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Splines, Operators, and Wavelets

Investigators: Michael Unser, John Paul Ward

Summary: We categorize (pseudo-)differential operators according to specific properties (e.g., invariance with respect to rotation, scale, or translation). These classes are then used to specify special brands of splines, as well as corresponding wavelets that approximate the behavior of the underlying operators. The following are some examples:

  • Exponential splines and wavelets correspond to convolution operators with a rational transfer function (ordinary differential equation).
  • Fractional splines and wavelets correspond to fractional derivatives, which are scale-invariant.
  • Polyharmonic splines and wavelets correspond to the fractional iterates of the Laplacian, which are both scale- and rotation-invariant.

Introduction

The L-spline formalism provides a direct connection between splines and differential operators. Similarily, it is well known from wavelet theory that a wavelet analysis can be qualitatively interpreted as a multiscale differentiation process. Our goal in this research is

  • to make this fundamental link more explicit—especially, the wavelet aspect of it;
  • to provide a general framework for the construction of spline and wavelet approximation spaces associated with a given operator and determine the structure and approximation properties of these spaces;
  • to take advantage of this framework to design new splines and wavelets with some desirable properties (e.g., controlled smoothness, fractional differentiation behavior, isotropy, directionality, among others).

Main Contributions

A high-level approach to splines is to define them via some suitable generalized differential operator L. Specifically, we say that the continuously defined function s(x) is a cardinal L-spline if and only if

L{s(x)}=∑ a[k]δ(x-k)

In essence, the operator L extracts the spline singularities (stream of Dirac impulses), while acting as an analog-to-digital converter: transformation of s(x) into a discrete sequence of weights a[k]. The simplest example is the derivative operator which is associated to piecewise-constant splines. The concept remains valid in higher dimensions and for arbitrary lattices.

Within this abstract framework, we have chosen to identify interesting classes of shift-invariant operators based on some general invariance principles. In the case of scale invariance, we have shown that the corresponding class of operators in 1D reduces to a two-parameter family of fractional derivatives. These generalized derivatives lead to the definition of an extended family of fractional splines that has the interesting property of being closed under fractional differentation. If we move to higher dimensions and add rotation invariance, we end up with fractional iterates of the Laplacian, the defining operators of polyharmonic splines. A nice side effect of the scale-invariance property is that the underlying spline functions satisfy scaling relations which makes them suitable for constructing wavelet bases.

Along the same line, we have shown that certain types of Box splines (on a hexagonal grid) could be associated with a special multidimensional iterated derivative operator. This formulation yields explicit formulas for the underlying B-splines, which were not available previously.

The proposed spline formalism also lends itself to the specification of wavelet basis functions that essentially behave like multiscale versions of the underlying operator. We have demonstrated the concept with the exponential-spline wavelets, which can be tuned to replicate the behavior of any ordinary (constant-coefficient) differential operator. We have also considered fractional iterates of the Laplacian which have led to the construction of new, nonseparable wavelets that are nearly isotropic, and act like smoothed versions of the Laplacian.

Past Investigators: Thierry Blu, Ildar Khalidov, Dimitri Van De Ville

Period: 2005-ongoing

Funding: Grant 200020-109415 from the Swiss Science Foundation

Major Publications

[1] 

I. Khalidov, J. Fadili, F. Lazeyras, D. Van De Ville, M. Unser, "Activelets: Wavelets for Sparse Representation of Hemodynamic Responses," Signal Processing, vol. 91, no. 12, pp. 2810-2821, December 2011.

[2] 

I. Khalidov, "Operator-Like Wavelets with Application to Functional Magnetic Resonance Imaging," Swiss Federal Institute of Technology Lausanne, EPFL Thesis no. 4257 (2009), 131 p., January 30, 2009.

[3] 

P.D. Tafti, D. Van De Ville, M. Unser, "Invariances, Laplacian-Like Wavelet Bases, and the Whitening of Fractal Processes," IEEE Transactions on Image Processing, vol. 18, no. 4, pp. 689-702, April 2009.

[4] 

M. Unser, D. Sage, D. Van De Ville, "Multiresolution Monogenic Signal Analysis Using the Riesz-Laplace Wavelet Transform," IEEE Transactions on Image Processing, vol. 18, no. 11, pp. 2402-2418, November 2009.

[5] 

B. Forster, T. Blu, D. Van De Ville, M. Unser, "Shift-Invariant Spaces from Rotation-Covariant Functions," Applied and Computational Harmonic Analysis, vol. 25, no. 2, pp. 240-265, September 2008.

[6] 

T. Blu, M. Unser, "Self-Similarity: Part II—Optimal Estimation of Fractal Processes," IEEE Transactions on Signal Processing, vol. 55, no. 4, pp. 1364-1378, April 2007.

[7] 

M. Unser, T. Blu, "Self-Similarity: Part I—Splines and Operators," IEEE Transactions on Signal Processing, vol. 55, no. 4, pp. 1352-1363, April 2007.

[8] 

I. Khalidov, M. Unser, "From Differential Equations to the Construction of New Wavelet-Like Bases," IEEE Transactions on Signal Processing, vol. 54, no. 4, pp. 1256-1267, April 2006.

[9] 

L. Condat, D. Van De Ville, "Three-Directional Box-Splines: Characterization and Efficient Evaluation," IEEE Signal Processing Letters, vol. 13, no. 7, pp. 417-420, July 2006.

[10] 

D. Van De Ville, T. Blu, M. Unser, "Isotropic Polyharmonic B-Splines: Scaling Functions and Wavelets," IEEE Transactions on Image Processing, vol. 14, no. 11, pp. 1798-1813, November 2005.

[11] 

I. Khalidov, T. Blu, M. Unser, "Generalized L-Spline Wavelet Bases," Proceedings of the SPIE Optics and Photonics 2005 Conference on Mathematical Methods: Wavelet XI, San Diego CA, USA, July 31-August 3, 2005, vol. 5914, pp. 59140F-1/59140F-8.

[12] 

M. Unser, T. Blu, "Wavelet Theory Demystified," IEEE Transactions on Signal Processing, vol. 51, no. 2, pp. 470-483, February 2003.

[13] 

T. Blu, M. Unser, "A Complete Family of Scaling Functions: The (α, τ)-Fractional Splines," Proceedings of the Twenty-Eighth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'03), Hong Kong SAR, People's Republic of China, April 6-10, 2003, vol. VI, pp. 421-424.

[14] 

M. Unser, T. Blu, "Fractional Splines and Wavelets," SIAM Review, vol. 42, no. 1, pp. 43-67, March 2000.

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