Complex BSpline Wavelets 
Investigator: Brigitte Forster 

Summary: By defining Bsplines 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. 

Complex wavelet transforms tend to have better shiftinvariance properties and directional sensitivity than the classical realvalued 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 fractallike 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. 

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) Bsplines with a complex exponent.
We have shown that these extended complex Bsplines 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 onesided predominance in the frequency domain. This latter property is especially interesting for signal analysis: it can yield a decomposition into onesided frequency bands, breaking the Hermitian symmetry (twosided frequency bands) of the traditional realvalued transforms. Another remarkable fact is that we can specify complex Bspline wavelets that closely approximate complexmodulated 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 rotationcovariant 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 imageprocessing 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 imageprocessing 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 realvalued wavelet approaches. 

Collaborations: Prof. Michael Unser, Dr. Thierry Blu 


Funding: Grant 200020101821 from the Swiss National Science Foundation, European HASSIP network 


[2]  B. Forster, T. Blu, M. Unser, "Complex BSplines," Applied and Computational Harmonic Analysis, vol. 20, no. 2, pp. 261282, March 2006.

[3]  B. Forster, T. Blu, M. Unser, "A New Family of Complex RotationCovariant Multiresolution Bases in 2D," Proceedings of the SPIE Conference on Mathematical Imaging: Wavelet Applications in Signal and Image Processing X, San Diego CA, USA, August 48, 2003, vol. 5207, part I, pp. 475479.

[4]  B. Forster, T. Blu, M. Unser, "Complex BSplines and Wavelets," Second International Conference on Computational Harmonic Analysis, Nineteenth Annual Shanks Lecture (CHA'04), Nashville TN, USA, May 2430, 2004.


