Wavelets on non-Euclidean manifolds

This dissertation investigates wavelets as a multiscale tool on non-Euclidean manifolds. The growing importance of using non-Euclidean manifolds as a geometric model for data comes from the diversity of the data collected. In this work we mostly deal with the sphere and the hyperboloid. First, given the recent success of the continuous wavelet transform on the sphere a natural extension is to build discrete frames. Then, from a more theoretical perspective, having already wavelets on the sphere, which is a non-Euclidean manifold of constant positive curvature, it is interesting and even challenging to build and prove the existence of wavelets on its dual manifold-the hyperboloid as non-Euclidean manifold of constant negative curvature. This dissertation starts with detailing the construction of one- and two-dimensional Euclidean wavelets in both continuous and discrete versions. Next, it continues with details on the construction of wavelets on the sphere. In the three cases (line, plane and sphere) the group theoretical approach for constructing wavelets is used. We develop discrete wavelet frames on the sphere by discretizing the existing spherical continuous wavelet transform. First, half-continuous wavelet frames are derived. Second, we show that a controlled frame may be constructed in order to get an easy reconstruction of functions from their decomposition coefficients. Finally we completely discretize the continuous wavelet transform on the sphere and give examples of frame decomposition of spherical data. As a close parent of the wavelet transform we also implement the Laplacian Pyramid on the sphere. Another important part of this dissertation is dedicated to the hyperboloid. We build a total family of functions, in the space of square-integrable functions on the hyperboloid, by picking a probe with suitable localization properties, applying on it hyperbolic motions and supplemented by appropriate dilations. Based on a minimal set of axioms, we define appropriate dilations for the hyperbolic geometry. Then, the continuous wavelet transform on the hyperboloid is obtained by convolution of the scaled wavelets with the signal. This transform is proved to be a well-defined invertible map, provided the wavelets satisfy an admissibility condition. As a final part in this dissertation, we discuss one possible application of non-Euclidean wavelets – the processing of non-Euclidean images. This leads to implementing some other basic non-Euclidean image processing techniques, for example scale-space analysis and active contour, that we apply to catadioptric image processing.

Optimizing Wavelets for the Analysis of fMRI Data

Thierry Blu, Michaël Unser

Ruttimann et al. have proposed to use the wavelet transform for the detection and localization of activation patterns in functional magnetic resonance imaging (fMRI). Their main idea was to apply a statistical test in the wavelet domain to detect the coefficients that are significantly different from zero. Here, we improve the original method in the case of non-stationary Gaussian noise by replacing the original z-test by a t-test that takes into account the variability of each wavelet coefficient separately. The application of a threshold that is proportional to the residual noise level, after the reconstruction by an inverse wavelet transform, further improves the localization of the activation pattern in the spatial domain. A key issue is to find out which wavelet and which type of decomposition is best suited for the detection of a given activation pattern. In particular, we want to investigate the applicability of alternative wavelet bases that are not necessarily orthogonal. For this purpose, we consider the various brands of fractional spline wavelets (orthonormal, B-spline, and dual) which are indexed by a continuously-varying order parameter α. We perform an extensive series of tests using simulated data and compare the various transforms based on their false detection rate (type I + type II errors). In each case, we observe that there is a strongly optimal value of α and a best number of scales that minimizes the error. We also find that splines generally outperform Daubechies wavelets and that they are quite competitive with SPM (the standard analysis method used in the field), although it uses much simpler statistics. An interesting practical finding is that performance is strongly correlated with the number of coefficients detected in the wavelet domain, at least in the orthonormal and B-spline cases. This suggest that it is possible to optimize the structural wavelet parameters simply by maximizing the number of wavelet counts, without any prior knowledge of the activation pattern. Some examples of analysis of real data are also presented.

SPIE2000

Guest Editorial: Wavelets in Medical Imaging

Michaël Unser

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 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, 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 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 two-dimensional 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 Z. 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. IV. Conclusion After having completed the tasks of putting this issue together and based on our collective knowledge in the field, it is our belief that wavelets are here to stay. The idea of decomposing a signal according to scale is as fundamental a notion for imaging as decomposing it into harmonic components (Fourier analysis); it is more intuitive and closer to what the visual system does. Another important point that has stimulated wavelet research is the quest for sparse representations of signals, as these offer obvious advantages for data compression, noise reduction and regularization purposes. As more progress is made in the years ahead, it is quite possible that the importance of wavelets as a research topic in its own right may diminish and that they will progressively reach the status of standard toolbox components for data processing (such as the fast Fourier transform). Indeed, we are seeing some of this maturity in the development of emerging commercial image analysis software and packages. Yet, given the flexibility of this powerful analysis tool, there is not much risk in betting that wavelets will continue to be widely used in a diversity of medical imaging applications. 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 K. Escher for her editorial assistance and patience. Finally, we want to close by dedicating this special issue to U. Ruttimann and J. Raz. References A. Haar, "Zur Theorie der Orthogonalen Funktionensysteme," Annals of Mathematics, vol. 69, pp. 331-371, 1910. S.G. Mallat, "A Theory of Multiresolution Signal Decomposition: The Wavelet Representation," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 11, no. 7, pp. 674-693, July 1989. I. Daubechies, "Orthogonal Bases of Compactly Supported Wavelets," Communications on Pure and Applied Mathematics, vol. 41, pp. 909-996, November 1988. B.B. Hubbard, The World According to Wavelets: The Story of a Mathematical Technique in the Making. A.K. Peters Ltd., Wellesley MA, USA, 286 p., 1998. J.B. Weaver, X. Yansun, D.M. Healy, Jr., L.D. Cromwell, "Filtering Noise from Images with Wavelet Transforms," Magnetic Resonance in Medicine, vol. 21, no. 2, pp. 288-295, 1991. A. Aldroubi, M.A. Unser, Wavelets in Medicine and Biology. CRC Press, Boca Raton FL, USA, 616 p., 1996. M. Unser, A. Aldroubi, "A Review of Wavelets in Biomedical Applications," Proceedings of the IEEE, vol. 84, no. 4, pp. 626-638, April 1996. A. Laine, "Wavelets in Temporal and Spatial Processing of Biomedical Images," Annual Review in Biomedical Engineering, vol. 2, pp. 511-550, 2000. U.E. Ruttimann, M. Unser, R.R. Rawlings, D. Rio, N.F. Ramsey, V.S. Mattay, D.W. Hommer, J.A. Frank, D.R. Weinberger, "Statistical Analysis of Functional MRI Data in the Wavelet Domain," IEEE Transactions on Medical Imaging, vol. 17, no. 2, pp. 142-154, April 1998. J. Raz, "FDR Wavelet Shrinkage Estimators for Inhomogeneous Poisson Processes," Proceedings of the SPIE Conference on Mathematical Imaging: Wavelet Applications in Signal and Image Processing VIII, San Diego CA, USA, July 31-August 4, 2000, vol. 4119, pp. . F.E. Turkheimer, J.A.D. Aston, R.B. Banati, C. Riddell, V.J. Cunningham , "A Linear Wavelet Filter for Parametric Imaging with Dynamic PET," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 289-301, March 2003. G.-A. Hossien-Zadeh, H. Soltanian-Zadeh, B.A. Ardekani, "Multiresolution fMRI Activation Detection Using Translation Invariant Wavelet Transform and Statistical Analysis Based on Resampling," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 302-314, March 2003. F.G. Meyer, "Wavelet Based Estimation of a Semi Parametric Generalized Linear Model of fMRI Time-Series," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 315-322, March 2003. A. Pizurica, W. Philips, "A Versatile Wavelet Domain Noise Filtration Technique for Medical Imaging," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 323-331, March 2003. R. Willett, R. Nowak, "Platelets: A Multiscale Approach for Recovering Edges and Surfaces in Photon Limited Medical Imaging," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 332-350, March 2003. J. Kalifa, A. Laine, P.D. Esser, "Regularization in Tomographic Reconstruction Using Thresholding Estimators," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 351-359, March 2003. S. Bonnet, F. Peyrin, F. Turjman, R. Prost, "Nonseparable Wavelet-Based Cone-Beam Reconstruction in 3-D Rotational Angiography," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 360-367, March 2003. O. Michailovich, D. Adam, "Robust Estimation of Ultrasound Pulses Using Outlier-Resistant De-Noising," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 368-381, March 2003. W.-L. Lee, Y.-C. Chen, K.-S. Hsei, "Ultrasonic Liver Tissues Classification by Fractal Feature Vector Based on M-Band Wavelet Transform," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 382-392, March 2003. G. Lemaur, K. Drouiche, J. DeConinck, "Highly Regular Wavelets for the Detection of Clustered Microcalcifications in Mammograms," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 393-401, March 2003. P. Heinlein, J. Drexl, W. Schneider, "Integrated Wavelets for Enhancement of Microcalcifications in Digital Mammography," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 402-413, March 2003. C. Davatzikos, X. Tao, D. Shen , "Hierarchical Active Shape Models, Using the Wavelet Transform," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 414-423, March 2003. G. Menegaz, J.-P. Thiran, "3D Encoding/2D Decoding of Medical Data," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 424-440, March 2003. P. Schelkens, A. Munteanu, J. Barbarien, M. Galca, X.G.I. Nieto, J. Cornelis, "Wavelet Coding of Volumetric Medical Datasets," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 441-458, March 2003. Z. Xiong, X. Wu, S. Cheng, J. Hua, "Lossy-To-Lossless Compression of Medical Volumetric Data Using Three-Dimensional Integer Wavelet Transforms," IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 459-470, March 2003.

2003

Transforms and Operators for Directional Bioimage Analysis: A Survey

Zsuzsanna Püspöki, Daniel Sage, Martin Kurt Storath, Michaël Unser

We give a methodology-oriented perspective on directional image analysis and rotation-invariant processing. We review the state of the art in the field and make connections with recent mathematical developments in functional analysis and wavelet theory. We unify our perspective within a common framework using operators. The intent is to provide image-processing methods that can be deployed in algorithms that analyze biomedical images with improved rotation invariance and high directional sensitivity. We start our survey with classical methods such as directional-gradient and the structure tensor. Then, we discuss how these methods can be improved with respect to robustness, invariance to geometric transformations (with a particular interest in scaling), and computation cost. To address robustness against noise, we move forward to higher degrees of directional selectivity and discuss Hessian-based detection schemes. To present multiscale approaches, we explain the differences between Fourier filters, directional wavelets, curvelets, and shearlets. To reduce the computational cost, we address the problem of matching directional patterns by proposing steerable filters, where one might perform arbitrary rotations and optimizations without discretizing the orientation. We define the property of steerability and give an introduction to the design of steerable filters. We cover the spectrum from simple steerable filters through pyramid schemes up to steerable wavelets. We also present illustrations on the design of steerable wavelets and their application to pattern recognition.