Wavelets have become so popular that distinct communities continue to have conferences and scientific journals entirely devoted to them. There are already historical anecdotes and folklore associated with them; an entertaining account of which can be found in the book of Barbara Burke Hubbard [4]. Readers who want to dive deeper into the subject have the daunting task of choosing among over 200 books written on wavelets. Our only advice in this regard is: in case of doubt, stick with the classics.

Given the size of the phenomenon, it is no surprise that wavelets have had an impact on a number of disciplines, medical imaging being no exception. A first record of activity in this particular area is the workshop on wavelets in medicine and biology that took place at the annual IEEE-EMBS meeting, Baltimore, 1992. The first journal paper describing a wavelet application in medical imaging—noise reduction in MRI by soft-thresholding in the wavelet domain—also appeared in 1992 [5]. Note that this work, which is often overlooked, provides the earliest description of a wavelet denoising method that has become extremely popular through the impulsion of Donoho et al. A large palette of wavelet applications in medical imaging is provided [6]. Two complementary review articles are also available; the first gives a complete account of the activity taking place from the beginning to 1996 [7], while the second covers the more recent papers until 2000 [8]. So far, the primary applications of wavelets in medical imaging have been the following:

• Compression of medical images
• CT reconstruction; local tomography
• Wavelet denoising (MRI, ultrasound)
• Wavelet-based feature extraction; texture and statistical descriptors
• Medical image enhancement (e.g. fluoroscopy, mammography)
• Analysis of functional images of the brain (PET, fMRI)
• Wavelet-encoded MRI

Most of these topics are still active areas of research, as illustrated by the papers that are published in this special issue.

Two other areas where wavelets have achieved great success is signal denoising (typically, by simple thresholding in the wavelet domain) and tomographic reconstruction, mainly, because the Radon operator is well localized in a wavelet basis. Pizurica et al. [14] propose a novel wavelet denoising method that uses a local statistical model for improved signal estimation and noise suppression. Willett and Nowak [15] introduce a new multi-scale image model, using piecewise planar basis functions, and apply their method to the reconstruction of photon-limited data (with Poisson noise). They develop penalized maximum likelihood methods for image denoising, deconvolution, and tomographic reconstruction. Kalifa et al. present a direct method for the efficient reconstruction of PET and SPECT data [16]. Their approach includes a non-linear noise reduction step that is implemented by thresholding in the transformed domain; the key here is to select a transform (wavelet packet) that is optimized for the problem and data at hand (sparse representation of the signal and near diagonalization of the Radon operator). Bonnet et al. [17] also develop a direct approach for the reconstruction of cone-beam data which is known to be challenging. In essence, their approach is a wavelet adaptation of the Feldkamp algorithm.

The special issue also features two contributions relating to ultrasound imaging. Michailovich and Adam consider the problem of the estimation of the spectrum of a ultrasound pulse [18]. Specifically, they develop a modified (outlier-resistant) wavelet estimator that they apply to the log-spectrum of the RF-sequence. Lee et al. present a pattern recognition system that uses an M-band wavelet filterbank to extract fractal and texture features from ultrasonic images of the liver [19]. They report promising classification results differientiating normal liver, cirrhosis, and hepatoma using a hierarchical classifier.

The recent development of commercial digital mammography imaging systems not only provides a significant improvement in image quality for traditional screening, but translates into a wealth of information for the analysis and detection of mammographic features by computer. The papers by Lemaur et al. [20]. and Heinlein et al. [21] focus on the goals of early detection and visual enhancement of microcalcifications, respectively. In the former, the regularity of a wavelet basis is used to identify microcalcification in clusters. The identification of microcalcifaction in clusters as opposed to individual occurrences is of clinical significance as clusters may suggest the likelihood of malignancy. In the later paper, a discretization of the continuous wavelet transform is development which allows a filterbank to be adapted for the enhancement of mammographic features. This implementation allows for the reconstruction of modified wavelet coefficients at arbitrary scales and orientations without the introduction of artifacts or loss of completeness. The integration of such an interactive enhancement tool into digital mammographic screening systems will be of great importance as the wealth of dynamic range (contrast) provided by digital detectors become generally available to radiologist through the introduction of lower cost softcopy display systems.

The paper of Davatzikos et al. [22] offers another illustration of the versatility of wavelets. Their proposal is to represent the contours of a shape in a wavelet bases and to use this sparse representation to derive active shape models. Their results are promising and significant in terms of providing an automated solution to problems in volume quantification.

The amount of data generated by modern imaging devices is often very large, and ever increasing. Thus, an important problem is to find efficient ways of compressing and encoding this information to facilitate its transmission, storage, and retrieval. Since wavelet transforms provide sparse representations of signals, one of their privileged areas of application has been coding and compression. In fact, wavelets have had so much success in this area that they have already become an integral part of the new JPEG2000 compression standard. Unlike other applications, however, biomedical image compression raises delicate issues; lossy compression must be done carefully so as to preserve all medically relevant information while eventually suppressing irrelevant features such as noise or background. Thus, research in this area must address these specific needs and pay great attention to the issue of validation. The paper by Menegaz and Thiran [23] uses wavelet transform techniques for encoding 3D volumetric data sets to optimally compress the volumetric images while allowing efficient 2D slices reconstruction. Schelkens et al. [24] propose new wavelet methods for coding volumetric medical datasets, and compares them to state-of-the-art compression techniques. The new methods perform favorably for both lossy and lossless compression. The paper of Zixiang Xiong et al. [25] also proposes a new method of wavelet compression that is competitive with other state-of-the-art compression techniques but suggests improved image quality.

A beneficial side effect of all the interdisciplinary activity taking place around wavelets has been to create a common language between mathematicians and engineers and to instill rigor in the formulation of imaging problems. Wavelets have been a cross-cutting magnet encouraging engineers to use more sophisticated mathematics and mathematicians to consider applying their knowledge to real world problems.

As Guest editors, we hope that the papers in this issue will stimulate further progress in this direction. We believe that the best is yet to come. We are particularly grateful to the authors for their timely contributions and to the reviewers who responded quickly with insightful and thorough comments. We would also like to thank Kathy Escher for her editorial assistance and patience. Finally, we want to close by dedicating this special issue to Urs Ruttimann and Jonathan Raz.