Ying Jui Chen

Integer fast Fourier transform

A 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 the properties that it is an integer-to-integer mapping, is power adaptable and is reversible. The lifting scheme is used to approximate complex multiplications appearing in the FFT lattice structures where the dynamic range of the lifting coefficients can be controlled by proper choices of lifting factorizations. Split-radix FFT is used to illustrate the approach for the case of 2/sup N/-point FFT, in which case, an upper bound of the minimal dynamic range of the internal nodes, which is required by the reversibility of the transform, is presented and confirmed by a simulation. The transform can be implemented by using only bit shifts and additions but no multiplication. A method for minimizing the number of additions required is presented. 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. Finally, they are applied to noise reduction applications, where the IntFFT provides significantly improvement over the FxpFFT at low power and maintains similar results at high power.

Multiplierless approximation of transforms with adder constraint

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.

Dyadic-based factorizations for regular paraunitary filterbanks and M-band orthogonal wavelets with structural vanishing moments

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 /spl ges/ 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.

Theory and factorization for a class of structurally regular biorthogonal filter banks

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 /spl dagger//+z/sup -1/UV/sup /spl dagger// for which U and V are M/spl times//spl gamma/ parameter matrices satisfying V/sup /spl dagger//U=I/sub /spl gamma//, 1/spl les//spl gamma//spl les/M, for any M/spl ges/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.

Lapped unimodular transforms: lifting factorization and structural regularity

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.

Structurally regular biorthogonal filter banks: theory, designs, and applications

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 /spl ges/ 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.

Structurally regular unimodular filter banks

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.