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ABSTRACT: In this paper, the lifting factorization and structural regularity of the lapped unimodular transforms (LUTs) are studied. The proposed M-channel lifting factorization is complete, is minimal in the McMillan sense, and has diagonal entries of unity. In addition to allowing for integer-to-integer mapping and guaranteeing perfect reconstruction even under finite precision, the proposed lifting factorization structurally ensures unimodularity. For regular LUT design, structural conditions that impose (1,1)-, (1,2)- and (2,1)-regularity onto the filter banks (FBs) are presented. Consequently, the optimal filter coefficients can be obtained through unconstrained optimizations. A special lifting-based lattice structure is used for parameterizing nonsingular matrices, which not only helps impose regularity but also has rational-coefficient unimodular FBs as a by-product. The regular LUTs can be transformed to the lifting domain with the proposed factorization for faster and multiplierless implementations. The lifting factorization and the regularity conditions are derived for two different (Type-I and Type-II) factorizations of the first-order unimodular FBs. Design examples are presented to confirm the proposed theory.
IEEE Transactions on Signal Processing 04/2006; · 2.63 Impact Factor
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ABSTRACT: Regularity is a fundamental and desirable property of wavelets and perfect reconstruction filter banks (PRFBs). Among others, it dictates the smoothness of the wavelet basis and the rate of decay of the wavelet coefficients. This paper considers how regularity of a desired degree can be structurally imposed onto biorthogonal filter banks (BOFBs) so that they can be designed with exact regularity and fast convergence via unconstrained optimization. The considered design space is a useful class of M-channel causal finite-impulse response (FIR) BOFBs (having anticausal FIR inverses) that are characterized by the dyadic-based structure W(z)=I-UV<sup>†</sup>+z<sup>-1</sup>UV<sup>†</sup> for which U and V are M×γ parameter matrices satisfying V<sup>†</sup>U=I<sub>γ</sub>, 1≤γ≤M, for any M≥2. Structural conditions for regularity are derived, where the Householder transform is found convenient. As a special case, a class of regular linear-phase BOFBs is considered by further imposing linear phase (LP) on the dyadic-based structure. In this way, an alternative and simplified parameterization of the biorthogonal linear-phase filter banks (GLBTs) is obtained, and the general theory of structural regularity is shown to simplify significantly. Regular BOFBs are designed according to the proposed theory and are evaluated using a transform-based image codec. They are found to provide better objective performance and improved perceptual quality of the decompressed images. Specifically, the blocking artifacts are reduced, and texture details are better preserved. For fingerprint images, the proposed biorthogonal transform codec outperforms the FBI scheme by 1-1.6 dB in PSNR.
IEEE Transactions on Signal Processing 03/2006; · 2.63 Impact Factor
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IEEE Transactions on Signal Processing. 01/2006; 54:921-931.
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IEEE Transactions on Signal Processing. 01/2006; 54:691-700.
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ABSTRACT: Paraunitary filterbanks (PUFBs) can be designed and implemented using either degree-one or order-one dyadic-based factorization. This work discusses how regularity of a desired degree is structurally imposed on such factorizations for any number of channels M ≥ 2, without necessarily constraining the phase responses. The regular linear-phase PUFBs become a special case under the proposed framework. We show that the regularity conditions are conveniently expressed in terms of recently reported M-channel lifting structures, which allow for fast, reversible, and possibly multiplierless implementations in addition to improved design efficiency, as suggested by numerical experience. M-band orthonormal wavelets with structural vanishing moments are obtained by iterating the resulting regular PUFBs on the lowpass channel. Design examples are presented and evaluated using a transform-based image coder, and they are found to outperform previously reported designs.
IEEE Transactions on Signal Processing 02/2005; · 2.63 Impact Factor
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IEEE Transactions on Signal Processing. 01/2005; 53:193-207.
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ABSTRACT: The paper studies how regularity of a desired degree can be imposed onto the dyadic-based building blocks of a class of M-channel (M ≥ 2) causal biorthogonal filter banks (BOFBs) having anti-causal inverses, without necessarily constraining the phase responses of the filters. Structural conditions for regularity are derived in terms of the elements of the dyadic-based building blocks, and regular BOFBs are designed accordingly and evaluated using a transform-based image coder. They are found to provide better objective performance and improved perceptual quality of the decompressed images, with reduced blocking artifacts and better preserved texture details.
Communications and Information Technology, 2004. ISCIT 2004. IEEE International Symposium on; 11/2004
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ABSTRACT: In this paper, we present the structural conditions for imposing regularity onto first-order unimodular filter banks (a.k.a. lapped unimodular transforms, LUT). For this purpose, we propose to parameterize non-singular matrices using a special lattice structure by which rational-coefficient unimodular filter banks can readily be designed. We consider two types of LUT factorizations and derive the corresponding structural conditions for regularity. Consequently, regular first-order unimodular filter banks can be designed using unconstrained optimizations, as the proposed structures always guarantee regularity. Design examples of regular first-order unimodular filter banks are presented to illustrate the proposed theory.
Digital Signal Processing Workshop, 2004 and the 3rd IEEE Signal Processing Education Workshop. 2004 IEEE 11th; 09/2004
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ABSTRACT: We re-examine the design of structurally regular linear-phase (LP) paraunitary filter banks (PUFBs) via a general framework without the LP constraint, and show that by a suitable choice of parameterization, the general regularity theory simplifies significantly under the LP assumption. A new geometric condition for regularity is presented which is essentially equivalent to but simpler than that in Oraintara et al. (2001). A connection between the general-purpose dyadic-based building block and the order-one LP building block used in the GenLOT structure is established.
Circuits and Systems, 2004. ISCAS '04. Proceedings of the 2004 International Symposium on; 06/2004
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ABSTRACT: This paper presents the complete parameterizations of the class of M-channel perfect reconstruction filter banks (PRFBs) having a prescribed admissible scaling filter H<sub>0</sub>(z). We show that in the biorthogonal case, such filter banks (BOFBs) are conveniently parameterized by the recently reported M-channel lifting factorization, which appropriately confines the degrees of freedom within the remaining M - 1 filters while guaranteeing H<sub>0</sub>(z) as prescribed. On the other hand, for the orthogonal case, such filter banks (PUFBs) with a prescribed scaling filter H<sub>0</sub>(z) of a certain length can be completed by the order-one factorization of an M × 1 lossless system proposed in this paper - the constraint on McMillian degree of the resulting PUFBs previously present in the literature has been relaxed, and PUFBs with better performance are thus obtained. Design examples for both classes are given.
Circuits and Systems, 2004. ISCAS '04. Proceedings of the 2004 International Symposium on; 06/2004
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ABSTRACT: We extend the method of lifting from M-channel paraunitary filter banks (PUFB) to M-channel unimodular filter banks. In particular, the lifting factorizations of the type I and type II building blocks for the first order unimodular matrices are presented, where the McMillian minimality is preserved. The proposed factorizations continue to have unity diagonal scaling, and perfect reconstruction (PR) is thus guaranteed under finite precision. Using degree-one unimodular building blocks, we present the design of first-order unimodular transforms (a.k.a. lapped unimodular transform, or LUT) and their lifting factorizations. Multiplierless (while reversible) implementations of the resulting LUT are obtained by dyadic approximation of the lifting coefficients.
Circuits and Systems, 2004. ISCAS '04. Proceedings of the 2004 International Symposium on; 06/2004
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ABSTRACT: The purpose of this paper is twofold: one is to establish a framework for general biorthogonal filter banks (BOFBs) with structural regularity; the other is to identify the connection between the general structure used here and the one commonly used for linear-phase biorthogonal filter banks (a.k.a. generalized lapped biorthogonal transform or GLBT). The latter also leads naturally to the same reduced number of free parameters as GLBT. We first revisit a minimal structure of BOFBs using order-one dyadic-based building blocks, by which BOFBs with length constraint can be designed. Conditions for filter bank regularity on the dyadic-based structure are derived and specialized to the case of GLBT. Design examples are presented.
Acoustics, Speech, and Signal Processing, 2004. Proceedings. (ICASSP '04). IEEE International Conference on; 06/2004 · 4.63 Impact Factor
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ABSTRACT: In this paper, the completion of linear-phase paraunitary filter banks (LPPUFBs) is presented. Given an M-band LPPUFB-admissible scaling filter H<sub>0</sub>(z), the proposed method finds a complete parameterization such that the resulting M - 1 bandpass/highpass filters Hi(z) and the given filter Ho(z) form an LPPUFB. Results from completion of a general PUFB are used in combination with the LP-generating dyadic-based structure, so that the scaling filter H<sub>0</sub>(z) of the resulting LPPUFB is as prescribed, and linear phase of the filter bank is guaranteed. The design procedure is demonstrated by an example.
Acoustics, Speech, and Signal Processing, 2004. Proceedings. (ICASSP '04). IEEE International Conference on; 06/2004 · 4.63 Impact Factor
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ABSTRACT: An intrinsic M-channel lifting factorization of perfect reconstruction filter banks (PRFBs) is presented as an extension of Sweldens' conventional two-channel lifting scheme. Given a polyphase matrix E(z) of a finite-impulse response (FIR) M- channel PRFB with det(E(z))=z<sup>-K</sup>, K∈Z, a systematic M-channel lifting factorization is derived based on the Monic Euclidean algorithm. The M-channel lifting structure provides an efficient factorization and implementation; examples include optimizing the factorization for the number of lifting steps, delay elements, and dyadic coefficients. Specialization to paraunitary building blocks enables the design of paraunitary filter banks based on lifting. We show how to achieve reversible, possibly multiplierless, implementations under finite precision, through the unit diagonal scaling property of the Monic Euclidean algorithm. Furthermore, filter-bank regularity of a desired order can be imposed on the lifting structure, and PRFBs with a prescribed admissible scaling filter are conveniently parameterized.
IEEE Transactions on Circuits and Systems II Analog and Digital Signal Processing 01/2004;
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ABSTRACT: We evaluate a wavelet-based algorithm to estimate the coil sensitivity modulation from surface coils. This information is used to improve the image homogeneity of magnetic resonance imaging when a surface coil is used for reception, and to increase image encoding speed by reconstructing images from under-sampled (aliased) acquisitions using parallel magnetic resonance imaging (MRI) methods for higher spatiotemporal image resolutions. The proposed algorithm estimates the spatial sensitivity profile of surface coils from the original anatomical images directly without using the body coil for additional reference scans or using coil position markers for electromagnetic model-based calculations. No prior knowledge about the anatomy is required for the application of the algorithm. The estimation of the coil sensitivity profile based on the wavelet transform of the original image data was found to provide a robust method for removing the slowly varying spatial sensitivity pattern of the surface coil image and recovering full FOV images from two-fold acceleration in 8-channel parallel MRI. The results, using bi-orthogonal Daubechies 97 wavelets and other members in this family, are evaluated for T1-weighted and T2-weighted brain imaging.
Human Brain Mapping 07/2003; 19(2):96-111. · 5.88 Impact Factor
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ABSTRACT: This paper presents a lifting-domain design of filter banks with a given McMillan degree. It is based on the M-channel lifting factorizations of the degree-0 and 1 building blocks I - 2uv<sup>†</sup> and I - uv<sup>†</sup> + z<sup>-1</sup>uv<sup>†</sup>, with v<sup>†</sup>u = 1. Paraunitariness further requires u = v. The proposed lifting factorization has a unity diagonal scaling throughout, and guarantees perfect reconstruction (PR) even when the parameters are quantized. It is shown to be minimal in terms of the minimum number of delays required. Based on the lifting factorization, regularity of the FB can be structurally imposed, and reversible, possibly multiplierless, implementation of the FB can readily be derived. Design examples are given to illustrate the versatility of the proposed approach.
Circuits and Systems, 2003. ISCAS '03. Proceedings of the 2003 International Symposium on; 06/2003
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ABSTRACT: This paper presents a lifting-domain design of filter banks with a given McMillan degree. It is based on the M-channel lifting factorizations of the degree-0 and 1 building blocks I 2uv + and uv + + z -1 uv + , with v + u = 1. Paraunitariness further requires u = v. The proposed lifting factorization has a unity diagonal scaling throughout, and guarantees perfect reconstruction (PR) even when the parameters are quantized. It is shown to be minimal in terms of the minimum number of delays required. Based on the lifting factorization, regularity of the FB can be structurally imposed, and reversible, possibly multiplierless, implementation of the FB can readily be derived. Design examples are given to illustrate the versatility of the proposed approach.
06/2003;
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ABSTRACT: This letter describes an algorithm for systematically finding a multiplierless approximation of transforms by replacing floating-point multipliers with VLSI-friendly binary coefficients of the form 2 . Assuming the cost of hardware binary shifters is negligible, the total number of binary adders employed to approximate the transform can be regarded as an index of complexity. Because the new algorithm is more systematic and faster than trial-and-error binary approximations with adder constraint, it is a much more efficient design tool. Furthermore, the algorithm is not limited to a specific transform; various approximations of the discrete cosine transform are presented as examples of its versatility.
01/2003;
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ABSTRACT: This letter describes an algorithm for systematically finding a multiplierless approximation of transforms by replacing floating-point multipliers with VLSI-friendly binary coefficients of the form k/2/sup n/. Assuming the cost of hardware binary shifters is negligible, the total number of binary adders employed to approximate the transform can be regarded as an index of complexity. Because the new algorithm is more systematic and faster than trial-and-error binary approximations with adder constraint, it is a much more efficient design tool. Furthermore, the algorithm is not limited to a specific transform; various approximations of the discrete cosine transform are presented as examples of its versatility.
IEEE Signal Processing Letters 12/2002; · 1.39 Impact Factor
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ABSTRACT: The concept of integer fast Fourier transform (IntFFT) for
approximating the discrete Fourier transform is introduced. Unlike the
fixed-point fast Fourier transform (FxpFFT), the new transform has
properties that it is an integer-to-integer mapping, power-adaptable and
also reversible. A lifting scheme is used to approximate complex
multiplications appearing in the FFT lattice structures. Split-radix FFT
is used to illustrate the approach for the case of 2<sup>N</sup>-point
FFT. The transform can be implemented by using only bit shifts and
additions but no multiplication. While preserving the reversibility, the
IntFFT is shown experimentally to yield the same accuracy as the FxpFFT
when their coefficients are quantized to a certain number of bits.
Complexity of the IntFFT is shown to be much lower than that of the
FxpFFT in terms of the numbers of additions and shifts
Acoustics, Speech, and Signal Processing, 2001. Proceedings. (ICASSP '01). 2001 IEEE International Conference on; 02/2001 · 4.63 Impact Factor
