In this paper we present a brief account of the use of the spectral theory of slanted
matrices in frame and sampling theory. Some abstract results on slanted matrices are also
presented.
We introduce the notion of operators localized with respect to frames and prove the boundedness of such operators
on families of Banach spaces. This generalizes the method used in proving boundedness of pseudodifferential
operators on various spaces. We also use this notion to provide sufficient conditions for the construction of frames
which have the localization property.
Nonrigid registration of medical images is an important procedure in many aspects of current biomedical and bioengineering research. For example, it is a necessary step for studying the variation of biological tissue properties, such as shape or diffusion properties across population, compute population averages, or atlas-based segmentation. Recently we have introduced the Adaptive Bases registration algorithm as a general method for performing nonrigid registration of medical images and we have shown it to be faster and more accurate than existing algorithms of the same class. The overall properties of the Adaptive Bases algorithm are reviewed here and the method is validated on applications that include the computation of average images, atlas based segmentation, and motion correction of video images. Results show the Adaptive Bases algorithm to be capable of producing high quality nonrigid matches for the applications above mentioned.
KEYWORDS: Image registration, Detection and tracking algorithms, Optimization (mathematics), Medical imaging, Image resolution, 3D acquisition, Magnetic resonance imaging, 3D image processing, Image fusion, Image segmentation
A number of methods have been proposed recently to solve nonrigid registration problems. One of these involves optimizing a Mutual Information (MI) based objective function over a regularly spaced grid of basis functions. This approach has produced good results but its computational complexity is inversely proportional to the compliance of the transformation. Transformations able to register two high resolution images on a very local scale need a large number of degrees of freedom. Finding an optimum in such a search space is lengthy and prone to convergence to local maxima. In this paper, we propose a modification to this class of algorithms that reduces their computational complexity and improves their convergence properties. The approach we propose adapts the compliance of the transformation locally. Registration is achieved iteratively, from a coarse to a fine scale. At each level, the gradient of the cost function with respect to the coefficients of a set of compactly supported radial basis functions spread over a regular grid is used to estimate a local adaptation of the grid. Optimization is then conducted over the estimated irregular grid one region at a time. Results show the advantage of the approach we propose over a method without local grid adaptation.
When performing registrations, it is often crucial to maintain certain structure of the template data T - the data being deformed into the subject data S - as well as to keep the deformation field smooth. Current approaches to registration often impose smoothness through heuristic means, but building it into the model has proven to be more difficult due mainly to computational constraints.
In this paper, we consider the p-frame property of the space V_p(\Phi) with the generator \Phi being compactly supported function. Moreover, for the one-dimensional case, we show that the p-frame and p-Riesz basis properties are essentially the same for the space V_p(\Phi).
High-resolution optical mapping is an emerging technique to record the activation and propagation of transmembrane potential on the surface of cardiac tissues. Important electrodynamic information previously not available from extracellular electric recording could be extracted from these detailed optical recordings. The noise contamination in the images is a major obstacle that prohibits higher level of information extraction. Because the patterns of interest contain sharp wavefronts and structures that we wish to detect and track in a series of frames, we seek to perform denoising based on wavelet decomposition approaches. Among the wavelet denoise methods that were tested in this preliminary study, the wavelet packet produced the best results that could be extended to denoise the entire image sequence for multi-dimensional information processing.
We discuss a method for initializing the multi-wavelet decomposition algorithm by pre-filtering. The proposed pre- filtering operation projects the input signal into the space defined by the multi-scaling function associated with the multi-wavelet. Since the approach is projection based, it is guaranteed to always have a solution. The space in which the original signal is contained is defined by multiple generating functions, making this work a generalization of our previous results.
Orthogonal, semiorthogonal and biorthogonal wavelet bases are special cases of oblique multiwavelet bases. One of the advantage of oblique multiwavelets is the flexibility they provide for constructing bases with certain desired shapes and/or properties. The decomposition of a signal in terms of oblique wavelet bases is still a perfect reconstruction filter bank. In this paper, we present several examples that show the similarity and differences between the oblique and other types of wavelet bases. We start with the Haar multiresolution to illustrate several examples of oblique wavelet bases, and then use the Cohen-Daubechies-Plonka multiscaling function to construct several oblique multiwavelets.
We present examples of a new type of wavelet basis functions that are orthogonal across shifts, but not across scales. The analysis functions are low order splines while the synthesis functions are polynomial splines of higher degree n2. The approximation power of these representations is essentially as good as that of the corresponding Battle- Lemarie orthogonal wavelet transform, with the difference that the present wavelet synthesis filters have a much faster decay. This last property, together with the fact that these transformation s are almost orthogonal, may be useful for image coding and data compression.
Often, the Discrete Wavelet Transform is performed and implemented with the Daubechies wavelets, the Battle-Lemarie wavelets or the splines wavelets whereas in continuous time wavelet decomposition a much larger variety of mother wavelets are used. Maintaining the dyadic time-frequency sampling and the recursive pyramidal computational structure, we present various methods to obtain any chosen analyzing wavelet (psi) w, with some desired shape and properties and which is associated with a semi-orthogonal multiresolution analysis or to a pair of bi-orthogonal multiresolutions. We explain in details how to design one's own wavelet, starting from any given Multiresolution Analysis or any pair of bi-orthogonal multiresolutions. We also explicitly derive, in a very general oblique (or bi-orthogonal) framework, the formulae of the filter bank structure that implements the designed wavelet. We illustrate these wavelet design, techniques with examples that we have programmed with Matlab routines, available upon request.
We construct oblique multi-wavelets bases which encompass the orthogonal multi-wavelets and the biorthogonal uni-wavelets of Cohen, Deaubechies and Feauveau. These oblique multi- wavelets preserve the advantages of orthogonal and biorthogonal wavelets and enhance the flexibility of wavelet theory to accommodate a wider variety of wavelet shapes and properties. Moreover, oblique multi-wavelets can be implemented with fast vector-filter-bank algorithms. We use the theory to derive a new construction of biorthogonal uni-wavelets.
We present a general framework for the design and efficient implementation of various types of running (or over-sampled) wavelet transforms (RWT) using polynomial splines. Unlike previous techniques, the proposed algorithms are not necessarily restricted to scales that are powers of two; yet they all achieve the lowest possible complexity: O(N) per scale, where N is signal length. In particular, we propose a new algorithm that can handle any integer dilation factor and use wavelets with a variety of shapes (including Mexican-Hat and cosine-Gabor). A similar technique is also developed for the computation of Gabor-like complex RWTs. We also indicate how the localization of the analysis templates (real or complex B-spline wavelets) can be improved arbitrarily (up to the limit specified by the uncertainty principle) by increasing the order of the splines. These algorithms are then applied to the analysis of EEG signals and yield several orders of magnitude speed improvement over a standard implementation.
An extrapolation problem for multiresolution is studied. Solutions involve Hankel type operators defined in terms of the generating sequence of the scaling function of the multiresolution.
We study the general problem of oblique projections in discrete shift-invariant spaces of l2 and we give error bounds on the approximation. We define the concept of discrete multiresolutions and wavelet spaces and show that the oblique projections on certain subclasses of discrete multiresolutions and their associated wavelet spaces can be obtained using perfect reconstruction filter banks. Therefore we obtain a discrete analog of the Cohen-Daubechies- Feauveau results on biorthogonal wavelets.
We present an iterative multiresolution algorithm for the translational and rotational alignment of digital images. An image is represented by an interpolating spline. Coarser versions of this continuous image model are obtained by using spline approximations at various scales (polynomial spline pyramid). We use a coarse-to-fine updating strategy to compute the alignment parameters iteratively, using a variation of the Levenberg-Marquardt non-linear least-squares optimization method. This approach yields very precise image registration with subpixel accuracy. It is also much faster and more robust than a comparable single-scale implementation, because the resolution of the underlying image mode is adapted to the step size of the algorithm.
We use approximation theory to generalize the classical sampling procedure of Shannon. We give a link between this theory and the theory of wavelet transforms. As an application, we give a general method for constructing scaling and wavelet functions with specifiable properties.
We present two methods for generating frames of a Hilbert space H. The first method uses bounded operators on H to transform a frame into another frame of H1 C H. The other method uses bounded linear operators on l2 to generate frames of H. We characterize all the mappings that transform frames into other frames. We also show how to construct all frames of a given Hilbert space H starting from any given frame. We show how to apply the theory to obtain shift-invariant tight frames, and shift-invariant tight multiresolution. We also show how to obtain scaling function and wavelets with prescribed properties. Finally, we discuss the noise reduction properties of frames.
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