Gradient Mapping Research Articles

Nearly Optimal First-Order Methods for Convex Optimization under Gradient Norm Measure: an Adaptive Regularization Approach

In the development of first-order methods for smooth (resp., composite) convex optimization problems, where smooth functions with Lipschitz continuous gradients are minimized, the gradient (resp., gradient mapping) norm becomes a fundamental optimality measure. Under this measure, a fixed iteration algorithm with the optimal iteration complexity for the smooth case is known, while determining this number of iteration to obtain a desired accuracy requires the prior knowledge of the distance from the initial point to the optimal solution set. In this paper, we report an adaptive regularization approach, which attains the nearly optimal iteration complexity without knowing the distance to the optimal solution set. To obtain further faster convergence adaptively, we secondly apply this approach to construct a first-order method that is adaptive to the Holderian error bound condition (or equivalently, the Łojasiewicz gradient property), which covers moderately wide classes of applications. The proposed method attains nearly optimal iteration complexity with respect to the gradient mapping norm.

Using floristic gradient mapping to assess seasonal thaw depth in interior Alaska

AbstractQuestionsIs it possible to map floristic gradients in heterogeneous boreal vegetation by using remote‐sensing data? Does a continuous vegetation map enable the creation of a spatially continuous map of seasonal permafrost soil thaw depth?LocationBonanza Creek LTER, Fairbanks, Alaska, USA.MethodsVegetation records are subjected to an ordination to extract the predominant floristic gradient. The ordination scores are then extrapolated using Sentinel 2 imagery and a digital elevation model (DEM). As the relation between vegetation pattern and seasonal thaw depth was confirmed in this study, the spatial distribution of ordination scores is then used to predict seasonal thaw depth over the same area.ResultsThe first dimension of the ordination space separates species corresponding to moist and cold soil conditions from species associated with well‐drained soils. This floristic gradient was successfully mapped within the sampled plant communities. The extrapolated thaw depths follow the typical distribution along a topographical and geomorphological gradient for this region. Besides vegetation information also DEM derivatives show high contributions to the thaw depth modeling.ConclusionWe demonstrate that floristic gradient mapping in boreal vegetation is possible. The accuracy of the thaw depth prediction model is comparable to that in previous analyses but uses a more parsimonious set of predictors, underlining the efficacy of this approach.

Let your maps be fuzzy!—Class probabilities and floristic gradients as alternatives to crisp mapping for remote sensing of vegetation

AbstractMapping vegetation as hard classes based on remote sensing data is a frequently applied approach, even though this crisp, categorical representation is not in line with nature's fuzziness. Gradual transitions in plant species composition in ecotones and faint compositional differences across different patches are thus poorly described in the resulting maps. Several concepts promise to provide better vegetation maps. These include (1) fuzzy classification (a.k.a. soft classification) that takes the probability of an image pixel's class membership into account and (2) gradient mapping based on ordination, which describes plant species composition as a floristic continuum and avoids a categorical description of vegetation patterns. A systematic and comprehensive comparison of these approaches is missing to date. This paper hence gives an overview of the state of the art in fuzzy classification and gradient mapping and compares the approaches in a case study. The advantages and disadvantages of the approaches are discussed and their performance is compared to hard classification (a.k.a. crisp or boolean classification). Gradient mapping best conserves the information in the original data and does not require an a priori categorization. Fuzzy classification comes close in terms of information loss and likewise preserves the continuous nature of vegetation, however, still relying on a priori classification. The need for a priori classification may be a disadvantage or, in other cases, an advantage because it allows using categorical input data instead of the detailed vegetation records required for ordination. Both approaches support spatially explicit accuracy analyses, which further improves the usefulness of the output maps. Gradient mapping and fuzzy classification offer various advantages over hard classification, can always be transformed into a crisp map and are generally applicable to various data structures. We thus recommend the use of these approaches over hard classification for applications in ecological research.

Dispersion of functional gradients across the adult lifespan

Ageing is commonly associated with changes to segregation and integration of functional brain networks, but, in isolation, current network-based approaches struggle to elucidate changes across the many axes of functional organisation. However, the advent of gradient mapping techniques in neuroimaging provides a new means of studying functional organisation in a multi-dimensional connectivity space. Here, we studied ageing and behaviourally-relevant differences in a three-dimensional connectivity space using the Cambridge Centre for Ageing Neuroscience cohort (n = 643). Building on gradient mapping techniques, we developed a set of measures to quantify the dispersion within and between functional communities. We detected a strong shift of the visual network across the adult lifespan from an extreme to a more central position in the 3D gradient space. In contrast, the dispersion distance between transmodal communities (dorsal attention, ventral attention, frontoparietal and default mode) did not change. However, these communities themselves were increasingly dispersed with increasing age, reflecting more dissimilar functional connectivity profiles within each community. Increasing dispersion of frontoparietal, attention and default mode networks, in particular, were associated negatively with cognition, measured by fluid intelligence. By using a technique that explicitly captures the ordering of functional systems in a multi-dimensional hierarchical framework, we identified behaviorally-relevant age-related differences of within and between network organisation. We propose that the study of functional gradients across the adult lifespan could provide insights that may facilitate the development of new strategies to maintain cognitive ability across the lifespan in health and disease.

The gradient projection algorithm for a proximally smooth set and a function with Lipschitz continuous gradient

We consider the minimization problem for a nonconvex function with Lipschitz continuous gradient on a proximally smooth (possibly nonconvex) subset of a finite-dimensional Euclidean space. We introduce the error bound condition with exponent for the gradient mapping. Under this condition, it is shown that the standard gradient projection algorithm converges to a solution of the problem linearly or sublinearly, depending on the value of the exponent . This paper is theoretical. Bibliography: 23 titles.

Susceptibility-based MR Imaging of Nitinol Stent.

Conventional MR techniques have difficulty to accurately localize the stent position and access the stent restenosis because of the effects of susceptibility and radiofrequency (RF) shielding artifacts caused by stent mesh. Previous studies have demonstrated that a susceptibility-based positive contrast MR method exhibits excellent efficacy for visualizing MR compatible metal devices by taking advantage of their high magnetic susceptibility. However, the method is not evaluated in the visualization of stents. Therefore, the purpose of this study is to prospectively assess whether the susceptibility-based positive contrast method can be used to visualize the nitinol stents, with the comparison of two typical MR positive contrast techniques, i.e., susceptibility gradient mapping using the original resolution (SUMO) and the gradient echo acquisition for super-paramagnetic particles with positive contrast (GRASP). The experiment results showed that the susceptibility-based method provided better visualization and more precise localization of the stent than SUMO and GRASP.

Sensitivity of GNSS tropospheric gradients to processing options

Abstract. An analysis of processing settings impacts on estimated tropospheric gradients is presented. The study is based on the benchmark data set collected within the COST GNSS4SWEC action with observations from 430 Global Navigation Satellite Systems (GNSS) reference stations in central Europe for May and June 2013. Tropospheric gradients were estimated in eight different variants of GNSS data processing using precise point positioning (PPP) with the G-Nut/Tefnut software. The impacts of the gradient mapping function, elevation cut-off angle, GNSS constellation, observation elevation-dependent weighting and real-time versus post-processing mode were assessed by comparing the variants by each to other and by evaluating them with respect to tropospheric gradients derived from two numerical weather models (NWMs). Tropospheric gradients estimated in post-processing GNSS solutions using final products were in good agreement with NWM outputs. The quality of high-resolution gradients estimated in (near-)real-time PPP analysis still remains a challenging task due to the quality of the real-time orbit and clock corrections. Comparisons of GNSS and NWM gradients suggest the 3∘ elevation angle cut-off and GPS+GLONASS constellation for obtaining optimal gradient estimates provided precise models for antenna-phase centre offsets and variations, and tropospheric mapping functions are applied for low-elevation observations. Finally, systematic errors can affect the gradient components solely due to the use of different gradient mapping functions, and still depending on observation elevation-dependent weighting. A latitudinal tilting of the troposphere in a global scale causes a systematic difference of up to 0.3 mm in the north-gradient component, while large local gradients, usually pointing in a direction of increasing humidity, can cause differences of up to 1.0 mm (or even more in extreme cases) in any component depending on the actual direction of the gradient. Although the Bar-Sever gradient mapping function provided slightly better results in some aspects, it is not possible to give any strong recommendation on the gradient mapping function selection.

Local convergence of the Levenberg–Marquardt method under Hölder metric subregularity

We describe and analyse Levenberg–Marquardt methods for solving systems of nonlinear equations. More specifically, we propose an adaptive formula for the Levenberg–Marquardt parameter and analyse the local convergence of the method under Holder metric subregularity of the function defining the equation and Holder continuity of its gradient mapping. Further, we analyse the local convergence of the method under the additional assumption that the Łojasiewicz gradient inequality holds. We finally report encouraging numerical results confirming the theoretical findings for the problem of computing moiety conserved steady states in biochemical reaction networks. This problem can be cast as finding a solution of a system of nonlinear equations, where the associated mapping satisfies the Łojasiewicz gradient inequality assumption.

An accelerated variance reducing stochastic method with Douglas-Rachford splitting

We consider the problem of minimizing the regularized empirical risk function which is represented as the average of a large number of convex loss functions plus a possibly non-smooth convex regularization term. In this paper, we propose a fast variance reducing (VR) stochastic method called Prox2-SAGA. Different from traditional VR stochastic methods, Prox2-SAGA replaces the stochastic gradient of the loss function with the corresponding gradient mapping. In addition, Prox2-SAGA also computes the gradient mapping of the regularization term. These two gradient mappings constitute a Douglas-Rachford splitting step. For strongly convex and smooth loss functions, we prove that Prox2-SAGA can achieve a linear convergence rate comparable to other accelerated VR stochastic methods. In addition, Prox2-SAGA is more practical as it involves only the stepsize to tune. When each loss function is smooth but non-strongly convex, we prove a convergence rate of $${\mathcal {O}}(1/k)$$ for the proposed Prox2-SAGA method, where k is the number of iterations. Moreover, experiments show that Prox2-SAGA is valid for non-smooth loss functions, and for strongly convex and smooth loss functions, Prox2-SAGA is prominently faster when loss functions are ill-conditioned.

Adaptive restart of accelerated gradient methods under local quadratic growth condition

Abstract By analyzing accelerated proximal gradient methods under a local quadratic growth condition, we show that restarting these algorithms at any frequency gives a globally linearly convergent algorithm. This result was previously known only for long enough frequencies. Then as the rate of convergence depends on the match between the frequency and the quadratic error bound, we design a scheme to automatically adapt the frequency of restart from the observed decrease of the norm of the gradient mapping. Our algorithm has a better theoretical bound than previously proposed methods for the adaptation to the quadratic error bound of the objective. We illustrate the efficiency of the algorithm on Lasso, regularized logistic regression and total variation denoising problems.

A Coaxial RF Applicator for Ultra-High Field Human MRI.

To develop a novel radio-frequency (RF) concept for ultra-high field (UHF) human magnetic resonance imaging (MRI) based on a coaxial resonant cavity. A two-channel slotted coaxial cavity RF applicator was designed for human head MRI at 9.4T. Physical dimensions made the proposed conducting structure resonant at the required frequency without tuning lumped elements. Numerical electromagnetic modeling was used to optimize the design. RF safety was assessed with two representative human body models. MR experiments on a 9.4T scanner included gradient echo images and mapping of a circularly polarized RF magnetic field in the human head phantom. The simulations and the phantom MR experiments agreed both qualitatively and quantitatively. The design was relatively simple, robust and required only a few additional reactive elements for the applicator's input impedance matching. The transmit efficiency and homogeneity of the excitation field were only 20% and 4% lower compared to a conventional 8-channel head array. The coaxial RF applicator was feasible for human MRI at UHF and required no lumped elements for its tuning. Imaging performance of the RF applicator was only moderately lower compared to the conventional transmit array, but would be sufficient to provide an anatomical reference for the heteronuclei MRI. An alternative approach with the minimal involvement of lumped elements becomes feasible to design volume-type RF coils for UHF human MRI.

Local multiscale blur estimation based on toggle mapping for sharp region extraction

In this study, a multiscale local blur estimation is proposed based on the existing local focus measure that combines gradient and toggle mapping. This method evaluates the quality of images regardless of their content (not in an autofocus context) and can predict Optical Character Recognition accuracy based on local blur. The resulting approach outperforms state of the art blur detection methods. Quantitative results are given on DIQA database. Moreover, the authors demonstrate its usefulness for extracting a region of interest from partially blurry images. Results are shown on images acquired by a project devoted to smartphone based text extraction for visually impaired people. In this study, sharp region extraction is essential since it allows warning the users when their picture is unusable. Moreover, it saves computing time.

Inverse images of open sets under gradient mapping

For every pair of sets $${F, U \subset \mathbb{R}^d}$$ , $${d \geq 2}$$ , F being of Borel class Fσ and U being nonempty, bounded and open, we construct a Frechet differentiable function $${f \colon \mathbb{R}^{d} \to \mathbb{R}}$$ such that $${F \subset (\nabla{f})^{-1}(U)}$$ and the Hausdorff dimension of $${(\nabla{f})^{-1}(U) \setminus F}$$ does not exceed 1. Moreover $${(\nabla{f})(\mathbb{R}^d) \subset \overline{U}}$$ . This generalizes both Zelený [10] and Deville–Matheron [8] results about the properties of open sets preimages under the gradient mapping.

Finite Element Analysis-Based Approach for Prediction of Aneurysm-Prone Arterial Segments

Arterial aneurysms represent a significant cause of morbidity and mortality worldwide; therefore, there is a growing need for the rapid and precise tool or method enabling to predict aneurysm emergence. The objective of this study was to develop a computer-aided method for aneurism prediction. Here we utilized a computer-based approach to establish a non-invasive, high-resolution and rapid method for the prediction of aneurysm-prone arterial segments based on combining microcomputed tomography (microCT) scanning with finite element analysis (FEA). We performed a microCT image binarization and designed a computing algorithm for FEA mesh construction, followed by application of gradient mapping and stress modeling to identify thin-walled high-stress areas responsible for the development of aneurysms. The fidelity of our computing algorithm for FEA mesh construction was similar to the commercially available software. The maximum possible error of our approach did not exceed that of either microCT or clinically available multislice computed tomography angiography scanning. Our computational approach revealed thin-walled arterial segments under a high stress, therefore potentially predicting aneurysm-prone sites. Here we demonstrate our approach for the prediction of arterial segments under a high risk of aneurysm occurrence, which should be further validated in pre-clinical models to be translated into the clinical practice.

Atomic Force Microscopy Force Mapping Analysis of an Adsorbed Surfactant above and below the Critical Micelle Concentration.

Force curves collected using an atomic force microscope (AFM) in the presence of adsorbed surfactants are often used to draw conclusions about adsorbed film packing, rigidity, and thickness. However, some noteworthy features of such force curve characteristics have yet to be thoroughly investigated and explained. In this work, we collected force curves from tetradecyltrimethylammonium bromide films adsorbed on highly oriented pyrolytic graphite (HOPG), silica, and silica that had been hydrophobized by functionalization with dichlorodimethyl silane. Breakthrough events in the force curves from several different trials were compared to show that the breakthrough distance, often reported as the adsorbed film thickness, increased with concentration below the critical micelle concentration (CMC) but was approximately 3.5 nm on all surfaces between 2× and 10× CMC; an unexpected result because of the different surface chemistries for the three surfaces. We employed an AFM probe with a different force constant ( k) value as well as a colloidal probe and the breakthrough distance remained approximately 3.5 nm in all cases. Gradient mapping, a variant of force mapping, was also implemented on the three surfaces and resulted in a new technique for visualizing adsorbed surfactant in situ. The resulting maps showed patches of adsorbed surfactant below the CMC and revealed that with increasing concentration, the size of the patches increased resulting in full coverage near and above the CMC. These results are, to our knowledge, the first time force mapping has been used to spatially track patches of adsorbed surfactant. Finally, layers of surfactants on an AFM tip were investigated by collecting a force map on a single AFM tip using the tip of a separate AFM probe. A breakthrough event was observed between the tips, indicating that a layer of surfactant was present on at least one, if not both tips.

ANOTHER LOOK AT THE FAST ITERATIVE SHRINKAGE/THRESHOLDING ALGORITHM (FISTA).

This paper provides a new way of developing the "Fast Iterative Shrinkage/Thresholding Algorithm (FISTA)" [3] that is widely used for minimizing composite convex functions with a nonsmooth term such as the ℓ1 regularizer. In particular, this paper shows that FISTA corresponds to an optimized approach to accelerating the proximal gradient method with respect to a worst-case bound of the cost function. This paper then proposes a new algorithm that is derived by instead optimizing the step coefficients of the proximal gradient method with respect to a worst-case bound of the composite gradient mapping. The proof is based on the worst-case analysis called Performance Estimation Problem in [11].

2-D inversion of P-wave polarization data to obtain maps of velocity gradient

Gradient mapping is a technique employed in the interpretation of tomographic velocity images for delineating geological structures. In this paper, a tomographic method is proposed for determining relative velocity gradient field from seismic polarization directions. This inverse problem is iteratively resolved by the damped least squares method. With Hamiltonian formulation of ray theory and under the assumption that the medium is weakly inhomogeneous, the problem formulation for polarization direction is approximately expressed asa function of relative velocity gradient. Explicit expressions of the Frechet derivatives of polarization directions with respect to model parameters are given. The proposed tomographic method is illustrated by conducting synthetic experiments for showing the ability of our method to recover relative velocity gradient field as well as its potential applicability to complex media. The test results demonstrate that the proposed method is a promising approach for imaging geological structures.

Bridge to success: A better method of cryoablation for atrioventricular nodal reentrant tachycardia in children

Cryoablation for atrioventricular nodal reentrant tachycardia (AVNRT) is associated with higher recurrence rates than radiofrequency ablation (RFA). Junctional tachycardia marks procedural success with RFA, but no such indicator exists for cryoablation. The purpose of this study as to determine the impact of voltage mapping plus longer ablation lesions on midterm success of cryoablation for children with AVNRT. We performed a single-center retrospective analysis of pediatric patients with AVNRT who underwent cryoablation from 2011 to 2015. Patients ablated using a standard electroanatomic approach (control) were compared with patients ablated using voltage mapping (voltage group). In the voltage group, EnSite NavX navigation and visualization technology (St Jude Medical, St Paul, MN) was used to develop a "bridge" of lower voltage gradients (0.3-0.8 mV) of the posteroseptal right atrium to guide cryoablation. Kaplan-Meier analysis was used to determine freedom from recurrence of supraventricular tachycardia. In all, 122 patients were included (71 voltage, 51 control). There was no difference between groups regarding age, sex, or catheter-tip size. Short-term success was similar in both groups (98.5% voltage vs 92% control; P = .159), but recurrence rates were lower in the voltage group (0% vs 11%, P = .006). Follow-up time was shorter in the voltage group (15 ± 7 months vs 22± 17 months, P < .05). The 1-year freedom from recurrence was lower in the voltage group (100% vs 91.5%, P <.05). Ablation times were longer in the voltage group (43.7 ± 20.9 minutes vs 34.3 ± 20.5 minutes, P = .01), but overall procedure times were shorter in the voltage group (157 ± 40 minutes vs 198 ± 133 minutes; P=.018). No significant complication was seen in either group. Voltage gradient mapping and longer lesion time can decrease recurrence rates in pediatric patients with AVNRT.

New Second-Order Karush–Kuhn–Tucker Optimality Conditions for Vector Optimization

In the present paper, we focus on the vector optimization problems with constraints, where objective functions and constrained functions are Frechet differentiable, and whose gradient mapping is locally Lipschitz. By using the second-order symmetric subdifferential and the second-order tangent set, we introduce some new types of second-order regularity conditions in the sense of Abadie. Then we establish some second-order necessary optimality conditions Karush–Kuhn–Tucker-type for local efficient (weak efficient, Geoffrion properly efficient) solutions of the considered problem. In addition, we provide some sufficient conditions for a local efficient solution to such problem.

Voltage gradient mapping and electrophysiologically guided cryoablation in children with AVNRT.

Recently, voltage gradient mapping of Koch's triangle to find low-voltage connections, or 'voltage bridges', corresponding to the anatomic position of the slow pathway, has been introduced as a method to ablate atrioventricular nodal reentry tachycardia (AVNRT) in children. Thus, we aimed to assess the effectiveness of voltage mapping of Koch's triangle, combined with the search for the slow potential signal in 'low-voltage bridges', to guide cryoablation of AVNRT in children. From June 2015 to May 2016, 35 consecutive paediatric patients (mean age 12.1 ± 4.5 years) underwent 3D-guided cryoablation of AVNRT at our Institution. Fifteen children were enrolled as control group (mean age 14 ± 4 years). A voltage gradient mapping of Koch's triangle was obtained in all patients, showing low-voltage connections in all children with AVNRT but not in controls. Prior to performing cryoablation, we looked for the typical 'hump and spike' electrogram, generally considered to be representative of slow pathway potential within a low-voltage bridge. In all patients the 'hump and spike' electrogram was found inside bridges of low voltage. Focal or high-density linear lesions, extended or not, were delivered guided by low-voltage bridge visualization. Acute success rate was 100%, and no recurrence was reported at a mean follow-up of 8 ± 3 months. Voltage gradient mapping of Koch's triangle, combined with the search for the slow potential signal in low-voltage bridges, is effective in guiding cryoablation of AVNRT in paediatric patients, with a complete acute success rate and no AVNRT recurrences at mid-term follow-up.