Guest Editorial: Wavelets in Medical Imaging (2003)

I. Introduction Wavelets are the result of collective efforts that recognized common threads between ideas and concepts that had been independently developed and investigated by distinct research communities. They provide a unifying framework for decomposing images, volumes, and time-series data into their elementary constituents across scale. Although a relatively recent construct, wavelets have become a tool of choice for engineers, physicists, and mathematicians, leading to efficient solutions in time and space frequency analysis problems, as well as a multitude of other applications. One of the consequences is that wavelet methods of analysis and representation are presently having a significant impact on the science of medical imaging and the diagnosis of disease and screening protocols. Because of a powerful underlying mathematical theory, they offer exciting opportunities for the design of new multiresolution image processing algorithms, and novel acquisition methods such as wavelet-encoded magnetic resonance imaging (MRI). This special issue of the IEEE Transactions on Medical Imaging focuses on these recent developments and highlights progress that has been accomplished in the areas related to medical imaging. II. Sizing the Wave: Some Facts and Figures Rarely has a mathematical concept generated so much response and enthusiasm within and between the engineering and mathematical research communities at large. To give a rough idea of the phenomenon, we provide a brief chronology. While wavelets have been traced all the way back to Alfred Haar in 1910 [1], for many, the starting point of their modern history coincides with two publications in the late 1980s by S. Mallat [2] and I. Daubechies [3]. These groundbreaking papers established a solid mathematical footing which would both shape and define the field. In a nutshell, S. Mallat identified the important concept of multiresolution analysis which is the corner stone of modern wavelet theory, while I. Daubechies constructed the first orthogonal wavelet bases that were compactly supported. These two contributions count among the most cited papers in the scientific literature (over 1500 SCI citations each). From that point on, the number of contributions relating to wavelet-applications and theory has increased steadily on the order of 9000 journal papers published to date. This trend is likely to continue as suggested by the strong response to the call for papers for this special issue (over 30 submissions). 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 B. 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, MD, 1992. The first journal paper describing a wavelet application in medical imagingânoise reduction in MRI by soft-thresholding in the wavelet domainâalso appeared in 1992 [5]. Note that this work, which is often overlooked, provided 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 in [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 [positron emission tomography (PET), functional MRI (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. III. Scanning Through the Issue Functional imaging is an area were methods of wavelet processing hold great promise. This particular line of research was initiated by U. Ruttimann, a creative researcher and good friend, who sadly passed away shortly before the publication of his paper in this very journal [9]. Another first rate statistician who was also active in this area at an early stage is J. Raz. By a sad coincidence, he also suffered a sudden death about a week before he was to present his latest results on wavelet analysis of fMRI [10]. Despite the tragic loss of these two pioneers, research in this area is alive and well as exemplified by the first three papers of this issue. Turkheimer et al. [11] consider the problem of the analysis of dynamic PET data; in particular, they advocate the use of a linear (James-Stein) wavelet estimator as an alternative to the more classical wavelet shrinkage or threshold detectors. Hossein-Zadeh et al. propose a wavelet-technique for the detection of activation in fMRI data [12]. Their contribution is twofold: first, the use of a redundant wavelet transform for better translation invariance, and second, a nonparametric detection method based on a randomization procedure. F. Meyer also considers fMRI time series but applies wavelets differently, within the context of a generalized linear model, to detrend the data [13]; that is, to get rid of signal drifts and disturbances that are not related to the stimulus. 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 multiscale 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 single photom emission computed tomography data [16]. Their approach includes a nonlinear 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 radio-freqency 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 developed 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 informat