Dinesh Bhati

Time–frequency localized three-band biorthogonal wavelet filter bank using semidefinite relaxation and nonlinear least squares with epileptic seizure EEG signal classification

Abstract In this paper, we design time–frequency localized three-band biorthogonal linear phase wavelet filter bank for epileptic seizure electroencephalograph (EEG) signal classification. Time–frequency localized analysis and synthesis low-pass filters (LPF) are designed using convex semidefinite programming (SDP) by transforming a nonconvex problem into a convex SDP using semidefinite relaxation technique. Three-band parameterized lattice biorthogonal linear phase perfect reconstruction filter bank (BOLPPRFB) is chosen and nonlinear least squares algorithm is used to determine its parameters values that generate the designed analysis and synthesis LPF such that the band-pass and high-pass filters are also well localized in time and frequency domain. The designed analysis and synthesis three-band wavelet filter banks are compared with the standard two-band filter banks like Daubechies maximally regular filter banks, Cohen–Daubechies–Feauveau (CDF) biorthogonal filter banks and orthogonal time–frequency localized filter banks. Kruskal–Wallis statistical test is employed to measure the statistical significance of the subband features obtained from the various two and three-band filter banks for epileptic seizure EEG signal classification. The results show that the designed three-band analysis and synthesis filter banks both outperform two-band filter banks in the classification of seizure and seizure-free EEG signals. The designed three-band filter banks and multi-layer perceptron neural network (MLPNN) are further used together to implement a signal classifier that provides classification accuracy better than the recently reported results for epileptic seizure EEG signal classification.

Design of Time---Frequency Localized Filter Banks: Transforming Non-convex Problem into Convex Via Semidefinite Relaxation Technique

We present a method for designing optimal biorthogonal wavelet filter banks (FBs). Joint time–frequency localization of the filters has been chosen as the optimality criterion. The design of filter banks has been cast as a constrained optimization problem. We design the filter either with the objective of minimizing its frequency spread (variance) subject to the constraint of prescribed time spread or with the objective of minimizing the time spread subject to the fixed frequency spread. The optimization problems considered are inherently non-convex quadratic constrained optimization problems. The non-convex optimization problems have been transformed into convex semidefinite programs (SDPs) employing the semidefinite relaxation technique. The regularity constraints have also been incorporated along with perfect reconstruction constraints in the optimization problem. In certain cases, the relaxed SDPs are found to be tight. The zero duality gap leads to the global optimal solutions. The design examples demonstrate that reasonably smooth wavelets can be designed from the proposed filter banks. The optimal filter banks have been compared with popular filter banks such as Cohen–Daubechies–Feauveau biorthogonal wavelet FBs, time–frequency optimized half-band pair FBs and maximally flat half-band pair FBs. The performance of optimal filter banks has been found better in terms of joint time–frequency localization.

Design of Time---Frequency Optimal Three-Band Wavelet Filter Banks with Unit Sobolev Regularity Using Frequency Domain Sampling

In this paper, we design three-band time–frequency-localized orthogonal wavelet filter banks having single vanishing moment. We propose new expressions to compute mean and variances in time and frequency from the samples of the Fourier transform of the asymmetric band-pass compactly supported wavelet functions. We determine discrete-time filter of length eight that generates the time–frequency optimal time-limited scaling and wavelet functions using cascade algorithm. Time–frequency product (TFP) of a function is defined as the product of its time variance and frequency variance. The TFP of the designed functions is close to 0.25 with unit Sobolev regularity. Three-band filter banks are designed by minimizing a weighted combination of TFPs of wavelets and scaling functions. Interestingly, empirical results show that time–frequency optimal, filter banks of length nine, designed with the proposed methodology, have unit Sobolev regularity, which is maximum achievable with single vanishing moment. Design examples for length six and length nine filter banks are given to demonstrate the effectiveness of the proposed design methodology.

A novel approach for time–frequency localization of scaling functions and design of three-band biorthogonal linear phase wavelet filter banks

Abstract Design of time–frequency localized filters and functions is a classical subject in the field of signal processing. Gabors uncertainty principle states that a function cannot be localized in time and frequency domain simultaneously and there exists a nonzero lower bound of 0.25 on the product of its time variance and frequency variance called time–frequency product (TFP). Using arithmetic mean (AM)– geometric mean (GM) inequality, product of variances and sum of variances can be related and it can be shown that sum of variances has lower bound of one. In this paper, we compute the frequency variance of the filter from its discrete Fourier transform (DFT) and propose an equivalent summation based discrete-time uncertainty principle which has the lower bound of one. We evaluate the performance of the proposed discrete-time time–frequency uncertainty measure in multiresolution setting and show that the proposed DFT based concentration measure generate sequences which are even more localized in time and frequency domain than that obtained from the Slepian, Ishii and Furukawas concentration measures. The proposed design approach provides the flexibility in which the TFP can be made arbitrarily close to the lowest possible lower bound of 0.25 by increasing the length of the filter. In the other proposed approach, the sum of the time variance and frequency variance is used to formulate a positive definite matrix to measure the time–frequency joint localization of a bandlimited function from its samples. We design the time–frequency localized bandlimited low pass scaling and band pass wavelet functions using the eigenvectors of the formulated positive definite matrix. The samples of the time–frequency localized bandlimited function are obtained from the eigenvector of the positive definite matrix corresponding to its minimum eigenvalue. The TFP of the designed bandlimited scaling and wavelet functions are close to the lowest possible lower bound of 0.25 and 2.25 respectively. We propose a design method for time–frequency localized three-band biorthogonal linear phase (BOLP) wavelet perfect reconstruction filter bank (PRFB) wherein the free parameters can be optimized for time–frequency localization of the synthesis basis functions for the specified frequency variance of the analysis scaling function. The performance of the designed filter bank is evaluated in classification of seizure and seizure-free electroencephalogram (EEG) signals. It is found that the proposed filter bank outperforms other existing methods for the classification of seizure and seizure-free EEG signals.

Design of two-channel linear phase biorthogonal wavelet filter banks via convex optimization

Filter banks can be designed via convex optimization techniques. We present a convex optimization based design method to design linear phase biorthogonal two-channel wavelet filter banks, with prescribed number of vanishing moments (VMs) or regularity. First we design an optimal low-pass filter (LPF) followed by the design of complementary optimal synthesis LPF. The optimization problem is formulated to minimize the convex combination of pass-band and stop-band errors, passband ripple and stop-band attenuation. The perfect reconstruction (PR) conditions are imposed as a set of linear equality constraints. The VM constraints are also incorporated in the design of the filters as a set of linear equality constraints. The wavelets are constructed from iterations of designed filter banks using the cascade algorithm. A few design examples are presented, which demonstrates that biorthogonal wavelets can be constructed using the proposed method. To the best of our knowledge, design of biorthogonal wavelet filter banks has not been addressed using convex optimization methods.

Design of Time–Frequency-Localized Two-Band Orthogonal Wavelet Filter Banks

In this paper, we design time–frequency-localized two-band orthogonal wavelet filter banks using convex semidefinite programming (SDP). The sum of the time variance and frequency variance of the filter is used to formulate a real symmetric positive definite matrix for joint time–frequency localization of filters. Time–frequency-localized orthogonal low-pass filter with specified length and regularity order is designed. For nonmaximally regular two-band filter banks of length twenty, it is found that, as we increase the regularity order, the solution of the SDP converges to the filters with time–frequency product (TFP) almost same as the Daubechies maximally regular filter of length twenty. Unlike the class of Daubechies maximally regular minimum phase wavelet filter banks, a rank minimization algorithm in a SDP is employed to obtain mixed-phase low-pass filters with TFP of the filters as well as the scaling and wavelet function better than the equivalent two-band Daubechies filter bank.

Optimal Design of Three-Band Orthogonal Wavelet Filter Bank with Stopband Energy for Identification of Epileptic Seizure EEG Signals

We design three-band orthogonal wavelet filter bank using unconstrained minimization of stopband energies of low-pass, band-pass, and high-pass filters. The analysis polyphase matrix of the orthogonal filter bank is represented by the parameterized structures such that the regularity condition is satisfied by the designed perfect reconstruction filter bank (PRFB). Dyadic and householder factorization of the analysis polyphase matrix is employed to impose perfect reconstruction, orthogonality, and regularity order of one. Three-band orthonormal scaling and wavelet functions are generated by the cascade iterations of the regular low-pass, band-pass, and high-pass filters. The designed three-band orthogonal filter bank of length 15 is used for feature extraction and classification of seizure and seizure-free electroencephalogram (EEG) signals. The classification accuracy of 99.33% is obtained from the designed filter bank which is better than the most of the recently reported results.

Congestion Control during Data Privacy in Secure Multiparty Computation

In this paper, we propose the methodology and design an algorithm to control congestion during Secure Multiparty Computation (SMC). As per our literature serve a lot of work has been done in SMC but they have worked only the main part of the SMC, privacy and correctness. Congestion control is one of the important components for the performance of the network and also is the most challenging one. This paper deals with the way of shaping the traffic to improve quality of service (QoS) in SMC. Under the congestion situation, the queue length may become very large in a short time, resulting in buffer overflow a\nd packet loss. So congestion control is necessary to ensure that users get the negotiated QoS. The objectives of traffic control and congestion control for SMC are: Support a set of QoS parameters and minimize network and endsystem complexity while maximizing network utilization. In our paper we carried out experiments for computing the FIFO length that ensures zero packet loss for different clock rate of the router.

Analysis of Data in Secure Multiparty Computation

A protocol is secure if the parties who want to compute their inputs hands it to the trusted parties. Trusted parties in turn compute the inputs using the function f and give the result to the respective parties after computation in such a way that no party can identify others party data. During computation of inputs, we had considered the factor, what if trusted third parties are malicious? Considering different probabilities for the malicious users, we have tried to find out the correctness of the result and percentage of system acceptability. We then tried to increase the number of TTPs in order to get the accuracy of the result. The aim of our proposed work is to identify what probability of malicious users will lead to the system in an unacceptable state.