. Sala, et al., EMDataBank unified data resource for 3DEM, Nucleic Acids Res., 44 (2016), pp. D396–D403.
28.
J.-Y. Lee and L. Greengard, The type 3 nonuniform FFT and its applications, J. Comput. Phys., 206 (2005), pp. 1–5.
29.
E. Nogales and S. H. W. Scheres, Cryo-EM: A unique tool for the visualization of macromolecular complexity, Mol. Cell, 58 (2015), pp. 677–689.
30.
M. O’Neil, F. Woolfe, and V. Rokhlin, An algorithm for the rapid evaluation of special function transforms, Appl. Comput. Harmon. Anal., 28 (2010), pp. 203–226, https://doi.org/10.1016/j.acha.2009.08.005.
31.
G. A. Papakostas, Y. S. Boutalis, D. A. Karras, and B. G. Mertzios, A new class of Zernike moments for computer vision applications, Inform. Sci., 177 (2007), pp. 2802–2819.
32.
G. Plonka, D. Potts, G. Steidl, and M. Tasche, Numerical Fourier Analysis, Springer, New York, 2018.
33.
A. Rangan, M. Spivak, J. Andén, and A. Barnett, Factorization of the translation kernel for fast rigid image alignment, Inverse Problems, 36 (2020), 024001.
34.
V. Rokhlin, A fast algorithm for the discrete Laplace transformation, J. Complexity, 4 (1988), pp. 12–32.
35.
T. R. Shaikh, H. Gao, W. T. Baxter, F. J. Asturias, N. Boisset, A. Leith, and J. Frank, SPIDER image processing for single-particle reconstruction of biological macromolecules from electron micrographs, Nat. Protoc., 3 (2008), pp. 1941–1974.
36.
Y.-h. Shih, G. Wright, J. Andén, J. Blaschke, and A. H. Barnett, cuFINUFFT: A load-balanced GPU library for general-purpose nonuniform FFTs, in 2021 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW), IEEE, 2021, pp. 688–697.
37.
D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty—I, Bell Syst. Tech. J., 40 (1961), pp. 43–63.
38.
N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, John Wiley & Sons, New York, 1996.
39.
A. Townsend, A fast analysis-based discrete Hankel transform using asymptotic expansions, SIAM J. Numer. Anal., 53 (2015), pp. 1897–1917, https://doi.org/10.1137/151003106.
40.
J. G. Van der Corput, Neue zahlentheoretische Abschätzungen, Math. Ann., 89 (1923), pp. 215–254.
41.
R. H. Wade, A brief look at imaging and contrast transfer, Ultramicroscopy, 46 (1992), pp. 145–156.
42.
M. Weiler, F. A. Hamprecht, and M. Storath, Learning steerable filters for rotation equivariant CNNs, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2018, pp. 849–858.
43.
von F. Zernike, Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode, Physica, 1 (1934), pp. 689–704.
44.
Z. Zhao, Y. Shkolnisky, and A. Singer, Fast steerable principal component analysis, IEEE Trans. Comput. Imaging, 2 (2016), pp. 1–12.
45.
Z. Zhao and A. Singer, Rotationally invariant image representation for viewing direction classification in cryo-EM, J. Struct. Biol., 186 (2014), pp. 153–166.
46.
R. Zhou and N. Grisouard, Spectral Solver for Cauchy Problems in Polar Coordinates Using Discrete Hankel Transforms, preprint, arXiv:2210.09736, 2022.
Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: A2431 - A2457
DOI: 10.1137/22M1542775
ISSN (online): 1095-7197
Copyright
© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 22 December 2022
Accepted: 19 May 2023
Published online: 22 September 2023
Keywords
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Funding Information
Air Force Office of Scientific Research (AFOSR): FA9550-20-1-0266
National Institute of General Medical Sciences (NIGMS): 1R01GM136780-01
National Science Foundation (NSF): DMS-1903015
National Science Foundation (NSF): IIS-1837992, DMS-2009753
Funding: The work of the first author was supported in part by NSF grant DMS-1903015. The work of the third author was supported in part by AFOSR grant FA9550-20-1-0266, the Simons Foundation Math+X Investigator Award, NSF BIGDATA Award IIS-1837992, NSF grant DMS-2009753, and NIH/NIGMS 1R01GM136780-01.
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