EPFL | Biomedical Imaging Group

Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid

D. Van De Ville, M. Unser

IEEE Transactions on Image Processing, vol. 17, no. 11, pp. 2063–2080, November 2008.

Our aim in this paper is to tighten the link between wavelets, some classical image-processing operators, and David Marr's theory of early vision. The cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the Gradient-Laplace operator. Starting from first principles, we show that a single-generator wavelet can be defined analytically and that it yields a semi-orthogonal complex basis of L₂(ℝ²), irrespective of the dilation matrix used. We also provide an efficient FFT-based filterbank implementation. We then propose a slightly redundant version of the transform that is nearly translation-invariant and that is optimized for better steerability (Gaussian-like smoothing kernel). We call it the Marr-like wavelet pyramid because it essentially replicates the processing steps in Marr's theory of early vision.We use it to derive a primal wavelet sketch which is a compact description of the image by a multiscale, subsampled edge map. Finally, we provide an efficient iterative algorithm for the reconstruction of an image from its primal wavelet sketch.

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AUTHOR="Van De Ville, D. and Unser, M.",
TITLE="Complex Wavelet Bases, Steerability, and the {M}arr-Like
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JOURNAL="{IEEE} Transactions on Image Processing",
YEAR="2008",
volume="17",
number="11",
pages="2063--2080",
month="November",
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