Research activities
I mostly work on wavelets, multiresolution and approximation with
applications like functional MRI, Optical Coherence Tomography and,
generally speaking, Image Processing. Below
is a commented selection of recent collaborated papers (see here for a more complete list of my publications):
Wavelet theory
- B. Forster, T. Blu, M. Unser, "Complex B-Splines," Applied and Computational Harmonic Analysis, vol. 20, no. 2, pp. 261-282, March 2006.
A follow-up of the fractional spline theory, in which the initially
real parameters are allowed to take complex values.
- M. Unser, T. Blu, "Wavelet Theory
Demystified," IEEE Transactions on Signal Processing, vol. 51, no.
2, pp. 470-483, February 2003. Which shows that, wavelet theory can
essentially be summarized by spline theory.
- T. Blu, M. Unser, "Wavelets,
Fractals, and Radial Basis Functions," IEEE Transactions on Signal
Processing, vol. 50, no. 3, pp. 543-553, March 2002. In
particular, we show here that every dyadic wavelet can be expressed as
a sum of "harmonic" splines. This paper has received a 2003
Best Paper Award from the IEEE Signal Processing Society (Signal
Processing Theory and Methods).
- M. Unser, T. Blu, "Fractional
Splines and Wavelets," SIAM Review, vol. 42, no. 1,
pp. 43-67, March 2000. This is an extension of splines to
non-integer (and even negative) degrees, with their
properties; a demo, a
MATLAB
software implementing the spline wavelet transform and some graphical art are also
included.
Image/Signal denoising
- T. Blu, F. Luisier, "The SURE-LET Approach to Image Denoising,"
IEEE Transactions on Image Processing, vol. 16, no. 11, pp. 2778-2786,
November 2007.
A generalization of the framework of the March 2007 paper below,
emphasizing image denoising with redundant transformations (not
necessarily wavelets).
- F. Luisier, T. Blu, M. Unser, "A New SURE
Approach to Image Denoising: Inter-Scale Orthonormal Wavelet
Thresholding," IEEE Transactions on Image Processing, vol. 16, no. 3, pp. 593-606, March 2007.
Thanks to a statistical unbiased estimate of the MSE (the SURE), we show how it is possible to
optimize the non-linear denoising of joint wavelet subbands, without making
any assumption on the non-noisy underlying image. This paper is listed in the Reader's Choice column
of the Signal Processing Magazine (September 2007 and January 2008 issues). Check our online demo.
Approximation and sampling
- P.L. Dragotti, M. Vetterli, T. Blu, "Sampling Moments and Reconstructing Signals of Finite Rate of Innovation: Shannon Meets Strang-Fix,"
IEEE Transactions on Signal Processing, vol. 55, no. 5, part 1, pp.
1741-1757, May 2007. New windows and related algorithms for sampling
signals with infinite bandwidth. Interestingly, we show that windows
that satisfy a well-known approximation condition (Strang-Fix) are also
particularly suited for sampling these types of signals.
- M. Unser, T. Blu, "Self-Similarity: Part I—Splines and Operators," IEEE Transactions on Signal Processing, vol. 55, no. 4, pp. 1352-1363, April 2007.
Where we show the intimate link between scale invariant operators and splines.
- T. Blu, M. Unser, "Self-Similarity: Part II—Optimal Estimation of Fractal Processes," IEEE Transactions on Signal Processing, vol. 55, no. 4, pp. 1364-1378, April 2007.
We show that the best estimate of a fractional Brownian
motion given its samples is a fractional spline and we compute its
expected approximation error.
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T. Blu, P. Thévenaz, M. Unser, "Linear Interpolation Revitalized,"
IEEE Transactions on Image Processing, , vol. 13, no. 5, pp. 710-719, May 2004. Which shows that
piecewise linear interpolation should be performed by shifting the
sampling knots by 0.21. A demo is available to exemplify this counterintuitive result.
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T. Blu, P. Thévenaz, M. Unser, "Complete Parameterization of Piecewise-Polynomial Interpolation Kernels,"
IEEE Transactions on Image Processing, vol. 12, no. 11, pp. 1297-1309,
November 2003. For practitioneers who need to tune interpolation
kernels to their own specific application.
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J. Kybic, T. Blu, M. Unser, "Generalized Sampling: A Variational Approach—Part I: Theory,Part II: Applications," IEEE Transactions on Signal Processing, vol. 50, no. 8, pp. 1965-1985, August 2002.
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M. Vetterli, P. Marziliano, T. Blu, "Sampling Signals with Finite Rate of Innovation,"
IEEE Transactions on Signal Processing, vol. 50, no. 6, pp. 1417-1428,
June 2002. A parametric approach to sampling/interpolation problems. This paper has received a 2006 Best Paper Award from the IEEE Signal Processing Society.
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M. Jacob, T. Blu, M. Unser, "Sampling of Periodic Signals: A Quantitative Error Analysis,"
IEEE Transactions on Signal Processing, vol. 50, no. 5, pp. 1153-1159,
May 2002. An application of the older 1999 results to the approximation
of periodic functions.
- T. Blu, P. Thévenaz, M. Unser, "MOMS:
Maximal-Order Interpolation of Minimal Support," IEEE Transactions
on Image Processing, vol. 10, no. 7, pp. 1069-1080, July
2001. Here, we compute the optimal wavelet-like (i.e., generated
by a shifted function) space for approximating low-pass signals, given
the support size of its generating function.
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P. Thévenaz, T. Blu, M. Unser, "Interpolation Revisited,"
IEEE Transactions on Medical Imaging, vol. 19, no. 7, pp. 739-758, July 2000.
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T. Blu, M. Unser, "Quantitative
Fourier Analysis of Approximation Techniques: Part I—Interpolators and
Projectors," IEEE Transactions on Signal Processing, vol. 47,
no. 10, pp. 2783-2795, October 1999. An extremely accurate
theory that is able to predict the approximation quality (based
on an L2 measure) of a wavelet-like space, in a way that is independent
of the function to approximate.
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T. Blu, M. Unser, "Quantitative
Fourier Analysis of Approximation Techniques: Part II—Wavelets," IEEE
Transactions on Signal Processing, vol. 47, no. 10, pp. 2796-2806, October 1999. The application of Part I to multiresolution
(or wavelet) spaces, with in particular the result that, asymptotically,
a spline approximation requires p-times less
samples than an approximation with Daubechies wavelets of identical order.
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T. Blu, M. Unser, "Approximation
Error for Quasi-Interpolators and (Multi-) Wavelet Expansions,"
Applied
and Computational Harmonic Analysis, vol. 6, no. 2,
pp. 219-251, March 1999. This paper includes the full mathematical
proofs of
the above IEEE Trans. on SP papers, in a more general setting since it
deals with multi-wavelet like spaces. In particular, we compute the
asymptotic
approximation constant for multi-scaling functions.
Others
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D. Van De Ville, T. Blu, M. Unser, "Integrated Wavelet
Processing and Spatial Statistical Testing of fMRI Data," NeuroImage,
vol. 23, no. 4, pp. 1472-1485, December 2004. Or how to provide a
statistical meaning to the reconstruction of wavelet detections of brain
activation in functional Magnetic Resonance Imaging.
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M. Liebling, T. Blu, M. Unser, "Complex Wave Retrieval from a
Single Off-Axis Hologram,"
Journal of the Optical Society of America A, vol. 21, no. 3, pp.
367-377, March 2004. A non-linear inversion approach to holography—as
opposed to the Fourier filtering approach.
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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.
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T. Blu, H. Bay, M. Unser, "A New High-Resolution Processing Method for the Deconvolution of Optical Coherence Tomography Signals,"
Proceedings of the First 2002 IEEE International Symposium on
Biomedical Imaging: Macro to Nano (ISBI'02), Washington DC, USA, July
7-10, 2002, vol. III, pp. 777-780. A new, parametric approach to OCT
imaging, as well as an exact solution to the inverse problem.
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A. Muñoz Barrutia, T. Blu, M. Unser, "Least-Squares Image Resizing Using Finite Differences," IEEE Transactions on Image Processing, vol. 10, no. 9, pp. 1365-1378, September 2001.