Summary

By defining B-splines with complex exponents, we are able to construct complex wavelet bases that are tunable in a continuous fashion, and that can closely approximate Gabor functions.

Introduction

Complex wavelet transforms tend to have better shift-invariance properties and directional sensitivity than the classical real-valued separable ones. This makes them particularly attractive for image processing. The complex wavelets that are known so far are all based on filterbank design; the major limitation of this type of approach is that the convergence and analytical properties of the underlying basis functions are not well understood—this means that the current complex wavelets are fractal-like and lack in regularity, and that it is difficult to control their analytical properties. Ideally, we would like their real part to be the Hilbert transform of the imaginary part.

Main Contribution

In this work, we take an alternative point of view and specify a whole new family of complex wavelets, starting from the properties of the underlying function spaces. The key idea lies in defining (fractional) B-splines with a complex exponent.

We have shown that these extended complex B-splines inherit most of the properties of their classical polynomial counterparts, and that they can generate a variety of multiresolution bases. The basis functions are tunable with respect to two parameters. The real part of the complex exponent (degree) is responsible for the smoothness, whereas the imaginary part induces a one-sided predominance in the frequency domain. This latter property is especially interesting for signal analysis: it can yield a decomposition into one-sided frequency bands, breaking the Hermitian symmetry (two-sided frequency bands) of the traditional real-valued transforms. Another remarkable fact is that we can specify complex B-spline wavelets that closely approximate complex-modulated Gaussians (Gabor functions), even for small degrees. This is extremely interesting because it is, to the best of our knowledge, the first example of wavelets that are optimally localized in time and frequency in the sense of Heisenberg's uncertainty principle.

Our second approach to complex wavelets concentrates on the idea of rotation invariance. Here, our elementary building blocks are generalized complex polyharmonic basis functions that are rotation-covariant in the sense that they can be rotated by simple multiplication by a phase factor. By localizing these functions, we are able to specify a wavelet transform with the following remarkable behavior: a rotation of the analyzed image results in a phase shift in the coefficients, but leaves the absolute value approximately unchanged. Thus, image rotations can be detected in the phase, but do not affect image-processing methods operating on the magnitudes of the coefficients. Also in this case, we introduce a degree of freedom that allows us to tune the regularity of the wavelet basis according to the requirements of the image-processing task.

In both cases, we have developed a fast implementation of the wavelet transform in the Fourier domain. We have tested both methods for denoising of MR images and found that they outperform classical real-valued wavelet approaches.