2D Signals Research Articles

A Non-Uniform Regional Partition Coding of One Dimension Signal and its Application

In this paper, a novel method for one-dimension-signal coding based on non-uniform regional partition is proposed. A given 1D signal can automatically be partitioned into different regions with different lengths and the one-variable polynomial is used to do the Least Square Approximation for signal values in each sub-region. When the approximation error and initial partition are specified, a specific signal partition result is obtained. Based on this algorithm, an effective denoising scheme of 1D signal is implemented and some of the experimental examples are illustrated to prove that the quality of the re-constructed signal, the denoising effect are all satisfactory and can be referenced by other researchers.

Three-dimensional morphological and signal intensity features for detection of intervertebral disc degeneration from magnetic resonance images

Advances in MRI hardware and sequences are continually increasing the amount and complexity of data such as those generated in high-resolution three-dimensional (3D) scanning of the spine. Efficient informatics tools offer considerable opportunities for research and clinically based analyses of magnetic resonance studies. In this work, we present and validate a suite of informatics tools for automated detection of degenerative changes in lumbar intervertebral discs (IVD) from both 3D isotropic and routine two-dimensional (2D) clinical T2-weighted MRI. An automated segmentation approach was used to extract morphological (traditional 2D radiological measures and novel 3D shape descriptors) and signal appearance (extracted from signal intensity histograms) features. The features were validated against manual reference, compared between 2D and 3D MRI scans and used for quantification and classification of IVD degeneration across magnetic resonance datasets containing IVD with early and advanced stages of degeneration. Combination of the novel 3D-based shape and signal intensity features on 3D (area under receiver operating curve (AUC) 0.984) and 2D (AUC 0.988) magnetic resonance data deliver a significant improvement in automated classification of IVD degeneration, compared to the combination of previously used 2D radiological measurement and signal intensity features (AUC 0.976 and 0.983, respectively). Further work is required regarding the usefulness of 2D and 3D shape data in relation to clinical scores of lower back pain. The results reveal the potential of the proposed informatics system for computer-aided IVD diagnosis from MRI in large-scale research studies and as a possible adjunct for clinical diagnosis.

Monitoring tunneling induced ground displacements using distributed fiber-optic sensing

Determination and monitoring of tunneling induced ground displacement is an important component in tunneling design and construction. In recent years several technologies for distributed strain measurement along fiber optics have been developed, namely the Brillouin Optical Time Domain Reflectometry (or Analysis) – BOTDR/A and the Rayleigh backscatter wavelength interferometry (OBR). This paper presents how these technologies could be used to monitor and define ground displacement models through an appropriate 2D and 3D optimization and signal analysis of information derived from a horizontally laid fiber above the tunnel. The suggested approach is evaluated in two field investigations, one involving excavation of a 3m diameter tunnel by TBM at depth of 18m, and the other installation of a 1m diameter water main by pipe-jacking at depth of 6m. Comparison between the results obtained by the different technologies shows that they are equally suitable for the suggest approach. The suggests approach allows reliable determination of the parameters involved in empirical ground displacement models, and allows field validation that the tunneling process lies within the design bounds. An interesting observation, supported by the analytical models, is that non-perpendicular alignment of the fiber, relatively to the tunnel line, results in a shift in the peak strain location as the tunnel advances. It was demonstrated that the rate of change in peak strain location, with tunnel advancement, can be used to obtain the settlement trough length parameter, without the need for complete evaluation of all other model parameters.

Least-squares phase estimation with wrapped measurements and branch points

A nonorthogonal model for 2D signals with rotational components is presented, which enables estimation of phase values from observations of its local gradients. In this research, the rotational components are caused by the presence of branch points, which indicates phase wrapping. Using the proposed model, the phase is estimated using standard least-squares or recently proposed wavelet techniques by processing a linear combination of the wrapped observed gradients and the curl generated by phase wrapping.

Compressive sensing using the modified entropy functional

In most compressive sensing problems, ℓ1 norm is used during the signal reconstruction process. In this article, a modified version of the entropy functional is proposed to approximate the ℓ1 norm. The proposed modified version of the entropy functional is continuous, differentiable and convex. Therefore, it is possible to construct globally convergent iterative algorithms using Bregmanʼs row-action method for compressive sensing applications. Simulation examples with both 1D signals and images are presented.

Hilbert and Riesz Transforms Using Atomic Function for Quaternionic Phase Computation

Complex and hyper-complex valued filtering play a substantial role in signal processing, especially to obtain local features in the frequency and phase domain. In the case of 1D signals, the analytic signal is typically computed using the Hilbert transform. Such complex representation allows us to compute the phase and magnitude of the signal. For high-dimension signals, the Riesz transform and partial Hilbert transforms have been used as extensions of the analytic signal to compute the local phase and local orientation. A major goal of this work is to highlight the role of an atomic function (AF) up(x) as a kernel for the Hilbert and Riesz transforms. It is well known that the use of phase information has a big potential because it is invariant to illumination and rotations. In this regard, we show the advantages to carry out computations of local Riesz phase using the atomic function instead of the classical global-phase approach. In addition, we explain how the atomic function up(x), formulated in the quaternionic algebra framework, can be used as a building block to perform multiple analytical operations commonly used in image processing, such as low-pass filter, derivatives, local phase processing, steering quaternionic filters, multi-resolution analysis and symmetries detection.

L-statistics based space/spatial-frequency filtering of 2D signals in heavy tailed noise

Most of the commonly used stationary filtering techniques, performed either in the spatial or frequency domains, fail to produce good results for noisy signals with fast varying non-stationary structures. The filtering results could be improved by using space/spatial-frequency based non-stationary filters. Hence, a robust approach to space/spatial-frequency analysis of two-dimensional noisy signals is proposed in this paper. It is based on the two-dimensional L-estimate forms of the short-time Fourier transform, the spectrogram and the S-method. The proposed space/spatial-frequency distributions are used to define the L-estimate space-varying filtering procedure. It is designed for denoising of 2D non-stationary signals affected by the strong impulsive or mixed heavy-tailed and Gaussian noise. The efficiency of the proposed procedure is tested on the examples with interferogram-like images, textures and satellite images.

Gaussian Blurring-Invariant Comparison of Signals and Images

We present a Riemannian framework for analyzing signals and images in a manner that is invariant to their level of blurriness, under Gaussian blurring. Using a well known relation between Gaussian blurring and the heat equation, we establish an action of the blurring group on image space and define an orthogonal section of this action to represent and compare images at the same blur level. This comparison is based on geodesic distances on the section manifold which, in turn, are computed using a path-straightening algorithm. The actual implementations use coefficients of images under a truncated orthonormal basis and the blurring action corresponds to exponential decays of these coefficients. We demonstrate this framework using a number of experimental results, involving 1D signals and 2D images. As a specific application, we study the effect of blurring on the recognition performance when 2D facial images are used for recognizing people.

Edge-preserving image deconvolution with nonlocal domain transform

In this paper, we propose a new approach for performing efficient edge-preserving image deconvolution algorithm based on a nonlocal domain transform (NLDT). We present the geodesic distance-preserving transforming procedure of a 1D signal embedded in 2D space into a new 1D domain via a transformation for simplicity. The nonlocal domain transform derives from the (1D) nonlocal means filter kernel and iteratively and separably applies 1D edge-aware operations. In order to solve the main issue with noisy images that is finding robust estimates for their derivatives, we develop an efficient joint nonlocal domain transform filter in the deblurring process. Furthermore, we derive the discrepancy principle to automatically adjust the regularization parameter at each iteration. We compare our deconvolution algorithm with many competitive deconvolution techniques in terms of ISNR and visual quality. 2013 Elsevier Ltd.

Robust Image Registration Using Mutual Information and Structural Features of Images

Medical image registration methods which use mutual information as similarity measure have been improved in recent decades. Mutual Information is a basic concept of Information theory which indicates the dependency of two random variables (or two images). In the most of these intensity based methods, images are treated as 1D signal and each pixel is considered independent from its neighbors. Although the location of pixels in an image includes more information than the intensity of them, it is ignored in most of the intensity based methods and they use only the intensity of a pixel to compute the images’ histogram. There are some other methods like Regional Mutual Information (RMI) which use both of intensity of pixels and the information of image structure for registration. In this paper using the intensity of neighbor pixels of a pixel in image, it is proposed to make a new feature matrix for any image and measure mutual information between these matrices to determine how much similar two images are. Because of the using structural information of images, this method is more robust against noise and intensity variation and it is more accurate in comparison with methods which use only intensity of pixels and also it is faster than methods like RMI. Experimental results of the rigid registration of clinical brain images (CT), show the superiority of the proposed scheme.

Permutation Meets Parallel Compressed Sensing: How to Relax Restricted Isometry Property for 2D Sparse Signals

Traditional compressed sensing considers sampling a 1D signal. For a multidimensional signal, if reshaped into a vector, the required size of the sensing matrix becomes dramatically large, which increases the storage and computational complexity significantly. To solve this problem, the multidimensional signal is reshaped into a 2D signal, which is then sampled and reconstructed column by column using the same sensing matrix. This approach is referred to as parallel compressed sensing, and it has much lower storage and computational complexity. For a given reconstruction performance of parallel compressed sensing, if a so-called acceptable permutation is applied to the 2D signal, the corresponding sensing matrix is shown to have a smaller required order of restricted isometry property condition, and thus, lower storage and computation complexity at the decoder are required. A zigzag-scan-based permutation is shown to be particularly useful for signals satisfying the newly introduced layer model. As an application of the parallel compressed sensing with the zigzag-scan-based permutation, a video compression scheme is presented. It is shown that the zigzag-scan-based permutation increases the peak signal-to-noise ratio of reconstructed images and video frames.

Reproducing Subgroups of $Sp(2,\mathbb{R})$ . Part I: Algebraic Classification

We classify the connected Lie subgroups of the symplectic group $Sp(2,\mathbb{R})$ whose elements are matrices in block lower triangular form. The classification is up to conjugation within $Sp(2,\mathbb{R})$. Their study is motivated by the need of a unified approach to continuous 2D signal analyses, as those provided by wavelets and shearlets.

ANALYSIS OF SIGNAL TREND BY A PSEUDO-EMD METHOD WITH APPLICATIONS TO WEATHER AND SPEECH DATA

Empirical mode decomposition (EMD) has been established as a valuable tool in determining nonlinear signal trend. EMD decomposes a one-dimensional (1D) signal into hierarchical components known as intrinsic mode functions (IMFs) and a residue, based on the local properties of the signal. The first IMF depicts the highest local oscillations, while the residue depicts the trend of a signal/data. In each iteration of the EMD process, interpolation is applied to some local maxima and minima points to form upper and lower envelopes, respectively. But, the application of interpolation methods causes huge computation time and other artifacts in the decomposition, which limits the use of EMD for many real life signals. This paper proposes an effective method that replaces the interpolation step by direct envelope estimation using order statistics filters, which results in decreased computation time, following a similar EMD approach that has been recently proposed for two-dimensional data or image analysis. The modified EMD of this paper called pseudo EMD (P-EMD) method is particularly useful in determining, analyzing, and/or modifying the trend of various signals to obtain and/or produce some desired results/outcomes. Several synthetic and real-life signals such as speech signal and sea level pressure and temperature are tested to verify the effectiveness of the P-EMD. From the results, P-EMD has been found as a superior alternative for trend analysis of signal/data, since it results in more accurate trend compared to the other interpolation based EMD methods such as classical EMD (CEMD) and a modified EMD (MEMD), and also facilitates faster computation.

Real-time precise detection of regular grids and matrix codes

The traditional approach in detecting sets of concurrent and/or parallel lines is to first detect lines in the image and then find such groups of them which meet the concurrence condition. The Hough Transform can be used for detecting the lines and variants of HT such as the Cascaded Hough Transform can be used to detect the vanishing points. However, these approaches disregard much of the information actually accumulated to the Hough space. This article proposes using the Hough space as a 2D signal instead of just detecting the local maxima and processing them. On the example of QRcode detection, it is shown that this approach is computationally cheap, robust, and accurate. The proposed algorithm can be used for efficient and accurate detection and localization of matrix codes (QRcode, Aztec, DataMatrix, etc.) and chessboard-like calibration patterns.

Adaptive Boundary Effect Processing For Empirical Mode Decomposition Using Template Matching

This paper is contributed to the boundary effect problem of the empirical mode decomposition algorithm, which results in a serious distortion in the EMD sifting process. An adaptive method for processing boundary effect in the empirical mode decomposition sifting process is presented, which has exploited the local time- or spatial- scales and the waveform or texture characteristics near boundary of the signal or image to extend the signal or image so that additional subsignal or subimage are obtained. The extended section is taken as the most suited subsignal or subimage to the inner signal or image by template matching operation. The multiple components of the original signal or image are available by applying EMD algorithm to the extended signal or image and then leaving out the extended parts of the decomposed components. Simulation results have proved that the proposed template matching based decomposition method outperforms the neural network extending method, the mirror extrema extending method and the AR model extending method for 1D signals, and perform texture extraction effectively for 2D natural images such as defect-free and defect fabrics.

A fast modified Newton's method for curvature based denoising of 1D signals

We propose a novel fast numerical method for denoising of 1D signals based on curvature minimization. Motivated by the primal-dual formulation for total variation minimization introduced by Chan, Golub, and Mulet, the proposed method makes use of some auxiliary variables to reformulate the stiff terms presented in the Euler-Lagrange equation which is a fourth-order differential equation. A direct application of Newton's method to the resulting system of equations often fails to converge. We propose a modified Newton's iteration which exhibits local superlinear convergence and global convergence in practical settings. The method is much faster than other existing methods for the model. Unlike all other existing methods, it also does not require tuning any additional parameter besides the model parameter. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Application Specific Integrated Circuit Implementation of Discrete Fractional Fourier Transform

Over the last two decades, discrete fractional Fourier transform (DFrFT), which is a generalization of the discrete Fourier transform (DFT), has attracted a lot of attention of the researchers. Several methods for computation of the DFrFT have been discussed in the literature but very little effort has been made towards realizing a dedicated hardware for its real time computation. In this paper we propose an application specific integrated circuit (ASIC) design for 16-point DFrFT computation of a 1D signal, which has been synthesized using Very High Speed Integrated Circuit hardware description language (VHDL). The same design can be easily extended for any number of signal samples. The simulation results of the proposed ASIC are very close to the theoretical DFrFT results obtained using the MATLAB software.

Comparison of various sampling schemes and accumulation profiles in covariance spectroscopy with exponentially decaying 2D signals

Two-dimensional covariance (COV2D) spectroscopy offers an alternative approach to 2D Fourier transformation (FT2D) to obtain homo-nuclear correlation NMR spectra. Considerable saving in experimental time, without loss of resolution, can be achieved by incorporating COV2D with: (i) uniform sampling with a t1 cut-off (CUO) or non-uniform sampling (NUS), and (ii) constant or Gaussian accumulation profiles. We find that covariance treatment, combined with the CUO sampling and Gaussian accumulation profile provides better gain in experimental time, with respect to that required with FT2D NMR. This is in contrast with the maximum entropy (MaxEnt) reconstruction for 2D spectra, which works better with the NUS scheme rather than with the CUO scheme. We further discuss the experimental conditions that define this optimum acquisition, and explain how these parameters can easily be optimized 'on the fly' according to the desired resolution, which is quite sample-dependent.

Robust Image Information Identification Algorithm Based on Time-Frequency Chirplet-Radon Transformation

In this paper, a robust image watermarking method in two-dimensional space/spatial-frequency distributions domain is proposed which is robust against geometric distortion. This watermarking is detected by a linear frequency change. The one-dimensional chirplet transformation and radon transformation are used to detect the watermark. The chirp signals are used as watermarks and this type of signals is resistant to all stationary filtering methods and exhibits geometrical symmetry. In the two-dimensional Radon-Wigner transformation domain, the chirp signals used as watermarks change only its position in space/spatial-frequency distribution, after applying linear geometrical attack, such as scale rotation and cropping. But the two-dimensional Radon-Wigner transformation needs too much difficult computing. So the image is put into a series of 1D signal by choosing scalable local time windows. The watermarks are embedded in the 1D Chirplet-Radon transformation domains. The watermark thus generated is invisible and performs well in StirMark test. Compared with other watermarking algorithms, this algorithm is more robust, especially against geometric distortion, while having excellent frequency properties.

Effect of form cues on 1D and 2D motion pooling

Local-motion information can provide either 1-dimensional (1D) or 2-dimensional (2D) solutions. 1D signals occur when the aperture problem has not been solved, so each signal is an estimate of the local-orthogonal component of the object’s motion. 2D signals occur when the aperture problem has been solved, so each signal is an estimate of the object’s motion. Previous research (JoV, 2009, 9, 1–25) has shown that 1D and 2D signals are pooled differently, via intersection-of-constraints (IOC) and vector-average processes, respectively. Previous research (e.g. Vis. Res., 2003, 2290–2301) has also indicated that form cues can influence how motion signals are perceived. We investigated whether forms cues can affect the pooling of motion signals and whether they differentially affect the pooling of 1D and 2D signals. Global-Gabor (GG) and global-plaid (GP) stimuli were used. These stimuli consist of multiple apertures that contain either Gabors or plaids, respectively. In the GG stimulus the global solution is defined by having the Gabor carriers move (1D signals) such that they are consistent with a single IOC-defined solution. In the GP stimuli the plaid motion (2D signals) are consistent with a vector-average solution defined by a Gaussian distribution. Form cues can be introduced by adding orientation information to the apertures that is either consistent (aligned with) or inconsistent (orthogonal to) with the global-solution. With the 1D stimuli, form cues affect how the motion signals are pooled, with motion being perceived in the direction defined by the orientation cue. Orientation cues had no direct effect on the pooling of the 2D signals.