Nonlinear Encoding Research Articles

Optimal Phase Transitions in Compressed Sensing

Compressed sensing deals with efficient recovery of analog signals from linear encodings. This paper presents a statistical study of compressed sensing by modeling the input signal as an i.i.d. process with known distribution. Three classes of encoders are considered, namely optimal nonlinear, optimal linear and random linear encoders. Focusing on optimal decoders, we investigate the fundamental tradeoff between measurement rate and reconstruction fidelity gauged by error probability and noise sensitivity in the absence and presence of measurement noise, respectively. The optimal phase transition threshold is determined as a functional of the input distribution and compared to suboptimal thresholds achieved by popular reconstruction algorithms. In particular, we show that Gaussian sensing matrices incur no penalty on the phase transition threshold with respect to optimal nonlinear encoding. Our results also provide a rigorous justification of previous results based on replica heuristics in the weak-noise regime.

Reconstruction of undersampled radial PatLoc imaging using total generalized variation

In the case of radial imaging with nonlinear spatial encoding fields, a prominent star-shaped artifact has been observed if a spin distribution is encoded with an undersampled trajectory. This work presents a new iterative reconstruction method based on the total generalized variation, which reduces this artifact. For this approach, a sampling operator (as well as its adjoint) is needed that maps data from PatLoc k-space to the final image space. It is shown that this can be realized as a type-3 nonuniform fast Fourier transform, which is implemented by a combination of a type-1 and type-2 nonuniform fast Fourier transform. Using this operator, it is also possible to implement an iterative conjugate gradient SENSE based method for PatLoc reconstruction, which leads to a significant reduction of computation time in comparison to conventional PatLoc image reconstruction methods. Results from numerical simulations and in vivo PatLoc measurements with as few as 16 radial projections are presented, which demonstrate significant improvements in image quality with the total generalized variation-based approach.

Constellation Shaping for Bit-Interleaved LDPC Coded APSK

An energy-efficient approach is presented for shaping a bit-interleaved low-density parity-check (LDPC) coded amplitude phase-shift keying (APSK) system. A subset of the interleaved bits output by a binary LDPC encoder are passed through a nonlinear shaping encoder whose output is more likely to be a zero than a one. The "shaping" bits are used to select from among a plurality of subconstellations, while the unshaped bits are used to select the symbol within the subconstellation. Because the shaping bits are biased, symbols from lower-energy subconstellations are selected more frequently than those from higher-energy subconstellations. An iterative decoder shares information among the LDPC decoder, APSK demapper, and shaping decoder. Information rates are computed for a discrete set of APSK ring radii and shaping bit probabilities, and the optimal combination of these parameters is identified for the additive white Gaussian noise (AWGN) channel. With the assistance of extrinsic-information transfer (EXIT) charts, the degree distributions of the LDPC code are optimized for use with the shaped APSK constellation. Simulation results show that the combination of shaping, degree-distribution optimization, and iterative decoding can achieve a gain in excess of 1 dB in AWGN at a rate of 3 bits/symbol compared with a system that does not use shaping, uses an unoptimized code from the DVB-S2 standard, and does not iterate between decoder and demodulator.

Improved conductivity reconstruction from multi-echo MREIT utilizing weighted voxel-specific signal-to-noise ratios

Magnetic resonance electrical impedance tomography (MREIT) is a non-invasive method to visualize cross-sectional electrical conductivity and/or current density by measuring a magnetic flux density signal when an electrical current is injected into a subject. In the MREIT system, it is crucial to reduce the scan duration while maintaining spatial resolution and contrast for practical in vivo implementation. The purpose of the study is to optimize the measured magnetic flux density using an interleaved multiple-echo pulse sequence (injected current nonlinear encoding) that acquires each spatial position multiple times, although these pixels vary between echoes in their signal-to-noise ratio due to (a) T*2 decay and (b) the current density passing through the pixel. Using the acquired multiple measured magnetic flux densities, the noise level for the measured magnetic flux density Bz at each pixel is estimated using the relationship between the intensity of the magnitude and the width of the injected current. We determine an optimal combination of the multiply acquired magnetic flux densities, which optimally reduces the random noise artifacts. We develop a new denoising technique and apply it to a recovered conductivity distribution with a known noise level of the recovered magnetic flux density, which is designed to provide a stable internal conductivity distribution, while sustaining resolution. The proposed method uses three key steps: the first step is optimizing the magnetic flux density by using the multiple-echo magnetic flux densities at each pixel, the second step is estimating the noise level of this optimized magnetic flux density and the third step is applying a denoising technique using the pixel-specific estimated noise level. Numerical simulation experiments using a three-dimensional cylindrical phantom model validated the proposed method. Multiple-echo Bz data were generated, including in short T*2 or low spin-density regions, as a function of T*2 and the temporal extent of the injected current. In the simulation experiment, comparing between an equally averaged and the optimized Bz methods, relative L2-mode errors were 0.053 and 0.024, respectively. In an actual imaging experiment of an agarose gel filled with objects of various conductivities and shapes, we acquired six echoes per repetition time. The optimal weighting factors minimized the effects of noise in Bz, and provided reconstructed conductivity maps that suppressed noise artifacts that normally accumulate in the low signal-to-noise-ratio defect regions.

Newton Method to Recover the Phase Accumulated during MRI Data Acquisition

For an internal conductivity image, magnetic resonance electrical impedance tomography (MREIT) injects an electric current into an object and measures the induced magnetic flux density, which appears in the phase part of the acquired MR image data. To maximize signal intensity, the injected current nonlinear encoding (ICNE) method extends the duration of the current injection until the end of the MR data reading. It disturbs the usual linear encoding of the MR k‐space data used in the inverse Fourier transform. In this study, we estimate the magnetic flux density, which is recoverable from nonlinearly encoded MR k‐space data by applying a Newton method.

Closed-Loop Measurements of Iso-Response Stimuli Reveal Dynamic Nonlinear Stimulus Integration in the Retina

Neurons often integrate information from multiple parallel signaling streams. How a neuron combines these inputs largely determines its computational role in signal processing. Experimental assessment of neuronal signal integration, however, is often confounded by cell-intrinsic nonlinear processes that arise after signal integration has taken place. To overcome this problem and determine how ganglion cells in the salamander retina integrate visual contrast over space, we used automated online analysis of recorded spike trains and closed-loop control of the visual stimuli to identify different stimulus patterns that give the same neuronal response. These iso-response stimuli revealed a threshold-quadratic transformation as a fundamental nonlinearity within the receptive field center. Moreover, for a subset of ganglion cells, the method revealed an additional dynamic nonlinearity that renders these cells particularly sensitive to spatially homogeneous stimuli. This function is shown to arise from a local inhibition-mediated dynamic gain control mechanism.

Dimensionality reduction of radio map with nonlinear autoencoder

Presented is a nonlinear autoencoder, which reduces the dimensionality of a radio map, and consequently contributes to the reduction of the power consumption of client devices in a fingerprint-based indoor localisation. The nonlinear encoder shows significantly better performance in reducing the dimensionality than principal component analysis, a representative linear technique, did.

Application of the fractional Fourier transform to image reconstruction in MRI

The classic paradigm for MRI requires a homogeneous B(0) field in combination with linear encoding gradients. Distortions are produced when the B(0) is not homogeneous, and several postprocessing techniques have been developed to correct them. Field homogeneity is difficult to achieve, particularly for short-bore magnets and higher B(0) fields. Nonlinear magnetic components can also arise from concomitant fields, particularly in low-field imaging, or intentionally used for nonlinear encoding. In any of these situations, the second-order component is key, because it constitutes the first step to approximate higher-order fields. We propose to use the fractional Fourier transform for analyzing and reconstructing the object's magnetization under the presence of quadratic fields. The fractional fourier transform provides a precise theoretical framework for this. We show how it can be used for reconstruction and for gaining a better understanding of the quadratic field-induced distortions, including examples of reconstruction for simulated and in vivo data. The obtained images have improved quality compared with standard Fourier reconstructions. The fractional fourier transform opens a new paradigm for understanding the MR signal generated by an object under a quadratic main field or nonlinear encoding.

Performance Analysis for Magnetic Resonance Imaging With Nonlinear Encoding Fields

Nonlinear spatial encoding fields for magnetic resonance imaging (MRI) hold great promise to improve on the linear gradient approaches by, for example, enabling reduced imaging times. Imaging schemes that employ general nonlinear encoding fields are difficult to analyze using traditional measures. In particular, the resolution is spatially varying, characterized by a position-dependent point spread function (PSF). Likewise, the use of nonlinear encoding fields creates an additional spatial dependence on the signal-to-noise ratio (SNR). Although the two properties of resolution and SNR are linked, in this work we focus on the latter. To this end, we examine the pixel variance, which requires a computation that is often not feasible for nonlinear encoding schemes. This paper presents a general formulation for the performance analysis of imaging schemes using arbitrary encoding fields. The analysis leads to the derivation of a practical and computationally efficient performance metric, which is demonstrated through simulation examples.

Centralized and Distributed Semiparametric Compression of Piecewise Smooth Functions

This paper introduces novel wavelet-based semiparametric centralized and distributed compression methods for a class of 1-D piecewise smooth functions. Classical centralized compression schemes are based on a relatively complex, nonlinear encoder and a simple, linear decoder. Recently, a new paradigm in compression called distributed source coding has emerged. This setup involves multiple encoders, where each one partially observes the source, and a centralized decoder. First, we focus on the dual situation of the centralized compression with a simple encoder and a complex decoder. We show that, by incorporating parametric estimation into the decoding procedure, it is possible to achieve the same rate-distortion performance as that of a conventional wavelet-based compression scheme. Second, we consider the distributed compression scenario, where each independent encoder partially observes the 1-D piecewise smooth function. We propose a new wavelet-based distributed compression scheme that uses parametric estimation to perform joint decoding. Our analysis shows that it is possible for the proposed scheme to achieve the same compression performance as that of a joint encoding scheme.

Encoding and decoding V1 fMRI responses to natural images with sparse nonparametric models

Functional MRI (fMRI) has become the most common method for investigating the human brain. However, fMRI data present some complications for statistical analysis and modeling. One recently developed approach to these data focuses on estimation of computational encoding models that describe how stimuli are transformed into brain activity measured in individual voxels. Here we aim at building encoding models for fMRI signals recorded in the primary visual cortex of the human brain. We use residual analyses to reveal systematic nonlinearity across voxels not taken into account by previous models. We then show how a sparse nonparametric method [bJ. Roy. Statist. Soc. Ser. B71 (2009b) 1009-1030] can be used together with correlation screening to estimate nonlinear encoding models effectively. Our approach produces encoding models that predict about 25% more accurately than models estimated using other methods [Nature452 (2008a) 352-355]. The estimated nonlinearity impacts the inferred properties of individual voxels, and it has a plausible biological interpretation. One benefit of quantitative encoding models is that estimated models can be used to decode brain activity, in order to identify which specific image was seen by an observer. Encoding models estimated by our approach also improve such image identification by about 12% when the correct image is one of 11,500 possible images.

Simultaneously driven linear and nonlinear spatial encoding fields in MRI

Spatial encoding in MRI is conventionally achieved by the application of switchable linear encoding fields. The general concept of the recently introduced PatLoc (Parallel Imaging Technique using Localized Gradients) encoding is to use nonlinear fields to achieve spatial encoding. Relaxing the requirement that the encoding fields must be linear may lead to improved gradient performance or reduced peripheral nerve stimulation. In this work, a custom-built insert coil capable of generating two independent quadratic encoding fields was driven with high-performance amplifiers within a clinical MR system. In combination with the three linear encoding fields, the combined hardware is capable of independently manipulating five spatial encoding fields. With the linear z-gradient used for slice-selection, there remain four separate channels to encode a 2D-image. To compare trajectories of such multidimensional encoding, the concept of a local k-space is developed. Through simulations, reconstructions using six gradient-encoding strategies were compared, including Cartesian encoding separately or simultaneously on both PatLoc and linear gradients as well as two versions of a radial-based in/out trajectory. Corresponding experiments confirmed that such multidimensional encoding is practically achievable and demonstrated that the new radial-based trajectory offers the PatLoc property of variable spatial resolution while maintaining finite resolution across the entire field-of-view.

Optimization of multiply acquired magnetic flux density Bz using ICNE-Multiecho train in MREIT

The aim of magnetic resonance electrical impedance tomography (MREIT) is to visualize the electrical properties, conductivity or current density of an object by injection of current. Recently, the prolonged data acquisition time when using the injected current nonlinear encoding (ICNE) method has been advantageous for measurement of magnetic flux density data, Bz, for MREIT in the signal-to-noise ratio (SNR). However, the ICNE method results in undesirable side artifacts, such as blurring, chemical shift and phase artifacts, due to the long data acquisition under an inhomogeneous static field. In this paper, we apply the ICNE method to a gradient and spin echo (GRASE) multi-echo train pulse sequence in order to provide the multiple k-space lines during a single RF pulse period. We analyze the SNR of the measured multiple Bz data using the proposed ICNE-Multiecho MR pulse sequence. By determining a weighting factor for Bz data in each of the echoes, an optimized inversion formula for the magnetic flux density data is proposed for the ICNE-Multiecho MR sequence. Using the ICNE-Multiecho method, the quality of the measured magnetic flux density is considerably increased by the injection of a long current through the echo train length and by optimization of the voxel-by-voxel noise level of the Bz value. Agarose-gel phantom experiments have demonstrated fewer artifacts and a better SNR using the ICNE-Multiecho method. Experimenting with the brain of an anesthetized dog, we collected valuable echoes by taking into account the noise level of each of the echoes and determined Bz data by determining optimized weighting factors for the multiply acquired magnetic flux density data.

MREIT conductivity imaging of canine head using multi-echo pulse sequence

In magnetic resonance electrical impedance tomography (MREIT), we measure induced magnetic flux densities subject to multiple injection currents to reconstruct cross-sectional conductivity images. Spin echo pulse sequence has been widely used in MREIT and produce postmortem and in vivo conductivity images of animal and human subjects. The image quality depends on the SNR of the measured magnetic flux density image. In order to reduce the scan time and current amplitude while keeping the image quality, we have developed a multi-echo pulse sequence for MREIT. In this study, we show results of canine head MREIT imaging experiments using the multi-echo pulse sequence. Compared to the injection current nonlinear encoding (ICNE) pulse sequence, it provides a higher SNR of MR magnitude images by combining multiple echo signals. Noise in measured magnetic flux density data is significantly reduced due to an increased current injection time over multiple echo signals. These allow us to significantly decrease the total scan time. Reconstructed conductivity images of a canine head show enhanced conductivity contrast between gray and white matter using the multi-echo pulse sequence. In our future work, we will focus on in vivo human and disease model animal experiments using the new MREIT pulse sequence.

On Linear Coding Schemes for Stabilizing LTI Control with Multiple Sensors

We examine the problem of designing the encoding and control policies of a linear stochastic control system, where the communication channel between the plant state observer (sensor) and the controller is a lossy wireless channel that is constrained in terms of transmit power and bandwidth. For a first-order ARMA modeled plant with Gaussian statistics, when there are two sensors observing the plant, nonlinear encoding is shown to result in smaller cost at time instant [Formula: see text] compared to the linear schemes, if transmissions are carried out over parallel Gaussian independent channels. In this paper, optimal linear coding schemes for the case of multiple sensors are examined. They are shown to minimize the control cost at the infinite time horizon, when the wireless channel is accessed using time division multiplexing. Our analysis is carried out for when separation between the state estimation and control is possible, and the optimal steady state control law is certainty equivalent. The distortion lower bound for estimating the plant state is derived, along with the necessary conditions on the transmit power that minimize the steady state control cost. We also propose a linear scheme that reaches the distortion bound asymptotically under relaxed conditions.

Improved Frey Chaotic Digital Encoder for Trellis-Coded Modulation

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this brief, nonlinear digital filters with finite precision are analyzed as recursive systematic convolutional (RSC) encoders. An infinite-impulse-response (IIR) digital filter with finite precision (wordlength of <formula formulatype="inline"><tex Notation="TeX">$N$</tex></formula> bits) is a rate-1 RSC encoder over a Galois field <formula formulatype="inline"><tex Notation="TeX">$GF(2^{N})$</tex></formula>. The Frey chaotic filter is analyzed for different wordlengths <formula formulatype="inline"><tex Notation="TeX">$N$</tex></formula>, and it is demonstrated that the trellis performances can be enhanced by proper filter design. Therefore, a modified definition for the encoding rate is provided, and a trellis design method is proposed for the Frey filter, which consists of reducing the encoding rate from 1 to 1/2. This trellis optimization partially follows Ungerboeck's rules, i.e., increasing the performances of the encoded chaotic transmission in the presence of noise. In fact, it is demonstrated that for the same spectral efficiency, the modified Frey encoder outperforms the original Frey encoder only for <formula formulatype="inline"> <tex Notation="TeX">$N = 2$</tex></formula>. To show the potential of these nonlinear encoders, it is demonstrated that a particular nonlinear digital filter over <formula formulatype="inline"><tex Notation="TeX">$GF(4)$</tex></formula> is equivalent to a <formula formulatype="inline"><tex Notation="TeX">$GF(2)$</tex></formula> conventional optimum RSC encoder. The symbol error rate (SER) is estimated for all the proposed schemes, and the results show the expected coding gains as compared to their equivalent nonencoded and linear versions. </para>

Noise analysis and MR pulse sequence optimization in MREIT using an injected current nonlinear encoding (ICNE) method

Magnetic resonance current density imaging (MRCDI) and magnetic resonance electrical impedance tomography (MREIT) visualize an internal distribution of current density and/or conductivity by injecting current into an electrically conductive object such as the human body using an MRI scanner. MREIT measures the induced magnetic flux density which appears in the phase part of the acquired MR image data. Recently, the injected current nonlinear encoding (ICNE) method in MREIT extended the duration of the current injection until the end of a reading gradient to maximize the signal intensity of the magnetic flux density. In this paper, we investigate the signal-to-noise ratio (SNR) of the magnetic flux density measured by the ICNE method in the presence of a zero-mean Gaussian random noise in measured k-space MR data. Based on the analysis of the noise standard deviation of the magnetic flux density, we determine an optimal combination between the current injection pulse width Tc and data acquisition time Ts which minimize the noise level of the measured magnetic flux density for a given echo time TE. On one hand, theoretically, the proposed ICNE MR pulse sequence using the optimal data acquisition time Ts* reduces the noise level of the measured magnetic flux density by about 42.3% compared with the optimal data acquisition time of the conventional MREIT pulse sequence. On the other hand, practically, the prolonged Ts* may result in undesirable artifacts including blurring, chemical shift and phase error along the phase encoding direction. We observe that the noise level is a function of the data acquisition time Ts and the rate of change in the noise level is slow near Ts = Ts*. Numerical phantom experiments show that a compromised Ts between the ordinary data acquisition time and the optimal Ts* reduces a relatively large amount of undesirable artifacts and almost maintains the optimized noise level of the measured magnetic flux density.

Measurement of induced magnetic flux density using injection current nonlinear encoding (ICNE) in MREIT

Magnetic resonance electrical impedance tomography (MREIT) measures induced magnetic flux densities subject to externally injected currents in order to visualize conductivity distributions inside an electrically conducting object. Injection currents induce magnetic flux densities that appear in phase parts of acquired MR image data. In the conventional current injection method, we inject currents during the time segment between the end of the first RF pulse and the beginning of the reading gradient in order to ensure the gradient linearity. Noting that longer current injections can accumulate more phase changes, we propose a new pulse sequence called injection current nonlinear encoding (ICNE) where the duration of the injection current pulse is extended until the end of the reading gradient. Since the current injection during the reading gradient disturbs the gradient linearity, we first analyze the MR signal produced by the ICNE pulse sequence and suggest a novel algorithm to extract the induced magnetic flux density from the acquired MR signal. Numerical simulations and phantom experiments show that the new method is clearly advantageous in terms of the reduced noise level in measured magnetic flux density data. The amount of noise reduction depends on the choice of the data acquisition time and it was about 24% when we used a prolonged data acquisition time of 10.8 ms. The ICNE method will enhance the clinical applicability of the MREIT technique when it is combined with an appropriate phase artefact minimization method.

The Optimal Method of Nonlinear Decoding for Nonlinear Encoding of QAM Signal

The Optimal Method of Nonlinear Decoding for Nonlinear Encoding of QAM Signal

Speech coding through adaptive combined nonlinear prediction

In this paper we propose a nonlinear predictive speech encoder based on an adaptive combiner with a neural net that weighs the prediction of several nonlinear predictors. Thus, we exploit the advantages of data fusion on a nonlinear prediction scheme, where it appears in a more natural way than for linear predictors. Experimental results reveal that this scheme outperforms the fixed combination (with mean, median, etc. operators) up to 1.5 dB in SEGSNR.