Optimal construction of subband coders using Lloyd-Max quantizers

A new method is presented for the analysis of the effects of Lloyd-Max quantization in subband filterbanks and for the optimal design of such filterbanks. A rigorous statistical model of a vector Lloyd-Max quantizer is established first, consisting of a linear time-invariant filter followed by additive noise uncorrelated/with the input. On the basis of this model, an expression for this variance of the error of a subband coder using Lloyd-Max quantizers is explicitly determined. Given analysis filters that statistically separate the subbands, it is shown that this variance is minimized if the synthesis filters are chosen, which mould achieve perfect reconstruction in lossless coding. The globally optimum of such a filterbank, minimizing the coder error variance, is further obtained by proper choice of its analysis filters. An alternative design method is also evaluated and optimized. In this, the errors correlated with the signal are set to zero, leaving a random error residue uncorrelated with the signal. This design method is optimized by choosing the analysis filters so as to minimize the random error variance. The results are evaluated experimentally in the realistic setting of a logarithmically split subband image coding scheme.     

Wavelet-based Rician noise removal for magnetic resonance imaging

It is well known that magnetic resonance magnitude image data obey a Rician distribution. Unlike additive Gaussian noise, Rician "noise" is signal-dependent, and separating signal from noise is a difficult task. Rician noise is especially problematic in low signal-to-noise ratio (SNR) regimes where it not only causes random fluctuations, but also introduces a signal-dependent bias to the data that reduces image contrast. This paper studies wavelet-domain filtering methods for Rician noise removal. We present a novel wavelet-domain filter that adapts to variations in both the signal and the noise.     

A Functional Analysis Approach to Subband System Approximation and Identification

The subband system identification method consists of identifying a linear system in the time-frequency domain. This technique can be also used to approximate a linear system in the same domain. In both cases, it has the advantage of having a very high numerical efficiency; however, analyzing such a technique is not trivial, and the best setup for subband system approximation and identification is not clear. In this paper, we propose a functional analysis setting to the analysis of the subband technique, which leads us to some results on both subband system approximation and identification. Concerning system approximation, we provide an analytical expression to calculate the optimal subband approximation of a given full-band system, when the quality of the approximation is measured by the power of the output error signal, assuming a white input signal. We also provide a tight approximation error bound, for a given subband configuration, which applies in the case where the unknown system to be approximated is known to be the finite-impulse response (FIR) of a given order. Concerning system identification, we provide a novel identification strategy that consists of identifying a "low quality" subband model and use it to build the required model for either subband or delayless reconstruction. This identification strategy reduces the computational complexity of the identification process and yields significantly smaller asymptotic residual errors, when compared with the existing methods     

Modeling, analysis, and optimum design of quantized M-band filterbanks

This paper provides a rigorous modeling and analysis of quantization effects in M-band subband codecs, followed by optimal filter bank design and compensation. The codec is represented by a polyphase decomposition of the analysis/synthesis filter banks and an embedded nonlinear gain-plus-additive noise model for the pdf-optimized scalar quantizers. We construct an equivalent time-invariant but nonlinear structure operating at the slow clock rate that allows us to compute the exact expression for the mean square quantization error in the reconstructed output. This error is shown to consist of two components: a distortion component and a dominant random noise component uncorrelated with the input signal. We determine the optimal paraunitary and biorthogonal FIR filter coefficients, compensators, and integer bit allocation to minimize this MSE subject to the constraints of filter length, average bit rate, and perfect reconstruction (PR) in the absence of quantizers. The biorthogonal filter bank results in a smaller MSE but the filter coefficients are very sensitive to signal statistics and to average bit constraints. By comparison, the paraunitary structure is much more robust. We also show that the null-compensated design that eliminates the distortion component is more robust than the optimally-compensated case that minimizes the total MSE, but only at nominal conditions. Both modeling and optimal design are validated by simulation in the two-channel case     

基于随机误差项的小波阈值去噪算法研究

针对振动环境下陀螺仪输出信号噪声干扰严重的问题,提出了一种用随机误差项改进小波阈值的去噪算法。通过对陀螺仪输出信号进行小波分解,根据频率成分将信号分解为多层;然后,对分解在各层的信号进行随机误差项辨识,进而利用随机误差项系数获取各层的噪声阈值;最后,利用获取的阈值进行小波去噪。改进阈值的提出,旨在解决Donoho全局阈值中因阈值选取过大或过小而产生的噪声误判或噪声残留问题,使噪声去除更彻底。通过实验分析,证明了本算法既能有效去除信号噪声,解决噪声残留的问题;又能保留输出的有效信号,解决噪声误判的问题。     

Two-dimensional filter bank design for optimal reconstruction usinglimited subband information

In this correspondence, we propose design techniques for analysis and synthesis filters of 2-D perfect reconstruction filter banks (PRFB's) that perform optimal reconstruction when a reduced number of subband signals is used. Based on the minimization of the squared error between the original signal and some low-resolution representation of it, the 2-D filters are optimally adjusted to the statistics of the input images so that most of the signal's energy is concentrated in the first few subband components. This property makes the optimal PRFB's efficient for image compression and pattern representations at lower resolutions for classification purposes. By extending recently introduced ideas from frequency domain principal component analysis to two dimensions, we present results for general 2-D discrete nonstationary and stationary second-order processes, showing that the optimal filters are nonseparable. Particular attention is paid to separable random fields, proving that only the first and last filters of the optimal PRFB are separable in this case. Simulation results that illustrate the theoretical achievements are presented.     

The Role of AM-to-PM Conversion in Memoryless Nonlinear Systems

A generalized proof is presented that AM-to-PM conversion can only degrade, never improve, the intermodulation-noise performance of memoryless nonlinear systems with random input signals having even probability density functions, and a measure of degradation is defined. It is also shown for such signals that AM-to-PM conversion causes a deterministic constant phase shift to be added to the argument of the signal component at the output but has no other effect on its phase. This class of inputs includes one or the sum of several PSK signals, as well as large ensembles that can be modeled as Gaussian noise. The latter are dealt with by using Bussgang's theorem on input-output cross correlation. In the proof, Bussgang's theorem is extended to the complex case, to include phase as well as amplitude nonlinearities, yielding a complex version of the theorem. For Gaussian inputs it is shown that the undistorted signal and the intermodulation noise at the output of such systems are uncorrelated.     

Estimation of Optimal Parameter of Regularization of Signal Recovery

In this paper there are researched regularizing properties of discretization in a space of output signals for some linear operator equation with noisy data. The essence of proposed method is selection of discretization level which is a parameter of the regularization in this context by the principle of equality of random and deterministic components of the input signal recovering error. It is shown the method, i.e. the solution which is discrete by input signal is stable to small inaccuracies in input signal. At that in case of definite level of output signal measurements inaccuracy the recovering error of input signal is unambiguously defined by input signal sampling increment that allows to select reasonably the regularization parameter for specific criterion, for example, for definite measurements inaccuracy. Specific calculations and examples are represented in explicit form for single-dimension case but this does not restricts generality of proposed method.     

Optimal filter banks for signal reconstruction from noisy subbandcomponents

Conventional design techniques for analysis and synthesis filters in subband processing applications guarantee perfect reconstruction of the original signal from its subband components. The resulting filters, however, lose their optimality when additive noise due, for example, to signal quantization, disturbs the subband sequences. We propose filter design techniques that minimize the reconstruction mean squared error (MSE) taking into account the second order statistics of signals and noise in the case of either stochastic or deterministic signals. A novel recursive, pseudo-adaptive algorithm is proposed for efficient design of these filters. Analysis and derivations are extended to 2-D signals and filters using powerful Kronecker product notation. A prototype application of the proposed ideas in subband coding is presented. Simulations illustrate the superior performance of the proposed filter banks versus conventional perfect reconstruction filters in the presence of additive subband noise     

System identification using nonstationary signals

The conventional method for identifying the transfer function of an unknown linear system consists of a least squares fit of its input to its output. It is equivalent to identifying the frequency response of the system by calculating the empirical cross-spectrum between the system's input and output, divided by the empirical auto-spectrum of the input process. However, if the additive noise at the system's output is correlated with the input process, e.g., in case of environmental noise that affects both system's input and output, the method may suffer from a severe bias effect. We present a modification of the cross-spectral method that exploits nonstationary features in the data in order to circumvent bias effects caused by correlated stationary noise. The proposed method is particularly attractive to problems of multichannel signal enhancement and noise cancellation, when the desired signal is nonstationary in nature, e.g., speech or image     

Improvements to the standard theory for photoreceiver noise

The standard theory for photoreceiver noise unrealistically defines the system transfer function solely in terms of the input and output pulse shapes, based on the assumption that equalization is provided at the receiver output. Most photoreceivers reported in the literature, however, are only front ends and do not include equalizers, making direct application of the conventional noise expressions inappropriate. Even if equalization is provided, a signal-dependent definition of the transfer function will be accurate only under certain limited conditions. Furthermore, it is unrealistic to assume a given pulse shape at the input. In this paper we consider the effect of incorporating a more realistic transfer function into the conventional noise theory. We choose the transimpedance amplifier for our analysis due to its widespread popularity; however, our approach is general and can he applied to a broad class of photoreceivers. Since our transfer function is based on a physical circuit, our results can be used to estimate photoreceiver noise performance without making any assumptions on the input or output pulse shapes     

Two-dimensional subband transforms: theory and applications

The one-dimensional subband FFT (SB-FFT) and one-dimensional SB-DCT were extended to the two-dimensional (2-D) case to obtain the 2-D SB-FFT and the 2-D SB-DCT. The two-dimensional subband transforms are based on subband decomposition of the input sequence in both dimensions. They use knowledge about the input signal to obtain an approximation to their transform by discarding the computations in bands that have little energy in both dimensions. Computational savings can be obtained from calculating only the remaining subbands. In many applications the computational speed is so important that some error in the calculated transform can be accepted. In image processing, due to the nature of most natural scenes, most of the energy content of the corresponding digitised images is concentrated predominantly in the low-low spatial frequency domain. The concentration of the energy in a localised region of the transform domain makes the approximate subband transform computation quite suitable for the calculation of the 2-D image spectra. The complexity and accuracy of both 2-D transforms are studied in detail in the paper. The approximation errors in both transforms are derived for a general case, in which any band out of M bands is to be computed. Both transforms are modified to be fully adaptive to select the band of interest to be computed. Image transform application examples are included. Savings in computational complexity of image transforms are shown. The efficiency of subband transforms of different images is indicated by computing the signal-to-noise ratio in the reconstructed images.     

Output signal-to-noise ratios of FM correlation systems

An analysis is presented on the output signal-to-noise ratios for the FM correlation systems having an FM detector in each input channel of a conventional correlator. The input consists of a frequency-modulated signal combined additively with stationary narrow-band Gaussian random noise. A general expression is derived for the output signal-to-noise ratio. A detailed calculation is made for the output signal-to-noise ratios when each input signal is a carrier frequency-modulated by a Gaussian random process and the integrating filter is of RC low pass. The dependence of the output signal-to-noise ratios on several parameters is discussed.     

The Theory of Operation of an Equal-Gain Predetection Regenerative Diversity Combiner with Rayleigh Fading Channels

The theory of operation of an equal-gain predetection diversity combiner employing the Granlund technique of utilizing a combination of feedback and feedforward is presented. It is shown that the time delays of the filters and the loop phase shifts uniquely determine the operating frequencies within the system. The random FM at the discriminator output is shown to be a weighted sum of that on the input signals, and an upper limit for additional random FM caused by phase-shift error between the loops is derived. At the combiner output, the noise is found to be the sum of the uncorrelated noise from each input channel while the carrier amplitude of the diversity signal is the sum of the carrier amplitudes of the input signals after being cophased.     

1/f分形噪声的一种多尺度Kalman滤波方法

针对淹没在1／f分形噪声中的有用信号恢复问题,提出了一种基于小波变换与Kalman滤波的多尺度滤波算法。首先将带有1／f分形噪声的信号分解成多尺度的子带信号,通过小波变换对1／f分形噪声的白化作用,消除了1／f分形噪声的自相似性和长程相关性。然后在小波域内,利用Kalman滤波实现了噪声和有用信号的分离,估计出了各子带中的有用信号。最后进行小波重构,较好地恢复出淹没在1／f分形噪声中的有用信号。仿真实验表明,使用多尺度Kalman滤波器能有效地抑制分形噪声,显著地提高了信噪比。     

Hard-Limiter Intermodulation Product Magnitudes for Three Signals in Random Noise

When two or more sinusoidal signals are passed through a hard limiter, intermodulation (IM) products are developed at the limiter output. The magnitudes of the IM products are a function of the relative levels of the input signals, as well as the input noise, if present. This paper considers the case of three input sinusoids plus random noise. Quantitative values for IM-product magnitudes are presented as a function of input signal-signal-to-noise ratio for two specific cases of relative input signal levels.     

Stochastic differential equations: an approach to the generation ofcontinuous non-Gaussian processes

The generation of continuous random processes with jointly specified probability density and covariation functions is considered. The proposed approach is based on the interpretation of the simulated process as a stationary output of a nonlinear dynamic system, excited by white Gaussian noise and described by a system of a first-order stochastic differential equations (SDE). The authors explore how the statistical characteristics of the equation's solution depends on the form of its operator and on the intensity of the input noise. Some aspects of the approximate synthesis of stochastic differential equations and examples of their application to the generation of non-Gaussian continuous processes are considered. The approach should be useful in signal processing when it is necessary to translate the available a priori information on the real random process into the language of its Markov model as well as in simulation of continuous correlated processes with the known probability density function     

A New Normalized Subband Adaptive Filter Algorithm with Individual Variable Step Sizes

Proposed is a novel variable step size normalized subband adaptive filter algorithm, which assigns an individual step size for each subband by minimizing the mean square of the noise-free a posterior subband error. Furthermore, a noniterative shrinkage method is used to recover the noise-free priori subband error from the noisy subband error signal. Simulation results using the colored input signals have demonstrated that the proposed algorithm not only has better tracking capability than the existing subband adaptive filter algorithms, but also exhibits lower steady-state error.     

Modified digital correlator and its estimation errors

The method of using auxiliary random noise to unbias the output of the multilevel digital correlator is investigated. It is shown that any random noise satisfying certain conditions may be used. The uniformly distributed noise that has been used previously in the polarity-coincidence correlator is a special case. Discretely distributed auxiliary noise, although not satisfying the conditions, has the same effect as quantizing the input signals more finely. The mean-square error of this modified digital correlator is analyzed.     

Image Super Resolution Based on Interpolation of Wavelet Domain High Frequency Subbands and the Spatial Domain Input Image

In this paper, we propose a new super‐resolution technique based on interpolation of the high‐frequency subband images obtained by discrete wavelet transform (DWT) and the input image. The proposed technique uses DWT to decompose an image into different subband images. Then the high‐frequency subband images and the input low‐resolution image have been interpolated, followed by combining all these images to generate a new super‐resolved image by using inverse DWT. The proposed technique has been tested on Lena, Elaine, Pepper, and Baboon. The quantitative peak signal‐to‐noise ratio (PSNR) and visual results show the superiority of the proposed technique over the conventional and state‐of‐art image resolution enhancement techniques. For Lena's image, the PSNR is 7.93 dB higher than the bicubic interpolation.