Wavelets

Wavelets are both a powerful conceptual tool and form the basis of many fast and robust algorithms. They are at the core of many of the digital image processing tools developed in the lab. Specifically, we developped wavelets and spline-like basis functions tailored to the specificities of digital holography (Fresnelets) and computerized tomography. We are particularly interested in applications related to optics, such as the modeling of optical systems, phase-retrieval, digital holography, interferometry, microscopy, or to all forms of biomedical imaging.

Fresnelets 0

Dancing Fresnelets. Fresnelets (in red) are wavelet-like functions that are generated from B-splines. B-splines can be constructed using a weighted sum of shifted polynomials (shown in black and blue). Here, Fresnelets are generated from a B-spline of degree 0 (centered indicator function) and their parameter τ=√(λd), where λ is the wavelength and d the propagation distance, is varied from 0 to 4.5… and back.

Fresnelets 1

In this example, Fresnelets are generated from a B-spline of degree 1, that is, the ‘hat’ function (recognizable when τ=0).

Fresnelets 3

As the degree of the B-spline is increased—here, the degree is 3—it converges to a Gaussian function. When compared to the Fresnelets generated from B-splines of smaller degree, the spatial spreading is kept to a minimum as the propagation parameter τ increases.

Related activites

ECE 258B: Multirate Digital Signal Processing: graduate class on wavelets and more.

Since 2005, Michael Liebling is a member of the technical committee for the Wavelets conference (held every other year as part of SPIE's Optics & Photonics symposium in San Diego, CA) during which he organized special sessions on Wavelets in Optics (2005) and Wavelets in Bio-imaging (2009; together with Jean-Christophe Olivo-Marin).

While a graduate student at EPFL, Michael Liebling organized the 2003 Catch-a-Wavelet seminar series.

Selected publications

M. Liebling, H. Ranganathan, "Wavelet Domain Mutual Information Synchronization of Multi-Modal Cardiac Microscopy Image Sequences," invited paper, Proceedings of the SPIE Conference Optics & Photonics, Wavelets XIII, San Diego CA, USA, August 2-4, 2009, vol. 7446, 744602, 6 pages, doi: 10.1117/12.826519.

M. Liebling, A. S. Forouhar, M. Gharib, S. E. Fraser, M. E. Dickinson, "Four-Dimensional Cardiac Imaging in Living Embryos via Postacquisition Synchronization of Nongated Slice Sequences," Journal of Biomedical Optics, vol. 10, no. 5, eid 054001, 10 pages, 2005.

M. Liebling, T.-F. Bernhard, A. H. Bachmann, L. Froehly, T. Lasser, M. Unser, "Continuous Wavelet Transform Ridge Extraction for Spectral Interferometry Imaging," Proceedings of the SPIE Conference BiOS 2005, Coherence domain optical methods and optical coherence tomography in biomedicine IX, San Jose CA, USA, January 22-27, 2005, vol. 5690, pp 397-402.

M. Liebling, T. Blu, M. Unser, "Fresnelets: New Multiresolution Wavelet Bases for Digital Holography," IEEE Transactions on Image Processing, vol. 12, no. 1, pp. 29-43, January 2003.