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Brushlets: A Tool for Directional Image Analysis and Image Compression
, 1997
"... This paper is organized as follows. In the next section we review the construction of orthonormal windowed Fourier bases. This is followed in section III by a description of biorthogonal windowed Fourier bases. The new brushlet basis is given in section IV. A biorthogonal brushlet basis is presented ..."
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Cited by 86 (12 self)
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This paper is organized as follows. In the next section we review the construction of orthonormal windowed Fourier bases. This is followed in section III by a description of biorthogonal windowed Fourier bases. The new brushlet basis is given in section IV. A biorthogonal brushlet basis is presented in section V. In section VI we describe the image compression algorithm based on a brushlet expansion of the image. Results of experiments are presented in Section VII
Image approximation and modeling via least statistically dependent bases
 PATTERN RECOGNITION 34 (2001)17651784
, 2001
"... Statistical independence is one of the most desirable properties of a coordinate system for representing and modeling images. In reality, however, truly independent coordinates may not exist for a given set of images, or it may be too di$cult to compute them in practice. Therefore, we propose a new ..."
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Cited by 12 (5 self)
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Statistical independence is one of the most desirable properties of a coordinate system for representing and modeling images. In reality, however, truly independent coordinates may not exist for a given set of images, or it may be too di$cult to compute them in practice. Therefore, we propose a new method to rapidly compute the least statistically dependent basis (LSDB)from a basis dictionary (e.g., the local cosine or wavelet packet dictionaries) containing a huge number of orthonormal (or biorthogonal) bases. Our new basis selection criterion is minimization of the mutual information of the distributions of the basis coe$cients as a measure of statistical dependence, which in turn is equivalent to minimization of the sum of the di!erential entropy of each coordinate in the basis dictionary. In this sense, we can view this LSDB algorithm as the bestbasis version of the Independent Component Analysis (ICA), which is increasingly gaining popularity. This criterion is di!erent from that of the Joint Best Basis (JBB)proposed by Wickerhauser, which can be viewed as the bestbasis version of the Karhunen}Loève basis (KLB). We demonstrate the usefulness of the LSDB for image approximation and modeling and compare its performance with that of KLB and JBB using a collection of real geophysical acoustic waveforms and an image database of human faces.
The polyharmonic local sine transform: A new tool for local image analysis and synthesis without edge effect
 Applied and Computational Harmonic Analysis
, 2006
"... We introduce a new local sine transform that can completely localize image information both in the space domain and in the spatial frequency domain. This transform, which we shall call the polyharmonic local sine transform (PHLST), first segments an image into local pieces using the characteristic f ..."
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Cited by 12 (9 self)
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We introduce a new local sine transform that can completely localize image information both in the space domain and in the spatial frequency domain. This transform, which we shall call the polyharmonic local sine transform (PHLST), first segments an image into local pieces using the characteristic functions, then decomposes each piece into two components: the polyharmonic component and the residual. The polyharmonic component is obtained by solving the elliptic boundary value problem associated with the socalled polyharmonic equation (e.g., Laplace’s equation, biharmonic equation, etc.) given the boundary values (the pixel values along the boundary created by the characteristic function). Subsequently this component is subtracted from the original local piece to obtain the residual. Since the boundary values of the residual vanish, its Fourier sine series expansion has quickly decaying coefficients. Consequently, PHLST can distinguish intrinsic singularities in the data from the artificial discontinuities created by the local windowing. Combining this ability with the quickly decaying coefficients of the residuals, PHLST is also effective for image approximation, which we demonstrate using both synthetic and real images. In addition, we introduce the polyharmonic local Fourier transform (PHLFT) by replacing the Fourier sine series above by the complex Fourier series. With a slight sacrifice of the decay rate of the expansion coefficients, PHLFT allows one to compute local Fourier magnitudes and phases without revealing the edge effect (or Gibbs phenomenon), yet is invertible and useful for various filtering, analysis, and approximation purposes.
NONLINEAR APPROXIMATION IN αMODULATION SPACES
, 2003
"... ABSTRACT. The αmodulation spaces are a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that brushlet bases can be constructed to form unconditional and even greedy bases for the αmodulation spaces. We study mterm nonlinear approximation with ..."
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Cited by 9 (2 self)
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ABSTRACT. The αmodulation spaces are a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that brushlet bases can be constructed to form unconditional and even greedy bases for the αmodulation spaces. We study mterm nonlinear approximation with brushlet bases, and give complete characterizations of the associated approximation spaces in terms of αmodulation spaces. 1.
Fast compression of seismic data with local trigonometric bases
 in Wavelet Applications in Signal and Image Processing VII
, 1999
"... Our goal in this paper is to provide a fast numerical implementation of the local trigonometric bases algorithm1 in order to demonstrate that an advantage can be gained by constructing a biorthogonal basis adapted to a target image. Different choices for the bells are proposed, and an extensive eval ..."
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Cited by 7 (0 self)
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Our goal in this paper is to provide a fast numerical implementation of the local trigonometric bases algorithm1 in order to demonstrate that an advantage can be gained by constructing a biorthogonal basis adapted to a target image. Different choices for the bells are proposed, and an extensive evaluation of the algorithm was performed on synthetic and seismic data. Because of its ability to reproduce textures so well, the coder performs very well, even at high bitrate.
Biorthogonal Local Trigonometric Bases
, 2000
"... Local trigonometric bases consist of cosines and sines multiplied by smooth, well localized window functions in order to have basis functions with good timefrequency localization. On the one hand, bases in the twooverlapping setting are considered. In particular, the development of such bases from ..."
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Cited by 2 (1 self)
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Local trigonometric bases consist of cosines and sines multiplied by smooth, well localized window functions in order to have basis functions with good timefrequency localization. On the one hand, bases in the twooverlapping setting are considered. In particular, the development of such bases from the orthonormal bases of Coifman and Meyer to the general approach for the construction of biorthogonal bases introduced by Chui and Shi is reviewed. On the other hand, a new generalized theory for biorthogonal Wilson bases is presented which includes former approaches. Connections between the twooverlapping bases and the Wilson bases are pointed out. Numerous examples illustrate the theoretical results.
From Local Cosine Bases to Global Harmonics
 Appl. Comput. Harmon. Anal
, 1999
"... In this paper, the reproduction of trigonometric polynomials with twooverlapping local cosine bases is investigated. This study is motivated by the need of representing most effectively a Fourier series in the form of a localized cosine series for the purpose of local analysis; thus providing a veh ..."
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Cited by 2 (1 self)
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In this paper, the reproduction of trigonometric polynomials with twooverlapping local cosine bases is investigated. This study is motivated by the need of representing most effectively a Fourier series in the form of a localized cosine series for the purpose of local analysis; thus providing a vehicle for the transition from classical harmonic analysis to analysis by Wilsontype wavelets. It is shown that there is one and only one class, which is a oneparameter family, of window functions that allow pointwise reproduction of all global harmonics, where the parameter is the order of smoothness of the window functions. It turns out that this class of window functions is also optimal in the sense that all global harmonics are reproduced by using a minimal number of the local trigonometric basis functions. 1 Introduction Following the approach of Sullivan, Rehr, Wilkins, and Wilson in [9] for the localization of trigonometric basis functions, Daubechies, Jaffard, and Journ'e [5] presen...
Bivariate Local Trigonometric Bases On Triangular Partitions
 In Wavelets and Multiscale Methods, Proceedings of the International Wavelet Conference, Tanger
, 1998
"... We construct bivariate local trigonometric bases on a "twooverlapping" triangular grid. A main result is the description of various trigonometric bases on triangles, satisfying parity conditions at the edges. Moreover, we introduce folding operators for the triangular grid. From these res ..."
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Cited by 1 (1 self)
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We construct bivariate local trigonometric bases on a "twooverlapping" triangular grid. A main result is the description of various trigonometric bases on triangles, satisfying parity conditions at the edges. Moreover, we introduce folding operators for the triangular grid. From these results we derive assertions on Riesz stability and the biorthogonal basis. 1. INTRODUCTION Since Daubechies, Jaffard, and Journ'e in [5] gave a method to construct an orthonormal basis of L 2 := L 2 (R) consisting of windowed trigonometric functions, local trigonometric bases have been investigated by many authors. In particular, the approach of Coifman and Meyer [4] to consider "twooverlapping" window functions turned out to be useful for applications. A detailed study of "twooverlapping" local trigonometric bases can be found in [1]. To include various desirable features in [6] and [7] orthogonality is replaced by biorthogonality. In particular, to improve the approximation properties bases ...