Perfect Reconstruction Property Research Articles

M-band wavelet network for machine anomaly detection from a frequency perspective

The autoencoder (AE) is widely utilized in deep anomaly detection, but it lacks explainability due to the complexity of nonlinear mapping. One approach to address this issue is incorporating wavelet theory, which shares similarities in decomposition and reconstruction procedures. However, the perfect reconstruction property of wavelet theory conflicts with AE-based anomaly detection. To tackle this problem, we introduce a novel deep anomaly detection method from a frequency perspective. A learnable M-band wavelet network (MWNet) is designed to offer a flexible frequency band structure for signal representation. Subsequently, with the aid of sparsity constraint, MWNet dynamically focuses on key components within each frequency band. A learnable hard threshold function with a threshold maximization constraint is proposed to retain the essential frequency band of normal signals. After training, the MWNet is exclusively capable of well reconstructing normal signals, thereby producing a noticeable reconstruction error difference between normal and abnormal signals. Extensive experiments on both simulated and experimental datasets validate the effectiveness of the proposed method. The corresponding Python codes are available at https://github.com/albertszg/MbandWaveletNet.

Medical Image Despeckling Using the Invertible Sparse Fuzzy Wavelet Transform with Nature-Inspired Minibatch Water Wave Swarm Optimization.

Speckle noise is a pervasive problem in medical imaging, and conventional methods for despeckling often lead to loss of edge information due to smoothing. To address this issue, we propose a novel approach that combines a nature-inspired minibatch water wave swarm optimization (NIMWVSO) framework with an invertible sparse fuzzy wavelet transform (ISFWT) in the frequency domain. The ISFWT learns a non-linear redundant transform with a perfect reconstruction property that effectively removes noise while preserving structural and edge information in medical images. The resulting threshold is then used by the NIMWVSO to further reduce multiplicative speckle noise. Our approach was evaluated using the MSTAR dataset, and objective functions were based on two contrasting reference metrics, namely the peak signal-to-noise ratio (PSNR) and the mean structural similarity index metric (MSSIM). Our results show that the suggested approach outperforms modern filters and has significant generalization ability to unknown noise levels, while also being highly interpretable. By providing a new framework for despeckling medical images, our work has the potential to improve the accuracy and reliability of medical imaging diagnosis and treatment planning.

WINNet: Wavelet-Inspired Invertible Network for Image Denoising.

Image denoising aims to restore a clean image from an observed noisy one. Model-based image denoising approaches can achieve good generalization ability over different noise levels and are with high interpretability. Learning-based approaches are able to achieve better results, but usually with weaker generalization ability and interpretability. In this paper, we propose a wavelet-inspired invertible network (WINNet) to combine the merits of the wavelet-based approaches and learning-based approaches. The proposed WINNet consists of K -scale of lifting inspired invertible neural networks (LINNs) and sparsity-driven denoising networks together with a noise estimation network. The network architecture of LINNs is inspired by the lifting scheme in wavelets. LINNs are used to learn a non-linear redundant transform with perfect reconstruction property to facilitate noise removal. The denoising network implements a sparse coding process for denoising. The noise estimation network estimates the noise level from the input image which will be used to adaptively adjust the soft-thresholds in LINNs. The forward transform of LINNs produces a redundant multi-scale representation for denoising. The denoised image is reconstructed using the inverse transform of LINNs with the denoised detail channels and the original coarse channel. The simulation results show that the proposed WINNet method is highly interpretable and has strong generalization ability to unseen noise levels. It also achieves competitive results in the non-blind/blind image denoising and in image deblurring.

Parallel Single-Pixel Imaging: A General Method for Direct\u2013Global Separation and 3D Shape Reconstruction Under Strong Global Illumination

We present parallel single-pixel imaging (PSI), a photography technique that captures light transport coefficients and enables the separation of direct and global illumination, to achieve 3D shape reconstruction under strong global illumination. PSI is achieved by extending single-pixel imaging (SI) to modern digital cameras. Each pixel on an imaging sensor is considered an independent unit that can obtain an image using the SI technique. The obtained images characterize the light transport behavior between pixels on the projector and the camera. However, the required number of SI illumination patterns generally becomes unacceptably large in practical situations. We introduce local region extension (LRE) method to accelerate the data acquisition of PSI. LRE perceives that the visible region of each camera pixel accounts for a local region. Thus, the number of detected unknowns is determined by local region area, which is extremely beneficial in terms of data acquisition efficiency. PSI possesses several properties and advantages. For instance, PSI captures the complete light transport coefficients between the projector–camera pair, without making specific assumptions on measured objects and without requiring special hardware and restrictions on the arrangement of the projector–camera pair. The perfect reconstruction property of LRE can be proven mathematically. The acquisition and reconstruction stages are straightforward and easy to implement in the existing projector–camera systems. These properties and advantages make PSI a general and sound theoretical model to decompose direct and global illuminations and perform 3D shape reconstruction under global illumination.

Time-Domain Audio Source Separation With Neural Networks Based on Multiresolution Analysis

We propose a time-domain audio source separation method based on multiresolution analysis, which we call multiresolution deep layered analysis (MRDLA). The MRDLA model is based on one of the state-of-the-art time-domain deep neural networks (DNNs), Wave-U-Net, which successively down-samples features and up-samples them to have the original time resolution. From the signal processing viewpoint, we found that the down-sampling (DS) layers of Wave-U-Net cause aliasing and may discard information useful for source separation because they are implemented with decimation. These two problems are due to the decimation; thus, to achieve a more reliable source separation method, we should design DS layers capable of simultaneously overcoming these problems. With this motivation, focusing on the fact that the successive DS architecture of Wave-U-Net resembles that of multiresolution analysis, we develop DS layers based on discrete wavelet transforms (DWTs), which we call the DWT layers, because the DWTs have anti-aliasing filters and the perfect reconstruction property. We further extend the DWT layers such that their wavelet basis functions can be trained together with the other DNN components while maintaining the perfect reconstruction property. Since a straightforward trainable extension of the DWT layers does not guarantee the existence of anti-aliasing filters, we derive constraints for this guarantee in addition to the perfect reconstruction property. Through music source separation experiments including subjective evaluations, we show the efficacy of the proposed methods and the importance of simultaneously considering both the anti-aliasing filters and the perfect reconstruction property.

A non‐uniform NSAF‐SF adaptive algorithm

Speech enhancement and acoustic noise reduction are two important tasks where adaptive filtering algorithms emerge as a competitive solution. Unfortunately, in such applications the convergence rate of the system identification is hampered when the excitation data is highly correlated. Subband adaptive algorithms have been developed to address such issue. A recently proposed low-complexity subband adaptive structure with sparse subfilters is generalised, in order to permit a non-uniform filter bank structure. This generalisation brings additional flexibility to the resulting critically decimated adaptive structure, which allows one to adapt it to the idiosyncrasies of the application of interest without losing the perfect-reconstruction property. A closed-form solution for the optimal values that the adaptive coefficient should assume in order to accurately emulate a given impulse response is derived. Simulations reveal that the resulting algorithm may outperform recently published subband adaptive filtering algorithms, thereby requiring even less computational effort.

An Adaptive Arbitrary Multiresolution Decomposition for Multiscale Geometric Analysis

The nonsubsampled Laplacian pyramid (NSLP) is widely used as a common multiresolution decomposition method in various nonsubsampled image transforms. However, the NSLP has a fixed spectrum partition, and thus cannot represent images accurately, and flexibly. We propose a new adaptive arbitrary multiresolution decomposition to solve these problems. First, using the affine characteristics of a pseudopolar Fourier transform (PPFT), we apply a 1-D nonuniform filter bank to the modulated PPFT to obtain a 2-D arbitrary resolution filter bank. This filter surpasses the limitation of fixed spectrum partitioning of the traditional tree structure. We then demonstrate that the proposed method satisfies the compact frame condition, and has translation invariance, and a linear phase. Furthermore, we propose an adaptive spectrum division approach at various scales based on image spectrum information based on a 2-D filter bank of arbitrary multiresolution, so our method can capture important visual information more accurately. Finally, we combine our method with a nonsubsampled directional filter bank of the nonsubsampled contourlet transform to create a new multiscale geometric analysis (MGA) method, and verify that the new method can have perfect reconstruction properties. The new MGA also performs better in image denoising, and recognition experiments than state-of-the-art MGA methods with translation invariance.

A Modified Tunable – Q Wavelet Transform Approach for Tamil Speech Enhancement

ABSTRACT In this paper, a new speech enhancement approach using the modified Tunable-Q Wavelet Transform (TQWT) for Tamil speaker recognition systems is proposed. Different forms of the existing wavelet transforms like continuous wavelet transform and discrete wavelet transform provide little ability to tune the Q factor and redundancy of the wavelet and have less reconstruction property. TQWT is fully discrete, has perfect reconstruction property and is modestly overcomplete. The Q-factor and redundancy can be easily tuned. Speech signals vary in both time and space. The entire speech signal is divided into different parts. Depending on the oscillatory contents of the speech signal, different Q factors are applied for each part instead of applying a single Q factor for the entire data. This gives more accuracy in decomposition. The proposed method is evaluated on several speakers and under various noise conditions including babble, street and exhibition noises. In order to tune the quality factor (Q) of the TQWT and to obtain good reconstruction property, speech enhancement has been conducted with varying values of Q. The reconstruction error, cost function, Signal to Noise Ratio(SNR) and Mean Opinion Score(MOS) show that the new approach highly improves the performance of speech enhancement based on the modified TQWT. Hardware implementation is also done using TMS320C6713 DSK and the results are similar to simulation.

Design of Two Channel Biorthogonal Filterbanks using Euler Frobenius Polynomial

This paper presents a technique to design a new class of biorthogonal perfect reconstruction (PR) filterbanks. In this technique, we use Euler Frobenius polynomial (EFP) to design maximally flat Euler Frobenius halfband polynomial (EFHP). This is obtained by imposing vanishing moments (VMs) and PR constraints on EFP. The resulted EFHP is used in three and four step lifting structure to determine analysis low-pass and high-pass filters. The lifting halfband kernels are designed using EFHP. It has been ensured that the proposed filters satisfy the linear phase and PR property. Also, the proposed filters have frame bound ratio very close to unity. Several design examples are presented and the properties of proposed filters are compared with existing filters. It has been ensured that the proposed filters give more regularity as compared to existing filterbanks.

Multiscale Discrete Framelet Transform for Graph-Structured Signals

Graph-structured signal enables rich description of data defined in the domain with irregular structure, which has seen its rapid growth in many applications including social, energy, transportation, sensor, neuronal networks, and many others. This paper aims at generalizing discrete framelet transform defined for regular grids in Euclidean space to finite undirected weighted graphs. By leveraging the intuition from classic framelet transform for signals on regular grids, we proposed an approach for constructing multiscale undecimal framelet transform for signals defined on finite graphs with a perfect reconstruction property. The proposed method is based on the definition of basic blocks involved in framelet transform, including graph shift operator, convolution, and band-limited down/up-sampling. These blocks enable a painless construction of a class of multilevel undecimal framelet transforms in vertex domain by directly calling wavelet filter banks of existing wavelet biframes and tight frames. The proposed discrete framelet transform on graphs keeps most desired properties of its counterpart on regular grids, and one can see its usage in applications.

FPGA implementation of UWB-IR impulse generator and its corresponding decoder based on discrete wavelet packet

In this paper, we propose an IR-UWB pulse generator and its corresponding decoder using respectively inverse discrete wavelet packet transform (IDWPT) and the discrete wavelet packet transform (DWPT) with various wavelets (Haar, Db2, Db4, Coif2, rbio1.5). These architectures being based on FIR filters, polyphase structure were chosen to easily suit in FPGA because of their parallel structures. The interpolator and decimator filter has been optimized using multiplexers to improve the delay, further the N-tap filter is divided into N/2-tap filters in parallel. The result comparisons in terms of BER has been shown for all wavelets through a Gaussian channel transmission. The results confirm the same transmission quality due to the perfect reconstruction property between the proposed IDWPT and DWPT implementation whatever the wavelet types that can be associated with sensors. We designed architectures of receiver and transmitter based on studied wavelets and we implemented them in FPGA to visualize generated pulses by experiences. The proposed FPGA implementations of IDWPT and DWPT architectures for Haar wavelet only require 3888 LUT/740 FF and 5359 LUT/996 FF, respectively, while providing low power consumption. The hardware and timing comparisons with existing techniques prove that our architectures are better. Our proposal permits a low complexity and good performance. Indeed, in transmitter part, one block can simultaneously generate pulses and modulate them. In receiver part, we replace a correlator used in a traditional IR-UWB system and generally complicated by our simple architecture.

Efficient Rationalization of Triplet Halfband Filter Banks and its Application to Image Compression

This paper presents a novel approach for rationalization of irrational coefficients of filters designed using three-step lifting scheme. The existing three-step lifting schemes result in the irrational filter coefficients which require infinite precision for implementation. In this work, Euler-Frobenius halfband polynomials are designed to obtain the required kernels of three-step lifting structure. Next, the efficient rationalization of the designed filter banks (FBs) is proposed to reduce the arithmetic complexity. The proposed rational FBs preserve perfect reconstruction, near-orthogonality and regularity properties of wavelet FBs. These filters with rational coefficients are then used in image compression algorithms to compress the test images in well-known Classic, EPFL, RAISE, FiveK datasets and chromosome image datasets. The performance of the proposed rational FBs is compared with the existing rational FBs. It is observed that the performance of the proposed rational FBs is improved in terms of computational cost, encoding-decoding time and compression ratio as compared to existing rational FBs.

Time-Frequency Localization optimization of PR FMT: Case of an overlapping factor equal to 2

In this paper, we analyze the Perfect Reconstruction (PR) property of Filtered MultiTone (FMT) systems with overlapping factor equal to 2. If small overlapping factors are not common for FMT systems, that originally targeted nearly perfect subchannels isolation, we show that they can nevertheless provide pretty good results when the optimization is carried out with the Time-Frequency Localization (TFL) criterion. Furthermore, it appears that the TFL criterion can ease the treatment of the PR equations. In this respect, different types of compact representation are proposed to efficiently optimize the TFL criterion and a simple and general construction algorithm is also provided for the computation of the PR FMT prototype filter coefficients. Numerical results and graphical displays illustrate the good behavior of our proposed design methods in terms of TFL performance and computational complexity. Finally, various FMT prototype filters are compared with regard to their resilience to symbol timing offset.

A perfect reconstruction property for PDE-constrained total-variation minimization with application in Quantitative Susceptibility Mapping

We study the recovery of piecewise constant functions of finite bounded variation (BV) from their image under a linear partial differential operator with unknown boundary conditions. It is shown that minimizing the total variation (TV) semi-norm subject to the associated PDE-constraints yields perfect reconstruction up to a global constant under a mild geometric assumption on the jump set of the function to reconstruct. The proof bases on establishing a structural result about the jump set associated with BV-solutions of the homogeneous PDE. Furthermore, we show that the geometric assumption is satisfied up to a negligible set of orthonormal transformations. The results are then applied to Quantitative Susceptibility Mapping (QSM) which can be formulated as solving a two-dimensional wave equation with unknown boundary conditions. This yields in particular that total variation regularization is able to reconstruct piecewise constant susceptibility distributions, explaining the high-quality results obtained with TV-based techniques for QSM.

Biorthogonal wavelet codes with prescribed code distance

Abstract We propose a scheme of construction of 2-circulant codes with given code distance on the basis of biorthogonal filters with the property of perfect reconstruction over a finite filed of odd characteristic. The corresponding algorithm for constructing biorthogonal filters utilizes the Euclidean algorithm for finding the gcd of polynomials.

Digital Affine Shear Filter Banks with 2-Layer Structure and Their Applications in Image Processing.

Digital affine shear filter banks with 2-layer structure (DAS-2 filter banks) are constructed and are shown to be with the perfect reconstruction (PR) property. The implementation of digital affine shear transforms using the transition and subdivision operators are given. The redundancy rate analysis shows that our digital affine shear transforms have redundancy rate no more than 8 and it decreases with respect to the number of directional filters. Numerical experiments on image processing demonstrate the advantage of our DAS-2 filter banks over many other state-of-the-art frame-based transforms. The connection between DAS-2 filter banks and affine shear tight frames with 2-layer structure is established. Characterizations and constructions of affine shear tight frames with 2-layer structure are provided.

Reconstruction of a Signal from the Real Part of Its Discrete Fourier Transform [Tips & Tricks

In this tutorial, we present a procedure for reconstructing a complex-valued, discrete-time signal from only partial Fourier transform (FT) information, more specifically, the real part of its discrete FT (RDFT). By applying a delay, coupled with appropriate zero-padding to ensure a sufficiently dense sampling of the frequency axis, we show that any signal can be reconstructed perfectly from the RDFT alone. The presented procedure can, in the case of a densely sampled DFT, provide a reduction in the computational complexity of analysis-modification-synthesis-based speech processing methods that independently process the real and imaginary (RI) parts temporally. Furthermore, the perfect reconstruction property of this method implies that the RDFT alone captures all of the information about the signal, which suggests that it may be a potentially useful frequencyderived domain for the processing of speech signals.

Adaptive wavelet tight frame construction for accelerating MRI reconstruction

The sparsity regularization approach, which assumes that the image of interest is likely to have sparse representation in some transform domain, has been an active research area in image processing and medical image reconstruction. Although various sparsifying transforms have been used in medical image reconstruction such as wavelet, contourlet, and total variation (TV) etc., the efficiency of these transforms typically rely on the special structure of the underlying image. A better way to address this issue is to develop an overcomplete dictionary from the input data in order to get a better sparsifying transform for the underlying image. However, the general overcomplete dictionaries do not satisfy the so-called perfect reconstruction property which ensures that the given signal can be perfectly represented by its canonical coefficients in a manner similar to orthonormal bases, resulting in time consuming in the iterative image reconstruction. This work is to develop an adaptive wavelet tight frame method for magnetic resonance image reconstruction. The proposed scheme incorporates the adaptive wavelet tight frame approach into the magnetic resonance image reconstruction by solving a l0-regularized minimization problem. Numerical results show that the proposed approach provides significant time savings as compared to the over-complete dictionary based methods with comparable performance in terms of both peak signal-to-noise ratio and subjective visual quality.

Time-Varying Weighted Optimal Empirical Mode Decomposition Using Multiple Sets of Basis Functions

Empirical mode decomposition (EMD) is a favorite tool for analyzing nonlinear and non-stationary signals. It decomposes any signal into a finite set of oscillation modes consisting of intrinsic mode functions and a residual function. Superimposing all these modes reconstructs the signal without any information loss. In addition to satisfying the perfect reconstruction property, however, there is no implication about the reconstruction optimality of the EMD. The lack of optimality restricts the signal recovery capability of the EMD in the presence of disturbances. Only a few attempts are made to meet this deficiency. In this paper, we propose a new algorithm named as time-varying weighted EMD. By this algorithm, original signal is reconstructed in the minimum mean-square error sense through the EMD followed by time-varying weightings of the oscillation modes. Determining the time-varying weights for the oscillation modes constitutes the backbone of the algorithm. Aiming to determine the time-varying weights of the oscillation modes; we use multiple sets of basis functions. The effectiveness of the proposed algorithm is demonstrated by computer simulations involving real biomedical signals. Simulation results show that the proposed algorithm exhibits better performance than that of its existing counterparts in terms of lower mean-square error and higher signal-to-error ratio.

Optimal signal reconstruction based on time-varying weighted empirical mode decomposition

Empirical mode decomposition (EMD) is a tool developed for analyzing nonlinear and non-stationary signals. It is capable of splitting any signal into a set of oscillation modes known as intrinsic mode functions and a residual function. Although the EMD satisfies the perfect signal reconstruction property by superimposing all the oscillation modes, it is not based on any optimality criterion. The lack of optimality limits the signal recovery performance of the EMD in the presence of disturbances such as noise and interference. In this paper, we propose a new algorithm, termed, time-varying weighted EMD, which gives the best estimate of a given signal in the minimum mean-square error sense. The main idea of the proposed algorithm is to reconstruct the original signal through the EMD followed by time-varying weightings of the oscillation modes. Simulations including two real-life signals are performed to show the superiority of the proposed algorithm.