Uniform Regularization Research Articles

LOCUS: A REGULARIZED BLIND SOURCE SEPARATION METHOD WITH LOW-RANK STRUCTURE FOR INVESTIGATING BRAIN CONNECTIVITY.

Network-oriented research has been increasingly popular in many scientific areas. In neuroscience research, imaging-based network connectivity measures have become the key for understanding brain organizations, potentially serving as individual neural fingerprints. There are major challenges in analyzing connectivity matrices, including the high dimensionality of brain networks, unknown latent sources underlying the observed connectivity, and the large number of brain connections leading to spurious findings. In this paper we propose a novel blind source separation method with low-rank structure and uniform sparsity (LOCUS) as a fully data-driven decomposition method for network measures. Compared with the existing method that vectorizes connectivity matrices ignoring brain network topology, LOCUS achieves more efficient and accurate source separation for connectivity matrices using low-rank structure. We propose a novel angle-based uniform sparsity regularization that demonstrates better performance than the existing sparsity controls for low-rank tensor methods. We propose a highly efficient iterative node-rotation algorithm that exploits the block multiconvexity of the objective function to solve the nonconvex optimization problem for learning LOCUS. We illustrate the advantage of LOCUS through extensive simulation studies. Application of LOCUS to Philadelphia Neurodevelopmental Cohort neuroimaging study reveals biologically insightful connectivity traits which are not found using the existing method.

Implementation of Takagi Sugeno Kang Fuzzy with Rough Set Theory and Mini-Batch Gradient Descent Uniform Regularization

The Takagi Sugeno Kang (TSK) fuzzy approach is popular since its output is either a constant or a function. Parameter identification and structure identification are the two key requirements for building the TSK fuzzy system. The input utilized in fuzzy TSK can have an impact on the number of rules produced in such a way that employing more data dimensions typically results in more rules, which causes rule complexity. This issue can be solved by employing a dimension reduction technique that reduces the number of dimensions in the data. After that, the resulting rules are improved with MBGD (Mini-Batch Gradient Descent), which is then altered with uniform regularization (UR). UR can enhance the classifier's fuzzy TSK generalization performance. This study looks at how the rough sets method can be used to reduce data dimensions and use Mini Batch Gradient Descent Uniform Regularization (MBGD-UR) to optimize the rules that come from TSK. 252 respondents' body fat data were utilized as the input, and the mean absolute percentage error (MAPE) was used to analyze the results. Jupyter Notebook software and the Python programming language are used for data processing. The analysis revealed that the MAPE value was 37%, falling into the moderate area. Doi: 10.28991/ESJ-2023-07-03-09 Full Text: PDF

Robust brain MR image compressive sensing via re-weighted total variation and sparse regression

Total variation (TV) and non-local self-similarity (NSS) are powerful tools for successfully enhancing compressive sensing performance. However, standard TV approaches often over-smooth detailed edges in the image, due to the uniform regularization of gradient magnitude. In this paper, a novel compressed sensing method for the reconstruction of medical images is proposed, the image edges are well preserved with the proposed reweighted TV. The redundancy of the NSS patch also is leveraged through the sparse regression model. The proposed model was solved with an efficient strategy of the Alternating Direction Method of Multipliers (ADMM) algorithm. Experimental results on thesimulated phantom, brain Magnetic resonance imaging (MRI) show that the proposed method outperforms the state-of-the-art compressed sensing approaches.

On Phase Correction in Tomographic Research

Under consideration is the problem of improving the contrast of the image obtained by processing tomographic projections with phase distortion. The study is based on the well-known intensity transfer equation. Unlike other works, this equation is solved in a bounded region of variation of the tomographic parameters. In a domain, a boundary value problem is posed for the intensity transfer equation which is then specialized for a three-dimensional parallel tomographic scheme. The case of two-dimensional tomography is also considered, together with the corresponding boundary value problem for the intensity transfer equation. We propose numerical methods for solving the boundary value problems of phase correction. The results are given of the numerical experiments on correction of tomographic projections and reconstruction of the structure of the objects under study (in particular, a slice of a geological sample) by using piecewise uniform regularization.

A regularized magnetotelluric inversion with a minimum support gradient constraint*

Magnetotelluric (MT) inversion is an ill-posed problem and the standard way to address it is through regularization, by adding a stabilizing functional to the data objective functional in order to obtain a stable solution. The traditional stabilizing functionals, in which a low-order differential operator is used, yield a smooth solution that may not be appropriate when anomalies occur in block patterns. In some cases the focused imaging of a sharp electrical boundary is necessary. Even though various experiments have used stabilizing functionals that are suitable to obtain a clear and sharp boundary, such as the minimum support (MS) and the minimum gradient support (MGS) functionals, there are still some limitations in practice. In this paper, the minimum support gradient (MSG) is proposed as the stabilizing functional. Under the uniform regularization framework, a regularized inversion with a variety of stabilizing functionals is performed and the inversion results are compared. This study shows that MSG inversion can not only obtain a clearly focused inversion but also a quite stable and robust one.

Optimize TSK Fuzzy Systems for Classification Problems: Minibatch Gradient Descent With Uniform Regularization and Batch Normalization

Takagi-Sugeno-Kang (TSK) fuzzy systems are flexible and interpretable machine learning models; however, they may not be easily optimized when the data size is large, and/or the data dimensionality is high. This paper proposes a mini-batch gradient descent (MBGD) based algorithm to efficiently and effectively train TSK fuzzy classifiers. It integrates two novel techniques: 1) uniform regularization (UR), which forces the rules to have similar average contributions to the output, and hence to increase the generalization performance of the TSK classifier; and, 2) batch normalization (BN), which extends BN from deep neural networks to TSK fuzzy classifiers to expedite the convergence and improve the generalization performance. Experiments on 12 UCI datasets from various application domains, with varying size and dimensionality, demonstrated that UR and BN are effective individually, and integrating them can further improve the classification performance.

Piecewise uniform regularization for the inverse problem of microtomography with a-posteriori error estimate

ABSTRACTAn inverse microtomography problem is under consideration in a class of functions with bounded V H variation. An algorithm for solving this problem is proposed based on Tikhonov's regularization with a special regularizer. The algorithm ensures piecewise uniform convergence of approximate solutions to exact solution of the inverse problem. In addition, the question of a-posteriori error estimate of approximate solutions obtained is considered. A new numerical algorithm for finding this estimate is proposed. Numerical experiments on solving a model inverse problem on the class of functions with bounded V H variation are presented along with the results of a-posteriori error estimate for approximate solutions obtained.

Estimating the Accuracy of a Method of Auxiliary Boundary Conditions in Solving an Inverse Boundary Value Problem for a Nonlinear Equation

An inverse boundary value problem for a nonlinear parabolic equation is considered. Two-sided estimates for the norms of values of a nonlinear operator in terms of those of a corresponding linear operator are obtained.On this basis, two-sided estimates for the modulus of continuity of a nonlinear inverse problem in terms of that of a corresponding linear problem are obtained. A method of auxiliary boundary conditions is used to construct stable approximate solutions to the nonlinear inverse problem. An accurate (to an order) error estimate for the method of auxiliary boundary conditions is obtained on a uniform regularization class.

Atlas-based reconstruction of high performance brain MR data

Image priors based on total variation (TV) and nonlocal patch similarity have shown to be powerful techniques for the reconstruction of magnetic resonance (MR) images from undersampled k-space measurements. However, due to the uniform regularization of gradients, standard TV approaches often over-smooth edges in the image, resulting in the loss of important details. This paper proposes a novel compressed sensing method which combines both external and internal information for the high-performance reconstruction of MRI data. A probabilistic atlas is used to model the spatial distribution of gradients that correspond to various anatomical structures in the image. This atlas is then employed to control the level of gradient regularization at each image location, within a weighted TV regularization prior. The proposed method also leverages the redundancy of nonlocal similar patches through a sparse representation model. Experiments on T1-weighted images from the ABIDE dataset show the proposed method to outperform state-of-the-art approaches, for different sampling rates and noise levels.

Adaptive regularization of earthquake slip distribution inversion

Regularization is a routine approach used in earthquake slip distribution inversion to avoid numerically abnormal solutions. To date, most slip inversion studies have imposed uniform regularization on all the fault patches. However, adaptive regularization, where each retrieved parameter is regularized differently, has exhibited better performances in other research fields such as image restoration. In this paper, we implement an investigation into adaptive regularization for earthquake slip distribution inversion. It is found that adaptive regularization can achieve a significantly smaller mean square error (MSE) than uniform regularization, if it is set properly. We propose an adaptive regularization method based on weighted total least squares (WTLS). This approach assumes that errors exist in both the regularization matrix and observation, and an iterative algorithm is used to solve the solution. A weight coefficient is used to balance the regularization matrix residual and the observation residual. An experiment using four slip patterns was carried out to validate the proposed method. The results show that the proposed regularization method can derive a smaller MSE than uniform regularization and resolution-based adaptive regularization, and the improvement in MSE is more significant for slip patterns with low-resolution slip patches. In this paper, we apply the proposed regularization method to study the slip distribution of the 2011 Mw 9.0 Tohoku earthquake. The retrieved slip distribution is less smooth and more detailed than the one retrieved with the uniform regularization method, and is closer to the existing slip model from joint inversion of the geodetic and seismic data.

Closed superstrings in a uniform magnetic field and regularization criterion

We summarize exact solutions of closed superstrings in a constant magnetic field, from a view point of the regularization criterion. Some models will be excluded according to this criterion. The spectrum-generating algebra is also constructed in these interacting models.

Spatially variant regularization of lateral displacement measurement using variance

The purpose of this work is to confirm the effectiveness of our proposed spatially variant displacement component-dependent regularization for our previously developed ultrasonic two-dimensional (2D) displacement vector measurement methods, i.e., 2D cross-spectrum phase gradient method (CSPGM), 2D autocorrelation method (AM), and 2D Doppler method (DM). Generally, the measurement accuracy of lateral displacement spatially varies and the accuracy is lower than that of axial displacement that is accurate enough. This inaccurate measurement causes an instability in a 2D shear modulus reconstruction. Thus, the spatially variant lateral displacement regularization using the lateral displacement variance will be effective in obtaining an accurate lateral strain measurement and a stable shear modulus reconstruction than a conventional spatially uniform regularization. The effectiveness is verified through agar phantom experiments. The agar phantom [60 mm (height) × 100 mm (lateral width) × 40 mm (elevational width)] that has, at a depth of 10 mm, a circular cylindrical inclusion (dia. = 10 mm) of a higher shear modulus (2.95 and 1.43 × 10 6 N/m 2, i.e., relative shear modulus, 2.06) is compressed in the axial direction from the upper surface of the phantom using a commercial linear array type transducer that has a nominal frequency of 7.5-MHz. Because a contrast-to-noise ratio (CNR) expresses the detectability of the inhomogeneous region in the lateral strain image and further has almost the same sense as that of signal-to-noise ratio (SNR) for strain measurement, the obtained results show that the proposed spatially variant lateral displacement regularization yields a more accurate lateral strain measurement as well as a higher detectability in the lateral strain image (e.g., CNRs and SNRs for 2D CSPGM, 2.36 vs 2.27 and 1.74 vs 1.71, respectively). Furthermore, the spatially variant lateral displacement regularization yields a more stable and more accurate 2D shear modulus reconstruction than the uniform regularization (however, for the regularized relative shear modulus reconstructions, slightly accurate, e.g., for 2D CSPGM, 1.51 vs 1.50). These results indicate that the spatially variant displacement component-dependent regularization will enable the 2D shear modulus reconstruction to be used as practical diagnostic and monitoring tools for the effectiveness of various noninvasive therapy techniques of soft tissue diseases (e.g., breast, liver cancers). Application of the regularization to the elevational displacement will also increase the stability of a three-dimensional (3D) reconstruction.

The Simulated Detection of the Objects at Variant Depth Using Depth-Adaptive Voxel Size Scheme in Diffuse Reflective Optical Tomography (DROT)

We study diffuse reflective optical tomography (DROT), aiming at the real-time imaging of the redox state change of the human cerebral cortex with the brain activity. In the uniform voxel size and simple regularization scheme we studied so far, the spatial resolution in the shallow region is unnecessarily good, whereas the noise and the sensitivity in the shallow region are too large to recover the signal in the depth. In this paper, we propose depth-adaptive voxel size reconstruction. By assigning wider lateral voxel size in the deep range, the relative sensitivity in the deep range is increased so that the signal in the deep range would be less masked by the presence of the unwanted target in the shallow range. We demonstrate that we can detect the deep objects without being masked by the shallow object using the proposed method.

Development and validation of an X-ray tomograph for two-phase flow.

This paper describes the development and validation of a high spatial resolution X-ray tomograph designed for the investigation of air-water two-phase flow. The device hardware mainly comprises a 60 keV X-ray source, a detector, and an accurate mechanical bench. Our study concentrated on accurate quantification with emphasis on the reconstruction procedure. As is well known, absorption gradients induce reconstruction artifacts when using standard algorithms based on uniform regularization. In the particular case of two-phase flow in a pipe, this leads to poor measurement accuracy in the vicinity of the walls. To overcome such effects, improved algorithms were developed during this study that involve spatially adaptive regularization methods. Preliminary calibration performed on static phantoms clearly exhibited the benefits of the advanced reconstruction algorithms. A validation procedure was carried out on an air-water bubble column, equipped with an optical probe, which could be translated in order to explore the 80 mm x 80 mm square cross section. Comparisons of local void fraction measurements were performed pixel by pixel. They demonstrate the accuracy improvement induced by the advanced reconstruction algorithms.

Spatial regularization of the electrocardiographic inverse problem and its application to endocardial mapping.

Numeric regularization methods for solving the inverse problem of electrocardiography in realistic volume conductor models have been mostly limited to uniform regularization in the spatial domain. A method of spatial regularization (SR) was developed and tested in canine, where each spatial spectral component of the volume conductor model was considered separately, and a SR operator was selected based on explicit a posteriori criterion at each time instant through the heartbeat. The inverse problem was solved in the left ventricle by reconstructing endocardial surface electrograms based on cavitary electrograms measured with the use of a noncontact, multielectrode probe. The results were validated based on electrograms measured in situ at the same endocardial locations using an integrated, multielectrode basket-catheter. A probe-endocardium three-dimensional model was determined from multiplane fluoroscopic images. The boundary element method was applied to solve the boundary value problem and derive the relationship between endocardial and probe potentials. Endocardial electrograms were reconstructed during both normal and paced rhythms using SR as well as standard, uniform, zeroth-order Tikhonov (ZOT) regularization. Compared to endocardial electrograms measured by the basket, electrograms reconstructed using SR [relative error (RE) = 0.32, correlation coefficient (CC) = 0.97, activation error = 3.3 ms] were superior to electrograms reconstructed using ZOT regularization (RE = 0.59, CC = 0.79, activation error = 4.9 ms). Therefore, regularization based on spatial spectral components of the model improves the solution of the inverse problem of electrocardiography compared to uniform regularization.

Application of functions of several variables with limited variations for piecewise uniform regularization of ill-posed problems

Application of functions of several variables with limited variations for piecewise uniform regularization of ill-posed problems

A fast scalable algorithm for discontinuous optical flow estimation

Multiple moving objects, partially occluded objects, or even a single object moving against the background gives rise to discontinuities in the optical flow field in corresponding image sequences. While uniform global regularization based moderately fast techniques cannot provide accurate estimates of the discontinuous flow field, statistical optimization based accurate techniques suffer from excessive solution time. A 'weighted anisotropic' smoothness based numerically robust algorithm is proposed that can generate discontinuous optical flow field with high speed and linear computational complexity. Weighted sum of the first-order spatial derivatives of the flow field is used for regularization. Less regularization is performed where strong gradient information is available. The flow field at any point is interpolated more from those at neighboring points along the weaker intensity gradient component. Such intensity gradient weighted regularization leads to Euler-Lagrange equations with strong anisotropies coupled with discontinuities in their coefficients. A robust multilevel iterative technique, that recursively generates coarse-level problems based on intensity gradient weighted smoothing weights, is employed to estimate discontinuous optical flow field. Experimental results are presented to demonstrate the efficacy of the proposed technique.

Estimate of the accuracy of solving an ill-posed problem using averaging functions

A linear ill-posed problem is considered with a large number of right-hand sides which are known with an identical level of error. A method of solving this problem of the uniform regularization type using averaging functions is considered. The connection between the properties of these functions and the properties of regularizing algorithms is examined. Estimates are obtained of the deviation of the approximate solutions from the averaged accurate solutions, which have the properties of continuous dependence on some parameters of the algorithms.

Comparison of methods of calculating atomic radial distribution functions

Tikhonov's method of weak and uniform regularization is used to solve the Fredholm integral equation of the first kind applied to the diffraction study of the structure of disordered condensed phases. The results of the calculation of the atomic radial distribution function are compared with those using numerical Fourier transforms for the case of liquid gallium.

Uniform regularization of the problem of calculating the values of an operator

Let X and Y be linear normed spaces, W a set in X, A an operator from W into Y, and\(\mathfrak{W}\) the set of all operators or the set ℒ of linear operators from X into Y. With δ>0 we put $$v\left( {\delta ,\mathfrak{M}} \right) = \mathop {\inf }\limits_{T \in \mathfrak{M}} \mathop {\sup }\limits_{x \in W} \mathop {\sup }\limits_{\left\| {\eta - x} \right\|_X \leqslant \delta } \left\| {Ax - T\eta } \right\|_Y $$ .