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Abstract. This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift-invariant spaces. It is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shiftinvariant spaces by bringing together wavelet theory, frame theory, reproducing kernel Hilbert spaces, approximation theory, amalgam spaces, and sampling. Inspired by applications taken from communication, astronomy, and medicine, the following aspects will be emphasized: (a) The sampling problem is well defined within the setting of shift-invariant spaces. (b) The general theory works in arbitrary dimension and for a broad class of generators. (c) The reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms. These algorithms converge geometrically and are robust in the presence of noise. (d) To model the natural decay conditions of real signals and images, the sampling theory is developed in weighted L p -spaces.

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Abstract-Nonrigid registration of medical images is important for a number of applications such as the creation of population averages, atlas-based segmentation, or geometric correction of functional magnetic resonance imaging (fMRI) images to name a few. In recent years, a number of methods have been proposed to solve this problem, one class of which involves maximizing a mutual information (MI)-based objective function over a regular grid of splines. This approach has produced good results but its computational complexity is proportional to the compliance of the transformation required to register the smallest structures in the image. Here, we propose a method that permits the spatial adaptation of the transformation's compliance. This spatial adaptation allows us to reduce the number of degrees of freedom in the overall transformation, thus speeding up the process and improving its convergence properties. To develop this method, we introduce several novelties: 1) we rely on radially symmetric basis functions rather than B-splines traditionally used to model the deformation field; 2) we propose a metric to identify regions that are poorly registered and over which the transformation needs to be improved; 3) we partition the global registration problem into several smaller ones; and 4) we introduce a new constraint scheme that allows us to produce transformations that are topologically correct. We compare the approach we propose to more traditional ones and show that our new algorithm compares favorably to those in current use.Index Terms-Adaptive bases algorithm, mutual information, nonrigid image registration.

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 f o r biomedical applications. In particular, we provide an interpretation of the continuous wavelet transfom (CWT) as a prewhitening multiscale matched filter. We also briefy 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 f o r the analysis of I-D physiological signals obtained by phonocardiography, electrocardiography (ECG), and electroencephalography (EEG), including evoked response potentials. Next, we provide a survey of recent wavelef developments in medical imaging. These include biomedical image processing algorithms (e.g., noise reduction, image enhancement, and detection of microcalciJcations in ntammograms), 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 ( f l R I ) ) . In each case, we provide the reader with some general background information and a brief explanation of how the methods work.