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Wavelets and Differential Operators: From Fractals to Marr's Primal Sketch

M. Unser

Plenary talk, proceedings of the Fourth French Biyearly Congress on Applied and Industrial Mathematics (SMAI'09), La Colle sur Loup, French Republic, May 25-29, 2009, p. 8.

Invariance is an attractive principle for specifying image processing algorithms. In this presentation, we promote affine invariance—more precisely, invariance with respect to translation, scaling and rotation. As starting point, we identify the corresponding class of invariant 2D operators: these are combinations of the (fractional) Laplacian and the complex gradient (or Wirtinger operator). We then specify some corresponding differential equation and show that the solution in the real-valued case is either a fractional Brownian field (Mandelbrot and Van Ness, 1968) [3] or a polyharmonic spline (Duchon, 1976) [2], depending on the nature of the system input (driving term): stochastic (white noise) or deterministic (stream of Dirac impulses). The affine invariance of the operator has two important consequences: (1) the statistical self-similarity of the fractional Brownian field, and (2) the fact that the polyharmonic splines specify a multiresolution analysis of L2(ℝ2) and lend themselves to the construction of wavelet bases [4, 1, 6]. The other fundamental implication is that the corresponding wavelets behave like multi-scale versions of the operator from which they are derived; this makes them ideally suited for the analysis of multidimensional signals with fractal characteristics (whitening property of the fractional Laplacian) [5].

The complex extension of the approach yields a new complex wavelet basis that replicates the behavior of the Laplace-gradient operator and is therefore adapted to edge detection [7]. We introduce the Marr wavelet pyramid which corresponds to a slightly redundant version of this transform with a Gaussian-like smoothing kernel that has been optimized for better steerability. We demonstrate that this multiresolution representation is well suited for a variety of image-processing tasks. In particular, we use it to derive a primal wavelet sketch—a compact description of the image by a multiscale, subsampled edge map—and provide a corresponding iterative reconstruction algorithm.


  1. B. Bacchelli, M. Bozzini, C. Rabut, M.-L. Varas, "Decomposition and Reconstruction of Multidimensional Signals Using Polyharmonic Pre-Wavelets," Applied and Computational Harmonic Analysis, vol. 18, no. 3, pp. 282-299, May 2005.

  2. J. Duchon, "Splines Minimizing Rotation-Invariant Semi-Norms in Sobolev Spaces," Proceedings of the First International Conference on Multivariate Approximation Theory, Oberwolfach, Germany, April 25-May 1, 1976, pp. 85-100.

  3. B.B. Mandelbrot, J.W. Van Ness, "Fractional Brownian Motions, Fractional Noises and Applications," SIAM Review, vol. 10, no. 4, pp. 422-437, October 1968.

  4. C.A. Micchelli, C. Rabut, F.I. Utreras, "Using the Refinement Equation for the Construction of Pre-Wavelets. III: Elliptic Splines," Numerical Algorithms, vol. 1, no. 4, pp. 331-352, November 1991.

  5. 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.

  6. 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.

  7. D. Van De Ville, M. Unser, "Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid," IEEE Transactions on Image Processing, vol. 17, no. 11, pp. 2063-2080, November 2008.

AUTHOR="Unser, M.",
TITLE="Wavelets and Differential Operators: {F}rom Fractals to {M}arr's
        Primal Sketch",
BOOKTITLE="Fourth French Biyearly Congress on Applied and Industrial
        Mathematics ({SMAI'09})",
address="La Colle sur Loup, French Republic",
month="May 25-29,",
note="Plenary talk")

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