Abstract
We consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on functions is a computationally intensive problem necessary for many practical problems. We analyze a technique called product-convolution expansion: The operator is locally approximated by a convolution, allowing to design fast numerical algorithms based on the fast Fourier transform. We design various types of expansions, provide their explicit rates of approximation and their complexity depending on the time-varying impulse response smoothness. This analysis suggests novel wavelet-based implementations of the method with numerous assets such as optimal approximation rates, low complexity and storage requirements as well as adaptivity to the kernels regularity. The proposed methods are an alternative to more standard procedures such as panel clustering, cross-approximations, wavelet expansions or hierarchical matrices.
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References
Arigovindan, M., Shaevitz, J., McGowan, J., Sedat, J.W., Agard, D.A.: A parallel product-convolution approach for representing the depth varying point spread functions in 3D widefield microscopy based on principal component analysis. Opt. Express 18(7), 6461–6476 (2010)
Beylkin, G.: On the representation of operators in bases of compactly supported wavelets. SIAM J. Numer. Anal. 29(6), 1716–1740 (1992)
Beylkin, G., Coifman, R., Rokhlin, V.: Fast wavelet transforms and numerical algorithms I. Appl. Comput. Harmon. Anal. 44(2), 141–183 (1991)
Bezhaev, A.Y., Vasilenko, V.A.: Variational Theory of Splines. Springer, Berlin (2001)
Bigot, J., Escande, P., Weiss, P.: Estimation of linear operators from scattered impulse responses. arXiv preprint arXiv:1610.04056 (2016)
Börm, S., Grasedyck, L., Hackbusch, W.: Introduction to hierarchical matrices with applications. Eng. Anal. Bound. Elem. 27(5), 405–422 (2003)
Busby, R., Smith, H.: Product-convolution operators and mixed-norm spaces. Trans. Am. Math. Soc. 263(2), 309–341 (1981)
Christensen, O.: An Introduction to Frames and Riesz Bases. Springer, New York (2013)
Chu, M., Funderlic, R., Plemmons, R.: Structured low rank approximation. Linear Algebra Appl. 366, 157–172 (2003)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems, vol. 40. SIAM, Philadelphia (2002)
Cohen, A., Dahmen, W., Daubechies, I., DeVore, R.: Harmonic analysis of the space BV. Rev. Mat. Iberoam. 19(1), 235–263 (2003)
Cohen, A., Daubechies, I., Vial, P.: Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1(1), 54–81 (1993)
Dahmen, W., Prößdorf, S., Schneider, R.: Wavelet approximation methods for pseudodifferential equations II: matrix compression and fast solution. Adv. Comput. Math. 1(3), 259–335 (1993)
Dahmen, W., Prössdorf, S., Schneider, R.: Wavelet approximation methods for pseudodifferential equations: I stability and convergence. Math. Z. 215(1), 583–620 (1994)
Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. pur. appl. math. 41(7), 909–996 (1988)
Daubechies, I.: Ten Lectures on Wavelets, vol. 61. SIAM, Philadelphia (1992)
Denis, L., Thiébaut, E., Soulez, F., Becker, J.-M., Mourya, R.: Fast approximations of shift-variant blur. Int. J. Comput. Vis. 115(3), 253–278 (2015)
DeVore, R.A.: Nonlinear approximation. Acta. Numer. 7, 51–150 (1998)
DeVore, R.A., Lorentz, G.G.: Constructive Approximation, vol. 303. Springer, Berlin (1993)
Escande, P., Weiss, P.: Sparse wavelet representations of spatially varying blurring operators. SIAM J. Imaging Sci. 8(4), 2976–3014 (2015)
Flicker, R.C., Rigaut, F.J.: Anisoplanatic deconvolution of adaptive optics images. J. Opt. Soc. Am. A 22(3), 504–513 (2005)
Gilad, E., Von Hardenberg, J.: A fast algorithm for convolution integrals with space and time variant kernels. J. Comput. Phys. 216(1), 326–336 (2006)
Griffiths, L.J., Smolka, F.R., Trembly, L.D.: Adaptive deconvolution: a new technique for processing time-varying seismic data. Geophysics 42(4), 742–759 (1977)
Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54(4), 463–491 (1989)
Halko, N., Martinsson, P.-G., Tropp, J.A.: Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53(2), 217–288 (2011)
Helemskii, A.Y.: Lectures and Exercises on Functional Analysis, vol. 233. American Mathematical Society, Providence (2006)
Hirsch, M., Sra, S., Scholkopf, B., Harmeling, S.: Efficient filter flow for space-variant multiframe blind deconvolution. In: Proc. CVPR IEEE (2010)
Hrycak, T., Das, S., Matz, G., Feichtinger, H.G.: Low complexity equalization for doubly selective channels modeled by a basis expansion. IEEE Trans. Signal Process. 58(11), 5706–5719 (2010)
Katznelson, Y.: An Introduction to Harmonic Analysis. Cambridge University Press, Cambridge (2004)
Kozek, W., Pfander, G.E.: Identification of operators with bandlimited symbols. SIAM J. Math. Anal. 37(3), 867–888 (2005)
Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, New York (1999)
Meyer, Y.: Wavelets and Operators, vol. 1. Cambridge University Press, Cambridge (1995)
Miraut, D., Portilla, J.: Efficient shift-variant image restoration using deformable filtering (Part I). EURASIP J. Adv. Signal Process. 2012, 100 (2012)
Nagy, J.G., O’Leary, D.P.: Restoring images degraded by spatially variant blur. SIAM J. Sci. Comput. 19(4), 1063–1082 (1998)
Oseledets, I., Tyrtyshnikov, E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432(1), 70–88 (2010)
Pinkus, A.: N-Widths in Approximation Theory, vol. 7. Springer, Berlin (2012)
Roh, J.C., Rao, B.D.: An efficient feedback method for MIMO systems with slowly time-varying channels. In: IEEE WCNC, vol. 2, pp. 760–764. IEEE (2004)
Sawchuk, A.A.: Space-variant image motion degradation and restoration. Proc. IEEE 60(7), 854–861 (1972)
Stockham, T.G.: High-speed convolution and correlation. In: Proceedings of the April 26–28, 1966, Spring joint computer conference, pp. 229–233. ACM (1966)
Trussell, J., Fogel, S.: Identification and restoration of spatially variant motion blurs in sequential images. IEEE Trans. Image Process. 1(1), 123–126 (1992)
Unser, M., Aldroubi, A., Eden, M.: B-spline signal processing. Part I—theory. IEEE Trans. Signal Process. 41(2), 821–833 (1993)
Acknowledgements
We began investigating a much narrower version of the problem in a preliminary version of [20]. An anonymous reviewer however suggested that it would be more interesting to make a general analysis and we therefore discarded the aspects related to product-convolution expansions from [20]. We thank the reviewer for motivating us to initiate this research. We also thank Sandrine Anthoine, Jérémie Bigot, Caroline Chaux, Jérôme Fehrenbach, Hans Feichtinger, Clothilde Mélot, Anh-Tuan Nguyen and Bruno Torrésani for fruitful discussions on related matters. The authors are particularly grateful to Hans Feichtinger for his numerous feedbacks on a preliminary version of this paper and for suggesting the name product-convolution.
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Escande, P., Weiss, P. Approximation of Integral Operators Using Product-Convolution Expansions. J Math Imaging Vis 58, 333–348 (2017). https://doi.org/10.1007/s10851-017-0714-8
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DOI: https://doi.org/10.1007/s10851-017-0714-8
Keywords
- Integral operators
- Wavelet
- Spline
- Structured low-rank decomposition
- Numerical complexity
- Approximation rate
- Fast Fourier transform