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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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TITLE="Complex Wavelet Bases, Steerability, and the {M}arr-Like
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JOURNAL="{IEEE} Transactions on Image Processing",
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