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Splines

Wavelets

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Sampling

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Calculer le charme discret de la continuité

P. Roth

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Horizon, le magazine suisse de la recherche scientifique, no. 79, Décembre 2008.

Le traitement des images médicales a pu être fortement amélioré et accéléré grâce à des fonctions mathématiques appelées splines. Michael Unser a apporté une contribution essentielle dans ce secteur, tant sur le plan de la théorie que des applications.

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Splines: A Perfect Fit for Signal and Image Processing

M. Unser

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IEEE Signal Processing Magazine, vol. 16, no. 6, pp. 22-38, November 1999.

IEEE Signal Processing Society's 2000 Magazine Award

The goals of this article are three-fold:
  • To provide a tutorial on splines that is geared to a signal processing audience.
  • To gather all their important properties, and to provide an overview of the mathematical and computational tools available; i.e., a road map for the practitioner with references to the appropriate literature.
  • To review the primary applications of splines in signal and image processing.

Image Processing with ImageJ

M. Abramoff, P. Magalhães, S. Ram

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Biophotonics International, vol. 11, no. 7, pp. 36-42, July 2004.

As the popularity of the ImageJ open-source, Java-based imaging program grows, its capabilities increase, too. It is now being used for imaging applications ranging from skin analysis to neuroscience.

Image Interpolation and Resampling

P. Thévenaz, T. Blu, M. Unser

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Handbook of Medical Imaging, Processing and Analysis, I.N. Bankman, Ed., Academic Press, San Diego CA, USA, pp. 393-420, 2000.

This chapter presents a survey of interpolation and resampling techniques in the context of exact, separable interpolation of regularly sampled data. In this context, the traditional view of interpolation is to represent an arbitrary continuous function as a discrete sum of weighted and shifted synthesis functions—in other words, a mixed convolution equation. An important issue is the choice of adequate synthesis functions that satisfy interpolation properties. Examples of finite-support ones are the square pulse (nearest-neighbor interpolation), the hat function (linear interpolation), the cubic Keys' function, and various truncated or windowed versions of the sinc function. On the other hand, splines provide examples of infinite-support interpolation functions that can be realized exactly at a finite, surprisingly small computational cost. We discuss implementation issues and illustrate the performance of each synthesis function. We also highlight several artifacts that may arise when performing interpolation, such as ringing, aliasing, blocking and blurring. We explain why the approximation order inherent in the synthesis function is important to limit these interpolation artifacts, which motivates the use of splines as a tunable way to keep them in check without any significant cost penalty.

A Review of Wavelets in Biomedical Applications

M. Unser, A. Aldroubi

Proceedings of the IEEE, vol. 84, no. 4, pp. 626-638, April 1996.

In this paper, we present an overview of the various uses of the wavelet transform (WT) in medicine and biology. We start by describing the wavelet properties that are the most important for biomedical applications. In particular, we provide an interpretation of the continuous WT as a prewhitening multi-scale matched filter. We also briefly indicate the analogy between the WT and some of the biological processing that occurs in the early components of the auditory and visual system. We then review the uses of the WT for the analysis of one-dimensional physiological signals obtained by phonocardiography, electrocardiography (ECG), and electroencephalography (EEG), including evoked response potentials. Next, we provide a survey of recent wavelet developments in medical imaging. These include biomedical image processing algorithms (e.g., noise reduction, image enhancement, and detection of microcalcifications in mammograms); image reconstruction and acquisition schemes (tomography, and magnetic resonance imaging (MRI)); and multiresolution methods for the registration and statistical analysis of functional images of the brain (positron emission tomography (PET), and functional MRI). In each case, we provide the reader with some general background information and a brief explanation of how the methods work. The paper also includes an extensive bibliography.

Wavelets in Medicine and Biology

A. Aldroubi, M.A. Unser, Eds.

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ISBN 0-8493-9483-X, CRC Press, Boca Raton FL, USA, 1996, 616 p.

For the first time, the field's leading international experts have come together to produce a complete guide to wavelet transform applications in medicine and biology. This book provides guidelines for all those interested in learning about waveletes and their applications to biomedical problems.

The introductory material is written for non-experts and includes basic discussions of the theoretical and practical foundations of wavelet methods. This is followed by contributions from the most prominent researchers in the field, giving the reader a complete survey of the use of wavelets in biomedical engineering.

The book consists of four main sections:

  • Wavelet Transform: Theory and Implementation
  • Wavelets in Medical Imaging and Tomography
  • Wavelets and Biomedical Signal Processing
  • Wavelets and Mathematical Models in Biology
  • BibTeX reference
  • Full review of this book
    Akram Aldroubi and Michael Unser, Eds., Wavelets in Medicine and Biology, CRC Press, Boca Raton, FL, 1996.
    A. Bultheel
    Journal of Approximation Theory, vol. 90, no. 3, pp. 458-459, September 1997.
    Wavelets have built a strong reputation in the context of signal and image processing. The editors of this book have invited several specialists to contribute a chapter illustrating this in the (bio)medical and biological sciences.

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