Null-space Components Research Articles

A novel least squares approach generating approximations orthogonal to the null space of the operator

We introduce a novel least squares functional specifically formulated to solve linear partial differential equations with operators that have a nonempty null space. Our method involves projecting the solution onto the orthogonal complement of the operator's null space to overcome challenges encountered by conventional numerical methods when nonzero null components are present. We describe the theoretical framework of the proposed method and validate it through numerical examples that show improved accuracy and usability in cases where traditional methods are less effective due to significant null space components. Overall, this approach provides a practical and reliable solution for partial differential equations with substantial null space components.

Null Space Matters: Range-Null Decomposition for Consistent Multi-Contrast MRI Reconstruction

Consistency and interpretability have long been the critical issues in MRI reconstruction. While interpretability has been dramatically improved with the employment of deep unfolding networks (DUNs), current methods still suffer from inconsistencies and generate inferior anatomical structure. Especially in multi-contrast scenes, different imaging protocols often exacerbate the concerned issue. In this paper, we propose a range-null decomposition-assisted DUN architecture to ensure consistency while still providing desirable interpretability. Given the input decomposed, we argue that the inconsistency could be analytically relieved by feeding solely the null-space component into proximal mapping, while leaving the range-space counterpart fixed. More importantly, a correlation decoupling scheme is further proposed to narrow the information gap for multi-contrast fusion, which dynamically borrows isotropic features from the opponent while maintaining the modality-specific ones. Specifically, the two features are attached to different frequencies and learned individually by the newly designed isotropy encoder and anisotropy encoder. The former strives for the contrast-shared information, while the latter serves to capture the contrast-specific features. The quantitative and qualitative results show that our proposal outperforms most cutting-edge methods by a large margin. Codes will be released on https://github.com/chenjiachengzzz/RNU.

Exposure to an incompatible virtual surgery impacts the null space components of the muscle patterns after re-adaptation but not the task performance.

Under the synergy hypothesis, novel muscle synergies may be required for motor skill learning. We have developed a "virtual surgery" experimental paradigm that alters the mapping of muscle activations onto virtual cursor motion during an isometric reaching task using myoelectric control. By creating virtual surgeries that are "incompatible" with the original synergies, we can investigate learning new muscle synergies in controlled experimental conditions. We have previously shown that participants are able to improve their task performance after an incompatible virtual surgery, using novel muscle patterns to overcome the perturbation. In this work, we investigated whether the activation of novel muscle patterns, that are required after an incompatible virtual surgery, affects task performance or the muscle patterns after re-adaptation to the unperturbed baseline mapping. We found that experiencing an incompatible virtual surgery did not affect the task performance during the baseline mapping. However, the adaptation to the incompatible virtual surgery resulted in changes in the null space components of the muscle patterns used in the unperturbed task.

GAN Prior Based Null-Space Learning for Consistent Super-resolution

Consistency and realness have always been the two critical issues of image super-resolution. While the realness has been dramatically improved with the use of GAN prior, the state-of-the-art methods still suffer inconsistencies in local structures and colors (e.g., tooth and eyes). In this paper, we show that these inconsistencies can be analytically eliminated by learning only the null-space component while fixing the range-space part. Further, we design a pooling-based decomposition (PD), a universal range-null space decomposition for super-resolution tasks, which is concise, fast, and parameter-free. PD can be easily applied to state-of-the-art GAN Prior based SR methods to eliminate their inconsistencies, neither compromise the realness nor bring extra parameters or computational costs. Besides, our ablation studies reveal that PD can replace pixel-wise losses for training and achieve better generalization performance when facing unseen downsamplings or even real-world degradation. Experiments show that the use of PD refreshes state-of-the-art SR performance and speeds up the convergence of training up to 2~10 times.

An L2-finite element approximation of the solution to div/curl-systems with nonempty null space

Partial differential equations involving divergence or curl operators often introduce nonempty null-space components in their solutions, causing conventional numerical methods to produce inaccurate results. In this paper, we present a new algorithm that addresses this challenge by explicitly accounting for the null-space component. The dual system least squares method is used to identify this component and remove it from the approximation using negative-norm minimization. Our approach retains the use of H1-conforming basis functions, ensuring that the resulting approximation is orthogonal to the null space of the operator. The effectiveness of our algorithm is demonstrated through numerical experiments with various partial differential equations, showing improved accuracy compared to conventional methods.

Deep Null Space Learning Improves Dataset Recovery for High Frame Rate Synthetic Transmit Aperture Imaging.

Synthetic transmit aperture (STA) imaging benefits from the two-way dynamic focusing to achieve optimal lateral resolution and contrast resolution in the full field of view, at the cost of low frame rate (FR) and low signal-to-noise ratio (SNR). In our previous studies, compressed sensing based synthetic transmit aperture (CS-STA) and minimal l2-norm least squares (LS-STA) methods were proposed to recover the complete STA dataset from fewer Hadamard-encoded plane wave (PW) transmissions. Results demonstrated that, compared with STA imaging, CS/LS-STA can maintain the high resolution of STA in the full field of view and improve the contrast in the deep region with increased FR. However, these methods would introduce errors to the recovered STA datasets and subsequently produce severe artifacts to the beamformed images, especially in the shallow region. Recently, we discovered that the theoretical explanation for the error introduced in the LS-STA-based recovery is that the LS-STA method neglects the null space component of the real STA dataset. To deal with this problem, we propose to train a convolutional neural network under the null space learning framework (CNN-Null) to estimate the missing null space component) for high-accuracy recovery of the STA dataset from fewer Hadamard-encoded PW transmissions. The mapping between the low-quality STA dataset (i.e., the range space component of the real STA dataset recovered using the LS-STA method) and the missing null space component of the real STA dataset was learned by the network with the high-quality STA dataset (obtained using full Hadamard-encoded STA imaging, HE-STA) as training labels. The performance of the proposed CNN-Null method was compared with the baseline LS-STA, conventional STA, and HE-STA methods, in terms of visual quality, normalized root-mean-square error (NRMSE), generalized contrast-to-noise ratio (gCNR), and lateral full width at half maximum (FWHM). The results demonstrate that the proposed method can greatly improve the recovery accuracy of the STA datasets (lower NRMSE) and therefore effectively suppress the artifacts presented in the images (especially in the shallow region) obtained using the LS-STA method (with a gCNR improvement of 0.4 in the cross-sectional carotid artery images). In addition, the proposed method can maintain the high lateral resolution of STA with fewer (as low as 16) PW transmissions, as LS-STA does.

Acoustic tomography of temperature and velocity fields by using the radial basis function and alternating direction method of multipliers

Considering the acoustic refraction effect, an investigation of simultaneously visualizing the temperature and velocity fields is carried out by using the combined meshfree radial basis function and alternating direction method of multipliers (RBF-ADMM). The radial basis function is introduced to recover the null-space component. It assumes that the reconstructed fields are continuous functions according to their nature, which, therefore, adds the prior information of the continuity in the fields’ reconstruction. Meanwhile, the alternating direction method of multipliers with improving anti-noise capacity is employed to iteratively resolve the inverse acoustic tomographic problem in low Mach number, which makes the parallel implementation and distributed optimization possible. Different from the traditional methods for solving boundary value problems, a fast two-point ray-tracing algorithm is introduced to trace the curved sound waves more efficiently. Proof-of-concept demonstrations including the representative temperature and aerodynamic fields are performed. For the two scenarios of two-peak and two-vortex inhomogeneous fields with 10% measurement errors, the averaged absolute errors of the reconstructed temperature fields are 120.7 K, and 39.9 K, respectively, and that of the reconstructed velocity fields are 0.24 m/s and 1.23 m/s, respectively. In addition, a data-based verification, which depicts the Karman vortex street phenomenon, is carried out to prove the ability of this method. All the results confirm the feasibility and effectiveness of the proposed RBF-ADMM in simultaneously reconstructing the temperature and velocity fields under different complicated environment. The proposed method for visualizing the arbitrary temperature and velocity fields will provide crucial input for control system such as the engines and boilers to optimize the thermal fluid and combustion process.

Simultaneous control of natural and extra degrees of freedom by isometric force and electromyographic activity in the muscle-to-force null space

Objective. Muscle activation patterns in the muscle-to-force null space, i.e. patterns that do not generate task-relevant forces, may provide an opportunity for motor augmentation by allowing to control additional end-effectors simultaneously to natural limbs. Here we tested the feasibility of muscular null space control for augmentation by assessing simultaneous control of natural and extra degrees of freedom. Approach. We instructed eight participants to control translation and rotation of a virtual 3D end-effector by simultaneous generation of isometric force at the hand and null space activity extracted in real-time from the electromyographic signals recorded from 15 shoulder and arm muscles. First, we identified the null space components that each participant could control more naturally by voluntary co-contraction. Then, participants performed several blocks of a reaching and holding task. They displaced an ellipsoidal cursor to reach one of nine targets by generating force, and simultaneously rotated the cursor to match the target orientation by activating null space components. We developed an information-theoretic metric, an index of difficulty defined as the sum of a spatial and a temporal term, to assess individual null space control ability for both reaching and holding. Main results. On average, participants could reach the targets in most trials already in the first block (72%) and they improved with practice (maximum 93%) but holding performance remained lower (maximum 43%). As there was a high inter-individual variability in performance, we performed a simulation with different spatial and temporal task conditions to estimate those for which each individual participants would have performed best. Significance. Muscular null space control is feasible and may be used to control additional virtual or robotics end-effectors. However, decoding of motor commands must be optimized according to individual null space control ability.

A Null-Space Compensation Control for Multi-input Redundant Plant Based on Model Reference Adaptive Control

In this paper, we proposed to design a null-space compensation control system for linear time invariant systems with multi-input redundant channels. In the input redundant systems, control input vector may have null-space component of the plant parameter vector. The null-space component does not contribute to the generation of the control force which drives the plant. If a control input that does not include the null-space component can be generated, efficient control is achieved in the sense of minimizing the norm of the control input. In the proposed method, a model reference adaptive control (MRAC) method was used to design the control system that compensates for the null-space component even if the plant parameters are unknown. The effectiveness of the proposed null-space compensation control system was shown by numerical examples.

Constrained Null Space Component Analysis for Semiblind Source Separation Problem.

The blind source separation (BSS) problem extracts unknown sources from observations of their unknown mixtures. A current trend in BSS is the semiblind approach, which incorporates prior information on sources or how the sources are mixed. The constrained independent component analysis (ICA) approach has been studied to impose constraints on the famous ICA framework. We introduced an alternative approach based on the null space component (NCA) framework and referred to the approach as the c-NCA approach. We also presented the c-NCA algorithm that uses signal-dependent semidefinite operators, which is a bilinear mapping, as signatures for operator design in the c-NCA approach. Theoretically, we showed that the source estimation of the c-NCA algorithm converges with a convergence rate dependent on the decay of the sequence, obtained by applying the estimated operators on corresponding sources. The c-NCA can be formulated as a deterministic constrained optimization method, and thus, it can take advantage of solvers developed in optimization society for solving the BSS problem. As examples, we demonstrated electroencephalogram interference rejection problems can be solved by the c-NCA with proximal splitting algorithms by incorporating a sparsity-enforcing separation model and considering the case when reference signals are available.

Null space component analysis for noisy blind source separation

We propose a novel operator-based model called the null space component analysis (NCA) to solve the noisy blind source separation (BSS) problem. Theoretically, we show that the NCA can resolve the rotation ambiguity in the BSS problem. In a set of m linearly independent source signals, the basic principle of the NCA is to associate each signal with a separating operator that includes the signal in its null space and repels other signals from the space. We show that the model can act as a constraint on the source signals in the noisy BSS problem. In contrast to the ICA-based and the sparsity-based approaches, NCA is a deterministic and data-adaptive algorithm that can solve both the under-determined and the over-determined BSS problems. To demonstrate the algorithm׳s efficiency, we process several types of signals, including real-life signals obtained from biomedical systems, and compare the results with those derived by other methods.

ALGAE: A Fast Algebraic Estimation of Interferogram Phase Offsets in Space-Varying Geometries

This paper deals with the estimation of terrain topography from multipass synthetic aperture radar (SAR) interferometry (InSAR), focusing on the case where variation of the system geometry within the imaged swath is relevant. A typical case is represented by airborne multipass interferometric campaigns where, due to the closeness between the radar sensor and the targets, the incidence-angle sensitivity undergoes a dramatic increase with respect to the spaceborne case, resulting in a high spatial variability of the normal baselines. The space-varying nature of the system geometry gives rise to a major issue in multipass InSAR analyses in that it prevents from compensating for the presence of interferogram phase offsets by simply phase locking the data stack to a reference point, therefore hindering the retrieval of terrain topography. To cope with this issue properly, we propose a novel approach that exploits the algebraic properties of the problem. Such an approach allows casting the problem in terms of identification of a null-space component for terrain topography after which both topography and the interferogram phase offsets are quickly obtained without exploiting calibration points. Experimental results are shown based on a P-band data set acquired by the Experimental SAR (E-SAR) airborne system, operated by the German Aerospace Center (DLR), in the framework of the European Space Agency (ESA) campaign BIOSAR 2008.

Control of Redundant Mechanical Systems Under Equality and Inequality Constraints on Both Input and Constraint Forces

The equality and inequality constraints on constraint force and/or the actuator force/torque arise in several robotic applications, for which different controllers have been specifically developed. This paper presents a unified approach to control a rather general class of robotic systems with closed loops under a set of linear equality and inequality constraints using the notion of projection operator. The controller does not require the kinematic constraints to be independent, i.e., systems with time-varying topology can be dealt with, while demanding minimum-norm actuation force or torque in the case that the system becomes redundant. The orthogonal decomposition of the control input force yields the null-space component and its orthogonal complement. The null-space component is obtained using the projected inverse dynamics control law, while the orthogonal complement component is found through solving a quadratic programming problem, in which the equality and inequality constraints are derived to be equivalent to the originally specified ones. Finally, a case study is presented to demonstrate how the control technique can be applied to multi-arms manipulation of an object.

Prior model evaluation from Null Space Compensation perspective with application to surface reconstruction from single images

Prior model is widely applied in the area of computer vision and computer graphics. However, there is still a lack of a general theoretical scheme for evaluating the performance of the priors and a guidance for choosing suitable models. In this paper, a general scheme is proposed for linear singular problems based on the idea of Null Space Compensation. It is proved that for a linear prior model the principal directions obtained from the singular value decomposition of the model shall not be parallel to those of the system matrix determined by the problem. It is also suggested that for a nonlinear prior, higher correlation between the null space components of the estimate data based on the given prior and those of the ground truth or controlled data indicate the better suitability of the prior. The proposed evaluation scheme is demonstrated through an application to a linearized shape from shading problem, where surface shall be reconstructed from single 2D images. Both linear model and nonlinear constraints are evaluated with experiments on both synthetic images and real images. The results validate the proposed evaluation scheme and its capability for guiding in choosing a good prior model structure.

Comment on “Two statistics for evaluating parameter identifiability and error reduction” by John Doherty and Randall J. Hunt

Doherty and Hunt (2009) present important ideas for first-order-second moment sensitivity analysis, but five issues are discussed in this comment. First, considering the composite-scaled sensitivity (CSS) jointly with parameter correlation coefficients (PCC) in a CSS/PCC analysis addresses the difficulties with CSS mentioned in the introduction. Second, their new parameter identifiability statistic actually is likely to do a poor job of parameter identifiability in common situations. The statistic instead performs the very useful role of showing how model parameters are included in the estimated singular value decomposition (SVD) parameters. Its close relation to CSS is shown. Third, the idea from p. 125 that a suitable truncation point for SVD parameters can be identified using the prediction variance is challenged using results from Moore and Doherty (2005). Fourth, the relative error reduction statistic of Doherty and Hunt is shown to belong to an emerging set of statistics here named perturbed calculated variance statistics. Finally, the perturbed calculated variance statistics OPR and PPR mentioned on p. 121 are shown to explicitly include the parameter null-space component of uncertainty. Indeed, OPR and PPR results that account for null-space uncertainty have appeared in the literature since 2000.

Underdetermined Blind Source Separation Based on Subspace Representation

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper considers the problem of blindly separating sub- and super-Gaussian sources from underdetermined mixtures. The underlying sources are assumed to be composed of two orthogonal components: one lying in the rowspace and the other in the nullspace of a mixing matrix. The mapping from the rowspace component to the mixtures by the mixing matrix is invertible using the pseudo-inverse of the mixing matrix. The mapping from the nullspace component to zero by the mixing matrix is noninvertible, and there are infinitely many solutions to the nullspace component. The latent nullspace component, which is of lower complexity than the underlying sources, is estimated based on a mean square error (MSE) criterion. This leads to a source estimator that is optimal in the MSE sense. In order to characterize and model sub- and super-Gaussian source distributions, the parametric generalized Gaussian distribution is used. The distribution parameters are estimated based on the expectation-maximization (EM) algorithm. When the mixing matrix is unavailable, it must be estimated, and a novel algorithm based on a single source detection algorithm, which detects time-frequency regions of single-source-occupancy, is proposed. In our simulations, the proposed algorithm, compared to other conventional algorithms, estimated the mixing matrix with higher accuracy and separated various sources with higher signal-to-interference ratio. </para>

Dynamic redundancy resolution in a nonholonomic wheeled mobile manipulator

SUMMARYWheeled Mobile Manipulators (WMM) possess many advantages over fixed-base counterparts in terms of improved workspace, mobility and robustness. However, the combination of the nonholonomic constraints with the inherent redundancy limits effective exploitation of end-effector payload manipulation capabilities. The dynamic-level redundancy-resolution scheme presented in this paper decomposes the system dynamics into decoupled task-space (end-effector motions/forces) and a dynamically consistent null-space (internal motions/forces) component. This simplifies the subsequent development of a prioritized task-space control (of end-effector interactions) and a decoupled but secondary null-space control (of internal motions) in a hierarchical WMM controller. Various aspects of the ensuing novel capabilities are illustrated using a series of simulation results.

Decomposition of images and objects into measurement and null components

We derive and discuss an algorithm for separating measurement-and null-space components of an object with respect to a given imaging system. This algorithm requires a discrete-to-discrete approximation of a typically continuous-to-discrete imaging system, and problems associated with such an approximation are examined. Situations where knowledge of the measurement and null spaces is crucial for analyzing imaging systems are discussed. We conclude with two examples demonstrating the usefulness of this algorithm, even in the discrete-to-discrete approximation.

Convergence of a substructuring method with Lagrange multipliers

We analyze the convergence of a substructing iterative method with Lagrange multipliers, propose recently by Farhat and Roux. The method decomposes finite element discretization of an elliptic boundary value problem into Neumann problems on the subdomains plus a coarse problem for the subdomain nullspace components. For linear conforming elements and preconditioning by the Dirichlet problems on the subdomains, we prove the asymptotic bound on the condition number C(1+log(H/h))^\alpha, \alpha=2 or 3, where h is the characteristic element size, H subdomain size, and C is independent of the number of subdomains.

Expectation-maximization algorithms, null spaces, and MAP image restoration

A computationally efficient, easily implementable algorithm for MAP restoration of images degraded by blur and additive correlated Gaussian noise using Gibbs prior density functions is derived. This algorithm is valid for a variety of complete data spaces. The constraints upon the complete data space arising from the Gaussian image formation model are analyzed and a motivation is provided for the choice of the complete data, based upon the ease of computation of the resulting EM algorithms. The overlooked role of the null space of the blur operator in image restoration is introduced. An examination of this role reveals an important drawback to the use of the simulated annealing algorithm in maximizing a specific class of functionals. An alternative iterative method for computing the nullspace component of a vector is given. The ability of a simple Gibbs prior density function to enable partial recovery of the component of an image within the nullspace of the blur operator is demonstrated.