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.
  


• 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. 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.
  
  
• 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.
  
  
• 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.
  

• 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.