It has been widely recognized that in compressed sensing, many restricted isometry property (RIP) conditions can be easily obtained by using the null space property (NSP) with its null space constant (NSC) 0<θ≤1 to construct a contradicted method for sparse signal recovery. However, the traditional NSP with θ=1 will lead to conservative RIP conditions. In this paper, we extend the NSP with 0<θ<1 to a scale NSP, which uses a factor τ to scale down all vectors belonged to the Null space of a sensing matrix. Following the popular proof procedure and using the scale NSP, we establish more relaxed RIP conditions with the scale factor τ, which guarantee the bounded approximation recovery of all sparse signals in the bounded noisy through the constrained l1 minimization. An application verifies the advantages of the scale factor in the number of measurements.

Homotopy algorithm provides a very powerful approach to select the best regularization term for the l1-norm minimization problem, but it is lack of provision for handling singularities. The singularity problem might be frequently encountered in practical implementations if the measurement matrix contains duplicate columns, approximate columns or columns with linear dependent in kernel space. The existing method for handling Homotopy singularities introduces a high-dimensional random ridge term into the measurement matrix, which has at least two shortcomings: 1) it is very difficult to choose a proper ridge term that applies to several different measurement matrices; and 2) the high-dimensional ridge term may accumulatively degrade the recovery performance for large-scale applications. To get around these shortcomings, a modified ridge-adding method is proposed to deal with the singularity problem, which introduces a low-dimensional random ridge vector into the l1-norm minimization problem directly. Our method provides a much simpler implementation, and it can alleviate the degradation caused by the ridge term because the dimension of ridge term in the proposed method is much smaller than the original one. Moreover, the proposed method can be further extended to handle the SVMpath initialization singularities. Theoretical analysis and experimental results validate the performance of the proposed method.

Linear Prediction (LP) analysis is commonly used in speech processing. LP is based on Auto-Regressive (AR) model and it estimates the AR model parameter from signals with l2-norm optimization. Recently, sparse estimation is paid attention since it can extract significant features from big data. The sparse estimation is realized by l1 or l0-norm optimization or regularization. Sparse LP analysis methods based on l1-norm optimization have been proposed. Since excitation of speech is not white Gaussian, a sparse LP estimation can estimate more accurate parameter than the conventional l2-norm based LP. These are time-invariant and real-valued analysis. We have been studied Time-Varying Complex AR (TV-CAR) analysis for an analytic signal and have evaluated the performance on speech processing. The TV-CAR methods are l2-norm methods. In this paper, we propose the sparse TV-CAR analysis based on adaptive LASSO (Least absolute shrinkage and selection operator) that is l1-norm regularization and evaluate the performance on F0 estimation of speech using IRAPT (Instantaneous RAPT). The experimental results show that the sparse TV-CAR methods perform better for a high level of additive Pink noise.

Image quality depends on the randomness of the k-space signal under-sampling in compressed sensing MRI (CS-MRI), especially for two-dimensional image acquisition. We investigate the feasibility of non-random signal under-sampling CS-MRI to stabilize the quality of reconstructed images and avoid arbitrariness in sampling point selection. Regular signal under-sampling for the phase-encoding direction is adopted, in which sampling points are chosen at equal intervals for the phase-encoding direction while varying the sampling density. Curvelet transform was adopted to remove the aliasing artifacts due to regular signal under-sampling. To increase the incoherence between the measurement matrix and the sparsifying transform function, the scale of the curvelet transform was varied in each iterative image reconstruction step. We evaluated the obtained images by the peak-signal-to-noise ratio and root mean squared error in localized 3×3 pixel regions. Simulation studies and experiments showed that the signal-to-noise ratio and the structural similarity index of reconstructed images were comparable to standard random under-sampling CS. This study demonstrated the feasibility of non-random under-sampling based CS by using the multi-scale curvelet transform as a sparsifying transform function. The technique may help to stabilize the obtained image quality in CS-MRI.

In this letter, a new semi-blind approach incorporating the bounded nature of communication sources with the distance between the equalizer outputs and the training sequence is proposed. By utilizing the sparsity property of l1-norm cost function, the proposed algorithm can outperform the semi-blind method based on higher-order statistics (HOS) criterion especially for transmitting sources with non-constant modulus. Experimental results demonstrate that the proposed method shows superior performance over the HOS based semi-blind method and the classical training-based method for QPSK and 16QAM sources equalization. While for 64QAM signal inputs, the proposed algorithm exhibits its superiority in low signal-to-noise-ratio (SNR) conditions compared with the training-based method.

We consider the problem of two-dimensional (2-D) angles of arrival estimation using a newly proposed structure of nonuniform linear array, referred to as nested coprime array with compressed inter-element spacing (CACIS). By constructing a cross-correlation matrix of the received signals, the nested CACIS exhibits a larger number of degrees of freedom. A two-step weighted l1-norm penalty strategy is proposed to fully utilize these degrees of freedom, where the weight matrices are constructed by MUSIC spectrum function and the threshold function, respectively. The proposed method has several salient advantages over the compared method, including increased resolution and accuracy, estimating many more number of sources and suppressing spurious peaks efficiently. Simulation results validate the superiority of the proposed method.

In this letter, a new subband adaptive filter (SAF) which is robust against impulsive noise in system identification is presented. To address the vulnerability of adaptive filters based on the L2-norm optimization criterion to impulsive noise, the robust SAF (R-SAF) comes from the L1-norm optimization criterion with a constraint on the energy of the weight update. Minimizing L1-norm of the a posteriori error in each subband with a constraint on minimum disturbance gives rise to robustness against impulsive noise and the capable convergence performance. Simulation results clearly demonstrate that the proposal, R-SAF, outperforms the classical adaptive filtering algorithms when impulsive noise as well as background noise exist.

We propose a novel algorithm for the recovery of non-sparse, but compressible signals from linear undersampled measurements. The algorithm proposed in this paper consists of two steps. The first step recovers the signal by the l1-norm minimization. Then, the second step decomposes the l1 reconstruction into major and minor components. By using the major components, measurements for the minor components of the target signal are estimated. The minor components are further estimated using the estimated measurements exploiting a maximum a posterior (MAP) estimation, which leads to a ridge regression with the regularization parameter determined using the error bound for the estimated measurements. After a slight modification to the major components, the final estimate is obtained by combining the two estimates. Computational cost of the proposed algorithm is mostly the same as the l1-nom minimization. Simulation results for one-dimensional computer generated signals show that the proposed algorithm gives 11.8% better results on average than the l1-norm minimization and the lasso estimator. Simulations using standard images also show that the proposed algorithm outperforms those conventional methods.

PCA-L1 (principal component analysis based on L1-norm maximization) is an approximate solution of L1-PCA (PCA based on the L1-norm), and has robustness against outliers compared with traditional PCA. However, the more dimensions the feature space has, the more calculation time PCA-L1 consumes. This paper focuses on an initialization procedure of PCA-L1 algorithm, and proposes a fast method of PCA-L1 using Gram-Schmidt orthogonalization. Experimental results on face recognition show that the proposed method works faster than conventional PCA-L1 without decrease of recognition accuracy.

Linear Discriminant Analysis (LDA) is a well-known feature extraction method for supervised subspace learning in statistical pattern recognition. In this paper, a novel method of LDA based on a new L1-norm optimization technique and its variances are proposed. The conventional LDA, which is based on L2-norm, is sensitivity to the presence of outliers, since it used the L2-norm to measure the between-class and within-class distances. In addition, the conventional LDA often suffers from the so-called small sample size (3S) problem since the number of samples is always smaller than the dimension of the feature space in many applications, such as face recognition. Based on L1-norm, the proposed methods have several advantages, first they are robust to outliers because they utilize the L1-norm, which is less sensitive to outliers. Second, they have no 3S problem. Third, they are invariant to rotations as well. The proposed methods are capable of reducing the influence of outliers substantially, resulting in a robust classification. Performance assessment in face application shows that the proposed approaches are more effectiveness to address outliers issue than traditional ones.

This paper addresses the estimation of time delay between two spatially separated noisy signals by system identification modeling with the input and output corrupted by additive white Gaussian noise. The proposed method is based on a modified adaptive Butler-Cantoni equalizer that decouples noise variance estimation from channel estimation. The bias in time delay estimates that is induced by input noise is reduced by an IIR whitening filter whose coefficients are found by the Burg algorithm. For step time-variant delays, a dual mode operation scheme is adopted in which we define a normal operating (tracking) mode and an interrupt operating (optimization) mode. In the tracking mode, only a few coefficients of the impulse response vector are monitored through L1-normed finite forward differences tracking, while in the optimization mode, the time delay optimized. Simulation results confirm the superiority of the proposed approach at low signal-to-noise ratios.