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Mathematical Imaging: Wavelet Applications in Signal and Image Processing 1993

作者： Unser, Michael Biomedical Engineering and Instrumentation Program Bldg 13 Room 3W13 National Center for Research Resources National Institutes of Health BethesdaMD20892 United States

Most orthogonal signal decompositions, including block transforms, wavelet transforms, wavelet packets, and perfect reconstruction filterbanks in general, can be represented by a paraunitary system matrix. This paper considers the general problem of finding the optimal PxP paraunitary transform that minimizes the approximation error when a signal is reconstructed from a reduced number of components Q © 1993 SPIE. All rights reserved.

关键词： Filter banks

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THE L(2) POLYNOMIAL SPLINE PYRAMID

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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 1993年第4期15卷 364-379页

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

The first part of this paper is concerned with the derivation of general methods for the L2 approximation of signals by polynomial splines. Such approximations can be represented in a variety of ways using different sets of shift-invariant basis functions (e.g., cardinal, dual, orthogonal or standard B-splines). The main result is that the expansion coefficients of the approximation are obtained by linear filtering and sampling. The second part applies those results to construct a L2 polynomial spline pyramid that is a parametric multiresolution representation of a signal. This hierarchical data structure is generated by repeated application of a REDUCE function (prefilter and down-sampler). A complementary EXPAND function (up-sampler and post-filter) allows a finer resolution mapping of any coarser level of the pyramid. Four equivalent representations of this pyramid are considered, and the corresponding REDUCE and EXPAND filters are determined explicitly for polynomial splines of any order n (odd). Some image processing examples are presented. The present formulation provides a number of interesting links with several other multiresolution techniques including the wavelet transform, scale-space filtering, and Burt's Gaussian and Laplacian pyramids. In particular, we demonstrate that the performance of the Laplacian pyramid can be improved significantly by using a modified EXPAND function associated with the dual representation of a cubic spline pyramid.

关键词： BINOMIAL FILTERS B-SPLINES DECIMATION INTERPOLATION MULTIGRID METHODS MULTIRESOLUTION POLYNOMIAL SPLINES PYRAMID RECURSIVE FILTERS SCALE SPACE SIGNAL APPROXIMATION SPLINE FILTERS WAVELET TRANSFORM

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Use of the wavelet transform to investigate differences in brain PET images between patient groups 2

Use of the wavelet transform to investigate differences in b...

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Mathematical Methods in Medical Imaging II 1993

作者： Ruttimann, Urs E. Unser, Michael Rio, Daniel Rawlings, Robert R. National Institute on Alcohol Abuse and Alcoholism National Institutes of Health BethesdaMD20892 United States Biomedical Engineering and Instrumentation Program National Institutes of Health BethesdaMD20892 United States

Suitability of the wavelet transform was studied for the analysis of glucose utilization differences between subject groups as displayed in PET images. To strengthen statistical inference, it was of particular interest investigating the tradeoff between signal localization and image decomposition into uncorrected components. This tradeoff is shown to be controlled by wavelet regularity, with the optimal compromise attained by third-order orthogonal spline wavelets. Testing of the ensuing wavelet coefficients identified only about 1.5% as statistically different (p © 1993 SPIE. All rights reserved.

关键词： Medical imaging

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The L/sub 2/-polynomial spline pyramid

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IEEE Transactions on Pattern Analysis and Machine Intelligence 1993年第4期15卷 364-379页

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

The authors are concerned with the derivation of general methods for the L/sub 2/ approximation of signals by polynomial splines. The main result is that the expansion coefficients of the approximation are obtained by linear filtering and sampling. The authors apply those results to construct a L/sub 2/ polynomial spline pyramid that is a parametric multiresolution representation of a signal. This hierarchical data structure is generated by repeated application of a REDUCE function (prefilter and down-sampler). A complementary EXPAND function (up-sampler and post-filter) allows a finer resolution mapping of any coarser level of the pyramid. Four equivalent representations of this pyramid are considered, and the corresponding REDUCE and EXPAND filters are determined explicitly for polynomial splines of any order n (odd). Some image processing examples are presented. It is demonstrated that the performance of the Laplacian pyramid can be improved significantly by using a modified EXPAND function associated with the dual representation of a cubic spline pyramid.< >

关键词： Spline Signal resolution Spatial resolution Nonlinear filters Polynomials Wavelet transforms Laplace equations Image processing Data structures Multigrid methods

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Electrical stimulation in a time-dependent, passive bidomain

Electrical stimulation in a time-dependent, passive bidomain

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Annual International Conference of the IEEE engineering in Medicine and Biology Society (EMBC)

作者： B.J. Roth N. Trayanova Biomedical Engineering and Instrumentation Program NCRR NIH Bethesda MD USA NSF ERC and Department of Biomedical Engineering Duke University Durham NC USA

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CABLE EQUATION FOR A MYELINATED AXON DERIVED FROM ITS MICROSTRUCTURE

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MEDICAL & BIOLOGICAL engineering & COMPUTING 1993年第1-Sup期31卷 S87-S92页

作者： BASSER, PJ 1. Biomedical Engineering & Instrumentation Program National Center of Research Resources National Institutes of Health 20892 Bethesda MD USA

A simplified cable equation that describes the subthreshold behaviour of a myelinated axon is derived from its microstructure. Specifically, a microcontinuum cable model of a composite axon is homogenised, yielding a familiar macrocontinuum cable equation of electrotonus, for which the space and time constants depend on microstructural electrical parameters. Activating functions for magnetic and electrical stimulation can be incorporated into this homogenised cable equation as sources or sinks of transmembrane potential. An integral solution to the forced cable equation is also presented for the subthreshold regime. Errors are introduced when myelin membrane resistance is assumed to be infinite.

关键词： CABLE EQUATION COMPOSITE MODEL HOMOGENIZATION MICROCONTINUUM MYELINATED AXON SUBTHRESHOLD RESPONSE

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A multiresolution image registration procedure using spline pyramids

A multiresolution image registration procedure using spline ...

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Mathematical Imaging: Wavelet Applications in Signal and Image Processing 1993

作者： Unser, Michael Aldroubi, Akram Gerfen, Charles R. Biomedical Engineering and Instrumentation Program Bldg 13 Room 3W13 National Center for Research Resources National Institutes of Health BethesdaMD20892 United States Laboratory of Cell Biology Bldg 36 Room 2D-10 National Institute of Mental Health National Institutes of Health BethesdaMD20892 United States

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 model is adapted to the step size of the algorithm. © 1993 SPIE. All rights reserved.

关键词： Image registration

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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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AN IMPROVED LEAST-SQUARES LAPLACIAN PYRAMID FOR IMAGE COMPRESSION

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SIGNAL PROCESSING 1992年第2期27卷 187-203页

作者： UNSER, M Biomedical Engineering and Instrumentation Program Bldg 13 Room 3W13 National Center for Research Resources National Institutes of Health Bethesda MD 20892 USA

This paper describes two ways of improving Burt and Adelson's Laplacian pyramid, a technique developed for image compression. The Laplacian pyramid is a multi-resolution image representation that captures the loss of information occurring through repeated reduction of the spatial resolution. The generation of this data structure involves the use of two complementary functions: EXPAND, which increases the size of an image by a factor of 2, and REDUCE, which performs the reverse operation. The first modification is the adjunction of a pre-filter to the initial EXPAND function in order to guarantee an image extrapolation that is an exact interpolation of the coarser resolution level. The second refinement is a REDUCE operation modified to. minimize information loss. The corresponding least squares Laplacian pyramid (LSLP) is generated by adding a post-filter to the initial REDUCE function. These new functions have an efficient implementation using recursive algorithms. Preliminary experiments indicate improved performance: for a Gaussian-like kernel (a=3/8), the new EXPAND function exhibits a 2 to 2.5 dB attenuation of the first level of the Laplacian pyramid, while the complete scheme (LSLP) leads to a 4.7 to 8.5 dB improvement in the two images used to test the procedure. For comparable compression ratios, the subjective image quality for the LSLP appears to be significantly better. A theoretical relationship between the present approach and the family of quadrature mirror filter image pyramids is also derived.

关键词： LAPLACIAN PYRAMID MULTIRESOLUTION REPRESENTATION IMAGE COMPRESSION IMAGE CODING GAUSSIAN PYRAMID LEAST SQUARES APPROXIMATION RECURSIVE FILTER QUADRATURE MIRROR FILTERS WAVELET TRANSFORM

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