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.
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• T. Blu, "Iterated Filter Banks with Rational Rate Changes—Connection with Discrete Wavelet Transforms," IEEE Transactions on Signal Processing, vol. 41, no. 12, pp. 3232-3244, December 1993. A generalization of discrete wavelet theory to fractional scaling factors, a subject on which I did my PhD thesis (in French). A rational filterbank design algorithm is available on the demo page.

Image/Signal restoration

• H. Pan, T. Blu, ""An Iterative Linear Expansion of Thresholds for $ell_1$-based Image Restoration", IEEE Transactions on Image Processing, Vol. 22, no. 9, pp. 3715-3728, September 2013. Solves generic l1-regularized restoration problems using an iterative approximation of the restoration function, that is chosen to belong to a "good" approximation space (i-LET).
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• F. Xue, F. Luisier, T. Blu, ""Multi-Wiener SURE-LET Deconvolution", IEEE Transactions on Image Processing, Vol. 22, no. 5, pp. 1954-1968, May 2013. Solves the deconvolution+denoising problem (with known PSF) by using several elementary deconvolvers (made of three fixed Wiener filters followed by adapted wavelet thresholds) and minimizing the SURE.
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• F. Luisier, T. Blu, P. Wolfe, "A CURE for Noisy Magnetic Resonance Images: Chi-Square Unbiased Risk Estimation", IEEE Transactions on Image Processing, Vol. 21, no. 8, pp. 3454-3466, August 2012. An extension of the SURE-LET denoising approach below to images corrupted by Rician noise (non-additive, typical of MRI images). This paper derives the first instance of an unbiased risk estimate for chi-square statistics (SURE is replaced by CURE).
• F. Luisier, T. Blu, M. Unser, "Image Denoising in Mixed Poisson-Gaussian Noise," IEEE Transactions on Image Processing, vol. 20, no. 3, pp. 696-708, March 2011. An extension of the redundant denoising approach below to images corrupted by Poisson noise (non-additive), further corrupted by a Gaussian additive noise (typical of fluorescence microscopy images).
• 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 has received a 2009 Young Author Best Paper Award from the IEEE Signal Processing Society and was listed in the Reader's Choice column of the Signal Processing Magazine (September 2007 and January 2008 issues). Check our online demo.

Finite Rate of Innovation representations

• H. Pan, T. Blu, P.L. Dragotti, "Sampling Curves with Finite Rate of Innovation", IEEE Transactions on Signal Processing, Vol. 62, no. 2, pp. 458-471, January 2014. Shows how the Finite Rate of Innovation framework can be used to represent (continuously-defined) curves in 2D, to sample edge images at a very low rate and to achieve high-quality superresolution from only one image.
• D. Kandaswamy, T. Blu, D. Van De Ville, "Analytic Sensing: Noniterative Retrieval of Point Sources from Boundary Measurements", SIAM Journal on Scientific Computing, Vol. 31 (4), pp. 3179-3194, 2009. The use of Green's theorem with analytic functions combined with a strong "Finite Rate of Innovation" hypothesis is shown to lead to an annihilation equation, from which the point source positions can be obtained exactly.
• 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. 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.
• 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.

Approximation and sampling

• 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.
• 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, 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.
• 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.
• 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.
• 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.
• P. Thévenaz, T. Blu, M. Unser, "Interpolation Revisited," IEEE Transactions on Medical Imaging, vol. 19, no. 7, pp. 739-758, July 2000.
• 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.
• 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.
• 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

• 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.
• 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.
• 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.
• 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.