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Approximation power of biorthogonal wavelet expansions

IEEE TRANSACTIONS ON SIGNAL PROCESSING 1996年第3期44卷 519-527页

作者： Unser, M Biomedical Engineering and Instrumentation Program National Institutes of Health Bethesda MD USA

This paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions, The Strang-Fix conditions imply that the error for an orthogonal wavelet approximation at scale alpha = 2(-i) globally decays as alpha(N), where N is the order of the transform, This is why, for a given number of scales, higher order wavelet transforms usually result in better signal approximations, We prove that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone, We also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to;zero, These results strengthen Sweldens' earlier analysis and confirm that the approximation power of biorthogonal and (semi-)orthogonal wavelet expansions is essentially the same, Finally, we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines, We also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error.

关键词： Wavelet transforms Wavelet analysis Approximation error Filtering theory Signal synthesis Constraint theory Continuous wavelet transforms Image coding Image analysis Fractals

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MECHANISMS FOR ELECTRICAL-STIMULATION OF EXCITABLE TISSUE

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CRITICAL REVIEWS IN biomedical engineering 1994年第3-4期22卷 253-305页

作者： ROTH, BJ Biomedical Engineering and Instrumentation Program National Institutes of Health Bethesda Maryland 20892 USA.

Electric fields excite electrically active tissue by several mechanisms. A long, straight, uniform fiber is polarized by an activating function, proportional to the axial gradient of the axial electric field. During unipolar anodal stimulation, the activating function results in two areas of depolarization (virtual cathodes) that are responsible for anode-make stimulation. During unipolar cathodal stimulation, the virtual anodes can be exploited to produce unidirectional propagation and physiological recruitment of axons. Anode-break stimulation of nerves arises from the intrinsic properties of the sodium channel kinetics;cathode-break stimulation in nerves is anode-break stimulation at a virtual anode. The activating function applies to magnetic stimulation as well as to electric stimulation. Other important mechanisms of stimulation arise if the fiber is terminated, nonuniform, or curved. In the brain, cortical neurons are excited when the electric field is directed from the dendrites toward the axon. Possible mechanisms for cortical excitation are the impedance mismatch between the axon and dendritic tree, and the axon bending as it enters the white matter. Transcranial magnetic stimulation differs from transcranial electric stimulation because during magnetic stimulation the electric field is parallel to the brain surface, whereas during electric stimulation the electric field has components both parallel and perpendicular to the brain surface. Cardiac tissue can be represented by use of the bidomain model. This model predicts that a point-source stimulus results in adjacent areas of depolarized and hyperpolarized tissue. The presence of virtual anodes during cathodal stimulation is analogous to the creation of virtual anodes along a one-dimensional fiber by the activating function. Anode- and cathode-break stimulation both occur in cardiac tissue, but the mechanism may be different than for nerve and may depend on diffusion of depolarization into a previously h

关键词： EXCITATION ELECTRIC FIELD ANODE-MAKE ANODE-BREAK CATHODE-MAKE CATHODE-BREAK ACTIVATING FUNCTION VIRTUAL ANODE VIRTUAL CATHODE MAGNETIC STIMULATION BIDOMAIN CRITICAL POINT DEFIBRILLATION

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CRYOELECTRON ENERGY-LOSS SPECTROSCOPY - OBSERVATIONS ON VITRIFIED HYDRATED SPECIMENS AND RADIATION-DAMAGE

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ULTRAMICROSCOPY 1995年第1-4期59卷 71-79页

作者： LEAPMAN, RD SUN, SQ Biomedical Engineering and Instrumentation Program NCRR National Institutes of Health Bethesda MD 20892 USA.

Valence electron energy loss spectroscopy (EELS) has been used to characterize the composition of frozen-hydrated specimens in the electron microscope. Fine structure in the energy range up to 30 eV provides a means of distinguishing between vitreous and crystalline ice. Some features of the ice spectrum can be understood in terms of transitions between molecular orbitals in the water molecule and by the existence of excitons in the solid. Spectra from hydrated biological specimens can be analyzed to obtain quantitative estimates of the water content by fitting contributions from the ice and organic components. EELS also provides information about the radiation chemistry that occurs when hydrated specimens are exposed to the electron beam. From the observation of the hydrogen K-edge at similar to 13 eV, it can be deduced that bubbles of molecular hydrogen are evolved during irradiation at doses of > 10(4) nm(-2), and that these bubbles contain gas at pressures in excess of one thousand atmospheres.

关键词： Animals Cryopreservation Cryopreservation Liver/radiation effects Mice Microscopy, Electron/methods Water

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A review of wavelets in biomedical applications

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PROCEEDINGS OF THE IEEE 1996年第4期84卷 626-638页

作者： Unser, M Aldroubi, A Biomedical Engineering and Instrumentation Program Nauonal Center for Research Resources National Institutes of Health Bethesda MD USA

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 for biomedical applications. In particular, we provide an interpretation of the continuous wavelet transform (CWT) as a prewhitening multiscale matched filter. Me also briefly 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 rises of the WT for the analysis of 1-D physiological signals obtained by phonocardiography, electrocardiography (ECC), and electroencephalography (EEG), including evoked response Next, we provide a survey of recent wavelet developments in medical imaging. These include biomedical image processing algorithms (e.g., noise reduction, image enhancement, and detection of microcalcifications in mammograms), 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 (fMRI)). In each case, we provide the reader with some general background information and a brief explanation of how the methods work.

关键词： Continuous wavelet transforms Magnetic resonance imaging Wavelet transforms biomedical imaging Electrocardiography Electroencephalography Positron emission tomography Matched filters Visual system Signal analysis

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ON THE OPTIMALITY OF IDEAL FILTERS FOR PYRAMID AND WAVELET SIGNAL APPROXIMATION

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 1993年第12期41卷 3591-3596页

作者： UNSER, M Biomedical Engineering and Instrumentation Program National Center for Research Resources National Institutes of Health Bethesda MD USA

The reconstructed lowpass component in a quadrature mirror filter (QMF) bank provides a coarser resolution approximation of the input signal. Since the outputs of the two QMF branches are orthogonal, the transformation that provides the maximum energy compaction in the lowpass channel is also the one that results in the minimum approximation error. This property is used as a common strategy for the optimization of QMF banks, orthogonal wavelet transforms, and least squares pyramids. A general solution is derived for the QMF bank that provides the optimal decomposition of an arbitrary wide sense stationary process. This approach is extended to the continuous case to obtain the minimum error approximation of a signal at a given sampling rate. In particular, it is shown that the sine-wavelet transform is optimal for the representation at all scales of signals with non-increasing spectral density.

关键词： Channel bank filters Least squares approximation Mirrors Filter bank Energy resolution Signal resolution Compaction Approximation error Continuous wavelet transforms Wavelet transforms

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CLASSIFICATION OF NATURAL TEXTURES BY MEANS OF 2-DIMENSIONAL ORTHOGONAL MASKS - COMMENTS

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 1991年第4期39卷 1005-1006页

作者： UNSER, M Biomedical Engineering and Instrumentation Program National Institutes of Health DHHS Bethesda MD USA

The correspondence of Cohen et al. describes a texture analysis method that is a special use of the approach reported in [3]. Moreover, the derivation of the optimum set of masks presented by these authors is based on an incorrect assumption;the correct solution should be the local Karhunen-Loeve transform which is generally different from a 2 x 2 Hadamard transform (2 x 2 DHT). However, this error has relatively minor effects from a practical point of view since the ability of the 2 x 2 DHT to decorrelate real texture images is usually excellent.

关键词： Signal processing algorithms Speech processing biomedical signal processing Frequency estimation Spectral analysis Inspection Prediction algorithms Algorithm design and analysis biomedical engineering Instruments

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A FAMILY OF POLYNOMIAL SPLINE WAVELET TRANSFORMS

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SIGNAL PROCESSING 1993年第2期30卷 141-162页

作者： UNSER, M ALDROUBI, A EDEN, M Biomedical Engineering and Instrumentation Program National Center for Research Resources National Institutes of Health Bethesda MD 20892 USA

This paper presents an extension of the family of orthogonal Battle/Lemarie spline wavelet transforms with emphasis on filter bank implementation. Spline wavelets that are not necessarily orthogonal within the same resoluton level, are constructed by linear combination of polynomial spline wavelets of compact support, the natural counterpart of classical B-spline functions. Mallat's fast wavelet transform algorithm is extended to deal with these non-orthogonal basis functions. The impulse and frequency responses of the corresponding analysis and synthesis filters are derived explicitly for polynomial splines of any order n (n odd). The link with the general framework of biorthogonal wavelet transforms is also made explicit. The special cases of orthogonal, B-spline, cardinal and dual wavelets are considered in greater detail. The B-spline (respectively dual) representation is associated with simple FIR binomial synthesis (respectively analysis) filters and recursive analysis (respectively synthesis) filters. The cardinal representation provides a sampled representation of the underlying continuous functions (interpolation property). The distinction between cardinal and orthogonal representation vanishes as the order of the spline is increased;both wavelets tend asymptotically to the bandlimited sinc-wavelet. The distinctive features of these various representations are discussed and illustrated with a texture analysis example.

关键词： WAVELET TRANSFORM BETA-SPLINES POLYNOMIAL SPLINES INTERPOLATION BINOMIAL FILTERS QUADRATURE MIRROR FILTERS (QMF) BIORTHOGONALITY MULTIRESOLUTION SIGNAL ANALYSIS

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POLYNOMIAL SPLINE SIGNAL APPROXIMATIONS - FILTER DESIGN AND ASYMPTOTIC EQUIVALENCE WITH SHANNONS SAMPLING THEOREM

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IEEE TRANSACTIONS ON INFORMATION THEORY 1992年第1期38卷 95-103页

作者： UNSER, M ALDROUBI, A EDEN, M Biomedical Engineering and Instrumentation Program National Center of Research Resources National Institutes of Health DHHS Bethesda MD USA

The least-squares polynomial spline approximation of a signal g(t) is-an-element-of L2(R) is obtained by projecting g(t) on S(n)(R) (the space of polynomial splines of order n). We show that this process can be linked to the classical problem of cardinal spline interpolation [1] by first convolving g(t) with a B-spline of order n. More specifically, the coefficients of the B-spline interpolation of order 2n + 1 of the sampled filtered sequence are identical to the coefficients of the least-squares approximation of g(t) of order n. We then show that this approximation can be obtained from a succession of three basic operations: prefiltering, sampling, and postfiltering, which confirms the parallel with the classical sampling/reconstruction procedure for bandlimited signals. We determine the frequency responses of these filters for three equivalent spline representations using alternative sets of shift-invariant basis functions of S(n)(R): the standard expansion in terms of B-spline coefficients, a representation in terms of sampled signal values, and a representation using orthogonal basis functions. For the two latter cases, we prove that the frequency response of these filters converge to the ideal lowpass filter pointwise and in all L(p)-norms with 1 less-than-or-equal-to p less-than-or-equal-to infinity as the order of the spline tends to infinity, which establishes the asymptotic equivalence with Shannon's sampling theorem.

关键词： INTERPOLATION B-SPLINES POLYNOMIAL SPLINES SPLINE FILTERS SAMPLING THEOREM LEAST-SQUARES APPROXIMATION ASYMPTOTIC CONVERGENCE

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ON THE ASYMPTOTIC CONVERGENCE OF B-SPLINE WAVELETS TO GABOR FUNCTIONS

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IEEE TRANSACTIONS ON INFORMATION THEORY 1992年第2期38卷 864-871页

作者： UNSER, M ALDROUBI, A EDEN, M Biomedical Engineering and Instrumentation Program National Center of Research Resources National Institutes of Health DHHS Bethesda MD USA

A family of nonorthogonal polynomial spline wavelet transforms is considered. These transforms are fully reversible and can be implemented efficiently. The corresponding wavelet functions have a compact support. It is proven that these B-spline wavelets converge to Gabor functions (modulated Gaussian) pointwise and in all L(p)-norms with 1 less-than-or-equal-to p < + infinity as the order of the spline (n) tends to infinity. In fact, the approximation error for the cubic B-spline wavelet (n = 3) is already less than 3%;this function is also near optimal in terms of its time/frequency localization in the sense that its variance product is within 2% of the limit specified by the uncertainty principle.

关键词： WAVELET TRANSFORM GABOR TRANSFORM UNCERTAINTY PRINCIPLE POLYNOMIAL SPLINE BETA-SPLINES TIME-FREQUENCY LOCALIZATION

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A GENERAL SAMPLING THEORY FOR NONIDEAL ACQUISITION DEVICES

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 1994年第11期42卷 2915-2925页

作者： UNSER, M ALDROUBI, A Biomedical Engineering and Instrumentation Program National Center for Research Resources National Institutes of Health DHHS Bethesda MD USA

We first describe the general class of approximation spaces generated by translation of a function rho(x), and provide a full characterization of their basis functions. We then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle. The theory puts no restrictions on the system input which can be an arbitrary finite energy signal;bandlimitedness is not required. In contrast to previous approaches, this formulation allows for an independent specification of the sampling (analysis) and approximation (synthesis) spaces. In particular, when both spaces are identical, the theorem provides a simple procedure for obtaining the least squares approximation of a signal. We discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited, and polynomial spline signal representations. We also define a spectral coherence function that measures the ''similarity'' between the sampling and approximation spaces, and derive a relative performance bound for the comparison with the least squares solution.

关键词： Sampling methods Least squares approximation Filters Polynomials Spline Signal sampling Interpolation Signal representations Kernel Signal resolution

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