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64
Image analysis using a dualtree Mband wavelet transform
 IEEE Trans. Image Process
, 2006
"... Abstract—We propose a twodimensional generalization to theband case of the dualtree decomposition structure (initially proposed by Kingsbury and further investigated by Selesnick) based on a Hilbert pair of wavelets. We particularly address: 1) the construction of the dual basis and 2) the result ..."
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Cited by 37 (24 self)
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Abstract—We propose a twodimensional generalization to theband case of the dualtree decomposition structure (initially proposed by Kingsbury and further investigated by Selesnick) based on a Hilbert pair of wavelets. We particularly address: 1) the construction of the dual basis and 2) the resulting directional analysis. We also revisit the necessary preprocessing stage in theband case. While several reconstructions are possible because of the redundancy of the representation, we propose a new optimal signal reconstruction technique, which minimizes potential estimation errors. The effectiveness of the proposedband decomposition is demonstrated via denoising comparisons on several image types (natural, texture, seismics), with variousband wavelets and thresholding strategies. Significant improvements in terms of both overall noise reduction and direction preservation are observed. Index Terms—Direction selection, dualtree, Hilbert transform, image denoising,band filter banks, wavelets.
Theory and Design of SignalAdapted FIR Paraunitary Filter Banks
 IEEE TRANS. SIGNAL PROCESSING
, 1998
"... We study the design of signaladapted FIR paraunitary filter banks, using energy compaction as the adaptation criterion. We present some important properties that globally optimal solutions to this optimization problem satisfy. In particular, we show that the optimal filters in the first channel ..."
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Cited by 33 (6 self)
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We study the design of signaladapted FIR paraunitary filter banks, using energy compaction as the adaptation criterion. We present some important properties that globally optimal solutions to this optimization problem satisfy. In particular, we show that the optimal filters in the first channel of the filter bank are spectral factors of the solution to a linear semiinfinite programming (SIP) problem. The remaining filters are related to the first through a matrix eigenvector decomposition. We discuss
Optimal Wavelet Representation Of Signals And The Wavelet Sampling Theorem
 IEEE Trans. Circuits Syst. II
, 1994
"... The wavelet representation using orthonormal wavelet bases has received widespread attention. Recently Mband orthonormal wavelet bases have been constructed and compactly supported Mband wavelets have been parameterized [15, 12, 32, 17]. This paper gives the theory and algorithms for obtaining th ..."
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Cited by 20 (0 self)
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The wavelet representation using orthonormal wavelet bases has received widespread attention. Recently Mband orthonormal wavelet bases have been constructed and compactly supported Mband wavelets have been parameterized [15, 12, 32, 17]. This paper gives the theory and algorithms for obtaining the optimal wavelet multiresolution analysis for the representation of a given signal at a predetermined scale in a variety of error norms [23]. Moreover, for classes of signals, this paper gives the theory and algorithms for designing the robust wavelet multiresolution analysis that minimizes the worst case approximation error among all signals in the class. All results are derived for the general Mband multiresolution analysis. An efficient numerical scheme is also described for the design of the optimal wavelet multiresolution analysis when the leastsquared error criterion is used. Wavelet theory introduces the concept of scale which is analogous to the concept of frequency in Fourier ana...
Symmetric orthonormal scaling functions and wavelets with dilation factor 4
 Adv. Comput. Math
, 1998
"... Abstract. It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric orthonormal scaling function except for the Haar function. In this paper we are concerned with the construction of symmetric orthonormal scaling f ..."
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Cited by 20 (9 self)
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Abstract. It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric orthonormal scaling function except for the Haar function. In this paper we are concerned with the construction of symmetric orthonormal scaling functions with dilation factor d = 4. Several examples of such scaling functions are provided in this paper. In particular, two examples of C1 orthonormal scaling functions, which are symmetric about 0 and 1, respectively, are presented. We will then discuss how to construct symmetric 6 wavelets from these scaling functions. We explicitly construct the corresponding orthonormal symmetric wavelets for all the examples given in this paper.
Theory and Applications of the ShiftInvariant, TimeVarying and Undecimated Wavelet Transforms
, 1995
"... In this thesis, we generalize the classical discrete wavelet transform, and construct wavelet transforms that are shiftinvariant, timevarying, undecimated, and signal dependent. The result is a set of powerful and efficient algorithms suitable for a wide variety of signal processing tasks, e.g., d ..."
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Cited by 17 (3 self)
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In this thesis, we generalize the classical discrete wavelet transform, and construct wavelet transforms that are shiftinvariant, timevarying, undecimated, and signal dependent. The result is a set of powerful and efficient algorithms suitable for a wide variety of signal processing tasks, e.g., data compression, signal analysis, noise reduction, statistical estimation, and detection. These algorithms are comparable and often superior to traditional methods. In this sense, we put wavelets in action.
Orthogonal complex filter banks and wavelets: some properties and design
 IEEE TRANS. ON SIGNAL PROC
, 1999
"... Recent wavelet research has primarily focused on realvalued wavelet bases. However, complex wavelet bases offer a number of potential advantageous properties. For example, it has been recently suggested that the complex Daubechies wavelet can be made symmetric. However, these papers always imply th ..."
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Cited by 15 (0 self)
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Recent wavelet research has primarily focused on realvalued wavelet bases. However, complex wavelet bases offer a number of potential advantageous properties. For example, it has been recently suggested that the complex Daubechies wavelet can be made symmetric. However, these papers always imply that if the complex basis has a symmetry property, then it must exhibit linear phase as well. In this paper, we prove that a linearphase complex orthogonal wavelet does not exist. We study the implications of symmetry and linear phase for both complex and realvalued orthogonal wavelet bases. As a byproduct, we propose a method to obtain a complex orthogonal wavelet basis having the symmetry property and approximately linear phase. The numerical analysis of the phase response of various complex and real Daubechies wavelets is given. Both real and complexsymmetric orthogonal wavelet can only have symmetric amplitude spectra. It is often desired to have asymmetric amplitude spectra for processing general complex signals. Therefore, we propose a method to design general complex orthogonal perfect reconstruct filter banks (PRFB’s) by a parameterization scheme. Design examples are given. It is shown that the amplitude spectra of the general complex conjugate quadrature filters (CQF’s) can be asymmetric with respect the zero frequency. This method can be used to choose optimal complex orthogonal wavelet basis for processing complex signals such as in radar and sonar.
A Nonlinear Stein Based Estimator for Multichannel Image Denoising
, 2007
"... The use of multicomponent images has become widespread with the improvement of multisensor systems having increased spatial and spectral resolutions. However, the observed images are often corrupted by an additive Gaussian noise. In this paper, we are interested in multichannel image denoising based ..."
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Cited by 13 (7 self)
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The use of multicomponent images has become widespread with the improvement of multisensor systems having increased spatial and spectral resolutions. However, the observed images are often corrupted by an additive Gaussian noise. In this paper, we are interested in multichannel image denoising based on a multiscale representation of the images. A multivariate statistical approach is adopted to take into account both the spatial and the intercomponent correlations existing between the different wavelet subbands. More precisely, we propose a new parametric nonlinear estimator which generalizes many reported denoising methods. The derivation of the optimal parameters is achieved by applying Steinâs principle in the multivariate case. Experiments performed on multispectral remote sensing images clearly indicate that our method outperforms conventional wavelet denoising techniques.
MBand wavelets: Application to texture segmentation for real life image analysis
 International Journal of Wavelets, Multiresolution and Information Processing
, 2003
"... This paper describes two examples of reallife applications of texture segmentation using Mband wavelets. In the first part of the paper, an efficient and computationally fast method for segmenting text and graphics part of a document image based on textural cues is presented. It is logical to assu ..."
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Cited by 10 (4 self)
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This paper describes two examples of reallife applications of texture segmentation using Mband wavelets. In the first part of the paper, an efficient and computationally fast method for segmenting text and graphics part of a document image based on textural cues is presented. It is logical to assume that the graphics part has different textural properties than the nongraphics (text) part. So, this is basically a twoclass texture segmentation problem. The second part of the paper describes a segmentation scheme for another reallife data such as remotely sensed image. Different quasihomogeneous regions in the image can be treated to have different texture properties. Based on this assumption the multiclass texture segmentation scheme is applied for this purpose. Keywords: Texture segmentation; Mband wavelets; document image; remotely sensed image. AMS Subject Classification: 22E46, 53C35, 57S20 1.
Shift Products And Factorizations Of Wavelet Matrices
, 1994
"... . A class of so called shift products of wavelet matrices is introduced. These products are based on circulations of columns of orthogonal banded block circulant matrices arising in applications of discrete orthogonal wavelet transforms (or paraunitary multirate filter banks) or, equivalently, on au ..."
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Cited by 9 (8 self)
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. A class of so called shift products of wavelet matrices is introduced. These products are based on circulations of columns of orthogonal banded block circulant matrices arising in applications of discrete orthogonal wavelet transforms (or paraunitary multirate filter banks) or, equivalently, on augmentations of wavelet matrices by zero columns (shifts). A special case is no shift; a product which is closely related to the Pollen product is then obtained. Known decompositions using factors formed by two blocks are described and additional conditions such that uniqueness of the factorization is guaranteed are given. Next it is shown that when nonzero shifts are used, an arbitrary wavelet matrix can be factorized into a sequence of shift products of square orthogonal matrices. Such a factorization, as well as those mentioned earlier, can be used for the parameterization and construction of wavelet matrices, including the construction from the first row. Moreover it is also suitable for ...
Lattice structure for regularparaunitary linearphase filter banks
 in Proc. 2nd Int. Workshop Transforms Filter Banks
, 2001
"... uniform filterbanks, which are also known as the generalized lapped orthogonal transforms (GenLOTs), can be designed and implemented using lattice structures. This paper discusses how to impose regularity constraints onto the lattice structure of PULP filterbanks. These conditions are expressed in t ..."
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Cited by 9 (2 self)
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uniform filterbanks, which are also known as the generalized lapped orthogonal transforms (GenLOTs), can be designed and implemented using lattice structures. This paper discusses how to impose regularity constraints onto the lattice structure of PULP filterbanks. These conditions are expressed in term of the rotation angles of the lattice components by which the resulting filterbanks are guaranteed to have one or two degrees of regularity. Iterating these new regular filterbanks on the lowpass subband generates a large family of symmetricband orthonormal wavelets. Design procedures with many design examples are presented. Smooth interpolation using regular PULP filterbanks is illustrated through image coding experiments where the novelband wavelets consistently yield smoother reconstructed images and better perceptual quality. I.