A new class of two-channel biorthogonal filter banks and wavelet bases

We propose a novel framework for a new class of two-channel biorthogonal filter banks. The framework covers two useful subclasses: i) causal stable IIR filter banks. ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for such a class. Filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity. The properties of such a class are discussed in detail. The design of the analysis/synthesis systems reduces to the design of a single transfer function. Very simple design methods are given both for FIR and IIR cases. Zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity property. In the IIR case, two new classes of IIR maximally flat filters different from Butterworth filters are introduced. The filter coefficients are given in closed form. The wavelet bases corresponding to the biorthogonal systems are generated. the authors also provide a novel mapping of the proposed 1-D framework into 2-D. The mapping preserves the following: i) perfect reconstruction; ii) stability in the IIR case; iii) linear phase in the FIR case; iv) zeros at aliasing frequency; v) frequency characteristic of the filters

Regularity scalable image coding based on wavelet singularity detection

In this paper, we propose an adaptive algorithm for scalable wavelet image coding, which is based on the general feature, the regularity, of images. In pattern recognition or computer vision, regularity of images is estimated from the oriented wavelet coefficients and quantified by the Lipschitz exponents. To estimate the Lipschitz exponents, evaluating the interscale evolution of the wavelet transform modulus sum (WTMS) over the directional cone of influence was proven to be a better approach than tracing the wavelet transform modulus maxima (WTMM). This is because the irregular sampling nature of the WTMM complicates the reconstruction process. Moreover, examples were found to show that the WTMM representation cannot uniquely characterize a signal. It implies that the reconstruction of signal from its WTMM may not be consistently stable. Furthermore, the WTMM approach requires much more computational effort. Therefore, we use the WTMS approach to estimate the regularity of images from the separable wavelet transformed coefficients. Since we do not concern about the localization issue, we allow the decimation to occur when we evaluate the interscale evolution. After the regularity is estimated, this information is utilized in our proposed adaptive regularity scalable wavelet image coding algorithm. This algorithm can be simply embedded into any wavelet image coders, so it is compatible with the existing scalable coding techniques, such as the resolution scalable and signal-to-noise ratio (SNR) scalable coding techniques, without changing the bitstream format, but provides more scalable levels with higher peak signal-to-noise ratios (PSNRs) and lower bit rates. In comparison to the other feature-based wavelet scalable coding algorithms, the proposed algorithm outperforms them in terms of visual perception, computational complexity and coding efficienc

Scalable Speech Coding for IP Networks

The emergence of Voice over Internet Protocol (VoIP) has posed new challenges to the development of speech codecs. The key issue of transporting real-time voice packet over IP networks is the lack of guarantee for reasonable speech quality due to packet delay or loss. Most of the widely used narrowband codecs depend on the Code Excited Linear Prediction (CELP) coding technique. The CELP technique utilizes the long-term prediction across the frame boundaries and therefore causes error propagation in the case of packet loss and need to transmit redundant information in order to mitigate the problem. The internet Low Bit-rate Codec (iLBC) employs the frame-independent coding and therefore inherently possesses high robustness to packet loss. However, the original iLBC lacks in some of the key features of speech codecs for IP networks: Rate flexibility, Scalability, and Wideband support. This dissertation presents novel scalable narrowband and wideband speech codecs for IP networks using the frame independent coding scheme based on the iLBC. The rate flexibility is added to the iLBC by employing the discrete cosine transform (DCT) and iii the scalable algebraic vector quantization (AVQ) and by allocating different number of bits to the AVQ. The bit-rate scalability is obtained by adding the enhancement layer to the core layer of the multi-rate iLBC. The enhancement layer encodes the weighted iLBC coding error in the modified DCT (MDCT) domain. The proposed wideband codec employs the bandwidth extension technique to extend the capabilities of existing narrowband codecs to provide wideband coding functionality. The wavelet transform is also used to further enhance the performance of the proposed codec. The performance evaluation results show that the proposed codec provides high robustness to packet loss and achieves equivalent or higher speech quality than state-of-the-art codecs under the clean channel condition

Discrete multitone modulation with principal component filter banks

Discrete multitone (DMT) modulation is an attractive method for communication over a nonflat channel with possibly colored noise. The uniform discrete Fourier transform (DFT) filter bank and cosine modulated filter bank have in the past been used in this system because of low complexity. We show in this paper that principal component filter banks (PCFB) which are known to be optimal for data compression and denoising applications, are also optimal for a number of criteria in DMT modulation communication. For example, the PCFB of the effective channel noise power spectrum (noise psd weighted by the inverse of the channel gain) is optimal for DMT modulation in the sense of maximizing bit rate for fixed power and error probabilities. We also establish an optimality property of the PCFB when scalar prefilters and postfilters are used around the channel. The difference between the PCFB and a traditional filter bank such as the brickwall filter bank or DFT filter bank is significant for effective power spectra which depart considerably from monotonicity. The twisted pair channel with its bridged taps, next and fext noises, and AM interference, therefore appears to be a good candidate for the application of a PCFB. This is demonstrated with the help of numerical results for the case of the ADSL channel

Compressive Space-Time Galerkin Discretizations of Parabolic Partial Differential Equations

We study linear parabolic initial-value problems in a space-time variational formulation based on fractional calculus. This formulation uses "time derivatives of order one half" on the bi-infinite time axis. We show that for linear, parabolic initial-boundary value problems on $(0,\infty)$, the corresponding bilinear form admits an inf-sup condition with sparse tensor product trial and test function spaces. We deduce optimality of compressive, space-time Galerkin discretizations, where stability of Galerkin approximations is implied by the well-posedness of the parabolic operator equation. The variational setting adopted here admits more general Riesz bases than previous work; in particular, no stability in negative order Sobolev spaces on the spatial or temporal domains is required of the Riesz bases accommodated by the present formulation. The trial and test spaces are based on Sobolev spaces of equal order $1/2$ with respect to the temporal variable. Sparse tensor products of multi-level decompositions of the spatial and temporal spaces in Galerkin discretizations lead to large, non-symmetric linear systems of equations. We prove that their condition numbers are uniformly bounded with respect to the discretization level. In terms of the total number of degrees of freedom, the convergence orders equal, up to logarithmic terms, those of best $N$-term approximations of solutions of the corresponding elliptic problems.Comment: 26 page

Neural Wavelet-domain Diffusion for 3D Shape Generation

This paper presents a new approach for 3D shape generation, enabling direct generative modeling on a continuous implicit representation in wavelet domain. Specifically, we propose a compact wavelet representation with a pair of coarse and detail coefficient volumes to implicitly represent 3D shapes via truncated signed distance functions and multi-scale biorthogonal wavelets, and formulate a pair of neural networks: a generator based on the diffusion model to produce diverse shapes in the form of coarse coefficient volumes; and a detail predictor to further produce compatible detail coefficient volumes for enriching the generated shapes with fine structures and details. Both quantitative and qualitative experimental results manifest the superiority of our approach in generating diverse and high-quality shapes with complex topology and structures, clean surfaces, and fine details, exceeding the 3D generation capabilities of the state-of-the-art models

Texture-adaptive mother wavelet selection for texture analysis

We discuss the use of texture-adaptive mother wavelets in an adaptive probabilistic wavelet packet approach to texture analysis. First, we present the use of adaptive biorthogonal wavelet packet bases in such ananalysis, thus combining the advantages of biorthogonal wavelets (FIR,linearphase) with those of a coherent texture model. In this case, the computation of the probability uses both the primal and dual coefficient of the adapted biorthogonal wavelet packet basis. The computation of the biorthogonal wavelet packet coefficient is done using a lifting scheme, which is very efficien in terms of reducing the computational complexity and achieving an intrinsic parameterization of wavelet filters Then we include the mother wavelet parameter into this model, in order to fin the optimal mother wavelet for a given texture using this model. The model is applied to the classificatio of mosaics of Brodatz textures, the results showing improvement over the performance of the corresponding orthogonal wavelets