Biorthogonal system

About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.

Random matrix ensembles involving Gaussian Wigner and Wishart matrices, and biorthogonal structure.

Abstract: We consider four nontrivial ensembles involving Gaussian Wigner and Wishart matrices. These are relevant to problems ranging from multiantenna communication to random supergravity. We derive the matrix probability density, as well as the eigenvalue densities for these ensembles. In all cases the joint eigenvalue density exhibits a biorthogonal structure. A determinantal representation, based on a generalization of Andreief's integration formula, is used to compactly express the r-point correlation function of eigenvalues. This representation circumvents the complications encountered in the usual approaches, and the answer is obtained immediately by examining the joint density of eigenvalues. We validate our analytical results using Monte Carlo simulations.

q-Series and Orthogonal Polynomials Associated with Barnes’ First Lemma

Abstract: We exploit symmetry (recurrence relation) techniques for the derivation of properties associated with families of basic hypergeometric functions. Similar methods have been used by Nikiforov, Suslov, and Uvarov. Here we apply these ideas to find new proofs of Barnes’ First Lemma and some of its q-analogues. We show that these integrals correspond to the weight functions determining the orthogonality relations for Hahn, q-Hahn, and big q-Jacobi polynomials. As another example of our method we introduce a biorthogonal system of rational functions whose weight function corresponds to the q-analogue of Kummer’s Theorem.

Neural Wavelet-domain Diffusion for 3D Shape Generation

Abstract: 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.

M-band biorthogonal interpolating wavelets via lifting scheme

Abstract: Recently, the lifting scheme was generalized to the multidimensional and multiband cases and was used to design M-band interpolating scaling filters and their duals. Based on this idea, we develop a new lifting pattern, namely, the progressive lifting pattern. This pattern allows us to pairwise generate M-band interpolating filterbanks and wavelets by the order from lowpass to highpass filters. A complete lifting procedure is divided into M - 1 simple steps, in each step, a pair of filters (the l'th filter and its dual) are generated. In this way, an M -band biorthogonal interpolating filterbank/wavelet is determined by M(M - 1) lifting filters. The first 2(M 1) lifting filters completely characterize the two scaling filters as well as the vanishing moments of bandpass and highpass filters; the residual (M - 1) (M - 2) lifting filters are used to pairwise optimize the bandpass and highpass filters in terms of the criterion of stopband energy minimization. The obtained M-band biorthogonal interpolating filterbanks and wavelets provide excellent frequency characteristics, in particular, low stopband sidelobes. Furthermore, the pattern is also utilized to design signal-adapted interpolating filterbanks and their rational coefficient counterparts in terms of subband coding gain. The obtained filterbanks achieve large subband coding gains. The rational coefficient filterbanks preserve the biorthogonality and allow wavelet transforms from integers to integers and a unifying lossy/lossless coding framework at the cost of a slight degradation in subband coding gain.

Optimized wavelets for fingerprint compression

Abstract: We present optimized biorthogonal and orthonormal wavelets for embedded zerotree wavelet compression of fingerprint images. These were obtained by simulated annealing over the wavelet filter coefficients using rms error between original and recovered images as the cost to be minimized. Linear-phase biorthogonal wavelets optimized for fingerprint image compression provide significantly improved fidelity compared either with standard wavelets or with wavelets optimized upon general images. These results are confirmed by psychovisual evaluation by two fingerprint experts whose independent rankings were in complete agreement. Filter coefficients for optimized orthonormal and linear-phase biorthogonal wavelets are tabulated.