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Isometric Projection (IsoP) is a linear dimensionality reduction method, which provides the best linear approximation to the true isometric embedding of data. However, IsoP and all its variants only consider the one-way mapping from high-dimensional space to low-dimensional space. The projected low-dimensional data may not "represent" the original sample accurately and effectively. In this paper, based on the "encoding-decoding" mechanism, a new IsoP method called IsoP-R (Isometric Projection with Reconstruction) has been proposed. In this method, the conventional projection of IsoP is viewed as the encoding stage, and the decoder is used to reconstruct the original high-dimensional data from the projected low-dimensional data. In this way, our algorithm makes the low-dimensional embedding data "represent" the original data more accurately and effectively. Experiment results on Handwritten Alphadigits, COIL-100, Olivetti Research Laboratory and Georgia Tech face datasets show that the proposed IsoP-R approach better represents the data and achieves much higher recognition accuracy. [ABSTRACT FROM AUTHOR]

We obtain an analog of Weyl's law for the Kohn Laplacian on lens spaces. We also show that two 3-dimensional lens spaces with fundamental groups of equal prime order are isospectral with respect to the Kohn Laplacian if and only if they are CR isometric. [ABSTRACT FROM AUTHOR]

Heterogeneous surface reactions exhibiting complex spatiotemporal dynamics and patterns can be studied as processes involving reaction-diffusion mechanisms. In many realistic situations, the surface has fractal characteristics. This situation is studied by isometric graphing and multidimensional scaling (IGMDS) of fractal surfaces for extracting geodesic distances (i.e., shortest scaled distances that obtain edges of neighboring surface nodes and their interconnections) and the results obtained used to model effects of surface diffusion with nonlinear reactions. Further analysis of evolved spatiotemporal patterns may be carried out by IGMDS because high-dimensional snapshot data can be efficiently projected to a transformed subspace with reduced dimensions. Validation of the IGMDS methodology is carried out by comparing results with reduction capabilities of conventional principal component analysis for simple situations of reaction and diffusion on surfaces. The usefulness of the IGMDS methodology is shown for analysis of complex patterns formed on both regular and fractal surfaces, and using generic nonlinear reaction-diffusion systems following FitzHugh Nagumo and cubic reaction kinetics. The studies of these systems with nonlinear kinetics and noise show that effects of surface disorder due to fractality can become very relevant. The relevance is shown by studying properties of dynamical invariants in IGMDS component space, viz., the Lyapunov exponents and the KS entropy for interesting situations of spiral formation and turbulent patterns. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Force enhancement (FE) is a phenomenon that is present in skeletal muscle. It is characterized by progressive forces upon active stretching—distinguished by a linear rise in force—and enhanced isometric force following stretching (residual FE (RFE)). In skeletal muscle, non-cross-bridge (XB) structures may account for this behaviour. So far, it is unknown whether differences between non-XB structures within the heart and skeletal muscle result in deviating contractile behaviour during and after eccentric contractions. Thus, we investigated the force response of intact cardiac trabeculae during and after isokinetic eccentric muscle contractions (10% of maximum shortening velocity) with extensive magnitudes of stretch (25% of optimum muscle length). The different contributions of XB and non-XB structures to the total muscle force were revealed by using an actomyosin inhibitor. For cardiac trabeculae, we found that the force–length dynamics during long stretch were similar to the total isometric force–length relation. This indicates that no (R)FE is present in cardiac muscle while stretching the muscle from 0.75 to 1.0 optimum muscle length. This finding is in contrast with the results obtained for skeletal muscle, in which (R)FE is present. Our data support the hypothesis that titin stiffness does not increase with activation in cardiac muscle. [ABSTRACT FROM AUTHOR]

Abstract It is known that the infimum of the sectional curvatures (on the regular part) of orbit spaces of isometric actions on unit spheres is bounded above by 4. We show that the infimum is 1 for "most" actions, and determine the cases in which it is bigger than 1. [ABSTRACT FROM AUTHOR]

Highlights • Bilinear isometric mapping is proposed to extract embedding of Riemannian manifold. • The proposed method can maximize the preservation of Riemannian geodesic distance. • A supervised classification algorithm based on extracted embedding is proposed. • High performances of proposed methods are supported by the experimental results. Abstract Because covariance features with the form of symmetric positive definite matrices lie on Riemannian manifold, classification on Riemannian manifold could possess high performance in many applications. Unfortunately, the applicability of classification methods developed on Riemannian manifold is limited by their huge computational complexities, particularly with the feature data on high-dimensional Riemannian manifold. To alleviate the problem of computational cost, in this paper, a simple yet efficient dimensionality reduction algorithm, bilinear isometric Riemannian embedding, is derived to construct a low-dimensional embedding from high-dimensional Riemannian manifold. To this end, we model the bilinear isometric mapping to identify a low-dimensional embedding that maximizes the preservation of Riemannian geodesic distance. A supervised classification method, embedding discriminant analysis, is then proposed based on the low-dimensional embedding. Experimental results on image and electroencephalogram reveal that the proposed algorithms can efficiently extract the distance-preserving embedding and obtain higher classification performance. [ABSTRACT FROM AUTHOR]

Objectives: There are several reports suggesting that forward head posture contributes to alterations in scapular kinematics and muscle activity, leading to the development of shoulder problems. Currently, it is unknown whether forward head posture alters the thickness of the scapular muscles. The aim of this study was to compare the thickness of the serratus anterior and upper and lower trapezius muscles at rest and during loaded isometric contractions in individuals with and without forward head posture. Methods: Twenty individuals with forward head posture and 20 individuals with normal head posture participated in this case‐control study. Three separate ultrasound images of the serratus anterior and upper and lower trapezius muscles were captured under 2 randomized conditions: at rest and during a loaded isometric contraction. Results: The thickness of each muscle significantly increased from rest to the loaded isometric contraction (P < .001). The only difference between the groups was that the thickness of the serratus anterior muscle at rest in the normal‐posture group was larger than that in the forward‐posture group (P = .01). Conclusions: Forward head posture appears to be related to atrophy of the serratus anterior muscle, which may contribute to the development of shoulder problems. Further research is required to identify more about the association of forward head posture with the imbalance of shoulder girdle muscles and the impact of head posture on upper quadrant pain. [ABSTRACT FROM AUTHOR]

Abstract We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on L p -spaces (affine isometric, and more generally (2 − 2 ϵ) 1 / 2 p -uniformly Lipschitz) with p varying in an interval increasing with the set of generators. In the same model, we establish a double inequality between the maximal p for which L p -fixed point properties hold and the conformal dimension of the boundary. In the Gromov density model, we prove that for every p 0 ∈ [ 2 , ∞) for a sufficiently large number of generators and for any density larger than 1/3, a random group satisfies the fixed point property for affine actions on L p -spaces that are (2 − 2 ϵ) 1 / 2 p -uniformly Lipschitz, and this for every p ∈ [ 2 , p 0 ]. To accomplish these goals we find new bounds on the first eigenvalue of the p -Laplacian on random graphs, using methods adapted from Kahn and Szemerédi's approach to the 2-Laplacian. These in turn lead to fixed point properties using arguments of Bourdon and Gromov, which extend to L p -spaces previous results for Kazhdan's Property (T) established by Żuk and Ballmann–Świa̧tkowski. [ABSTRACT FROM AUTHOR]

Some important factors of female work arc analysed, and the importance and use of knowledge about maximum muscular strength of women in theory and praxis of Ergonomics, Biomechanics is demonstrated. With the term of maximum isometric muscular strength of position isodynes are defined as curves which join points of application of forces within the arm reach and which indicate strengths of the same type and amount. Isodynes are used for complex describing the dynamo-metric structure of the arm reach. Some examples of isodyne-diagrams demonstrate their practical applicability for work study practitioners and designers of work places and layout. [ABSTRACT FROM AUTHOR]

Abstract We present characterizations of polyhedrality (I), (I I) and (I I I) in a separable L 1 -predual space in terms of its representing matrix. We also continue our study of diagrammatic representations of separable L 1 -predual spaces and show that in a polyhedral (I V) predual of L 1 , every M -ideal corresponds to a directed sub-diagram. [ABSTRACT FROM AUTHOR]

Let X, Y, Z be Banach spaces. We show a formula for a minimal projection from onto . It is a generalization of a result of Cheney and Light who showed a formula for a minimal projection from onto . [ABSTRACT FROM AUTHOR]

To cope with the huge expenditure associated with the fast growing sampling rate, compressed sensing (CS) is proposed as an effective technique of signal processing. In this paper, first, we construct a type of CS matrix to process signals based on singular linear spaces over finite fields. Second, we analyze two kinds of attributes of sensing matrices. One is the recovery performance corresponding to compressing and recovering signals. In particular, we apply two types of criteria, error-correcting pooling designs (PD) and restricted isometry property (RIP), to investigate this attribute. Another is the sparsity corresponding to storage and transmission signals. Third, in order to improve the ability associated with our matrices, we use an embedding approach to merge our binary matrices with some other matrices owing low coherence. At last, we compare our matrices with other existing ones via numerical simulations and the results show that ours outperform others. [ABSTRACT FROM AUTHOR]

One of the fundamental assumptions of traditional manifold learning algorithms is that the embedded manifold is globally or locally isometric to Euclidean space. Under this assumption, these algorithms divided manifold into a set of overlapping local patches which are locally isometric to linear subsets of Euclidean space. Then the learnt manifold would be a flat manifold with zero Riemannian curvature. But in the general cases, manifolds may not have this property. To be more specific, the traditional manifold learning does not consider the curvature information of the embedded manifold. In order to improve the existing algorithms, we propose a curvature-aware manifold learning algorithm called CAML. Without considering the local and global assumptions, we will add the curvature information to the process of manifold learning, and try to find a way to reduce the redundant dimensions of the general manifolds which are not isometric to Euclidean space. The experiments have shown that CAML has its own advantage comparing to other traditional manifold learning algorithms in the sense of the neighborhood preserving ratios (NPR) on synthetic databases and classification accuracies on image set classification. [ABSTRACT FROM AUTHOR]

We provide new upper bounds for mixing times of general finite Markov chains. We use these bounds to show that the total variation mixing time is robust under rough isometry for bounded degree graphs that are roughly isometric to trees. [ABSTRACT FROM AUTHOR]

Skeletal muscle force is generated by cross-bridge interactions between the overlapping contractile proteins, actin and myosin. The geometry of this overlap gives us the force–length relationship in which maximum isometric force is generated at an intermediate, optimum, length. However, the force–length relationship is not constant; optimum length increases with decreasing muscle activation. This effect is not predicted from actin–myosin overlap. Here we present evidence that this activation-dependent shift in optimum length may be due to a series compliance within muscles. As muscles generate force during fixed-end contractions, fibers shorten against series compliance until forces equilibrate and they become isometric. Shortening against series-compliance is proportional to activation, and creates conditions under which shortening-induced force depression may suppress full force development. Greater shortening will result in greater force depression. Hence, optimum length may decrease as activation rises due to greater fiber shortening. We discuss explanations of such history dependence, giving a review of previously proposed processes and suggesting a novel mechanistic explanation for the most likely candidate process based on tropomyosin kinetics. We suggest this mechanism could change the relationship between actin–myosin overlap and cross-bridge binding potential, not only depressing force at any given length, but also altering the relationship between force and length. This would have major consequences for our understanding of in vivo muscle performance. [ABSTRACT FROM AUTHOR]

Abstract Most human expression variations cause a non-rigid deformation of face scans, which is a challenge today. In this article, we present a novel framework for 3D face recognition that uses a geometry and local shape descriptor in a matching process to overcome the distortions caused by expressions in faces. This algorithm consists of four major components. First, the 3D face model is presented at different scales. Second, isometric-invariant features on each scale are extracted. Third, the geometric information is obtained on the 3D surface in terms of radial and level facial curves. Fourth, the feature vectors on each scale are concatenated with their corresponding geometric information. We conducted a number of experiments using two well-known and challenging datasets, namely, the GavabDB and Bosphorus datasets, and superior recognition performance has been achieved. The new system displays an overall rank-1 identification rate of 98.9% for all faces with neutral and non-neutral expressions on the GavabDB database. [ABSTRACT FROM AUTHOR]

A set D of vertices in a graph G is a dominating set if every vertex of G , which is not in D , has a neighbor in D . A set of vertices D in G is convex (respectively, isometric), if all vertices in all shortest paths (respectively, all vertices in one of the shortest paths) between any two vertices in D lie in D . The problem of finding a minimum convex dominating (respectively, isometric dominating) set is considered in this paper from algorithmic point of view. For the class of weak dominating pair graphs (i.e., the graphs that contain a dominating pair, which is a pair of vertices x , y ∈ V ( G ) such that vertices of any path between x and y form a dominating set), we present an efficient algorithm that finds a minimum isometric dominating set of such a graph. On the other hand, we prove that even if one restricts to weak dominating pair graphs that are also chordal graphs, the problem of deciding whether there exists a convex dominating set bounded by a given arbitrary positive integer is NP-complete. By further restricting the class of graphs to chordal dominating pair graphs (i.e., the chordal graphs in which every connected induced subgraph has a dominating pair) we are able to find a polynomial time algorithm that determines the minimum size of a convex dominating set of such a graph. [ABSTRACT FROM AUTHOR]

We construct a form of the swallowtail singularity in which uses coordinate transformation on the source and isometry on the target. As an application, we classify configurations of asymptotic curves and characteristic curves near swallowtails. [ABSTRACT FROM AUTHOR]

Highlights • An algorithm for achieving CNN based object-instance recognition across imaging conditions. • An active learning algorithm that can generate images of failure conditions in order to actively adapt the trained model on-the-fly. • A characterization of the extent to which CNNs can learn and cope with different imaging conditions. We dub this the learnability of imaging conditions. • Extensive experiments on both synthetic and real datasets showing the effectiveness of our algorithm. Abstract In the recent past, algorithms based on Convolutional Neural Networks (CNNs) have achieved significant milestones in object recognition. With large examples of each object class, standard datasets train well for inter-class variability. However, gathering sufficient data to train for a particular instance of an object within a class is impractical. Furthermore, quantitatively assessing the imaging conditions for each image in a given dataset is not feasible. By generating sufficient images with known imaging conditions, we study to what extent CNNs can cope with hard imaging conditions for instance-level recognition in an active learning regime. Leveraging powerful rendering techniques to achieve instance-level detection, we present results of training three state-of-the-art object detection algorithms namely, Fast R-CNN, Faster R-CNN and YOLO9000, for hard imaging conditions imposed into the scene by rendering. Our extensive experiments produce a mean Average Precision score of 0.92 on synthetic images and 0.83 on real images using the best performing Faster R-CNN. We show for the first time how well detection algorithms based on deep architectures fare for each hard imaging condition studied. [ABSTRACT FROM AUTHOR]

The strong geodetic problem is a recent variation of the classical geodetic problem. For a graph G, its strong geodetic number sg(G) is the cardinality of a smallest vertex subset S, such that each vertex of G lies on one fixed shortest path between a pair of vertices from S. In this paper, some general properties of the strong geodetic problem are studied, especially in connection with the diameter of a graph. The problem is also solved for balanced complete bipartite graphs. [ABSTRACT FROM AUTHOR]

This paper is a companion to a series of papers devoted to the study of the spectral distribution of the free Jacobi process associated with a single projection. Actually, we note that the flow derived in [N. Demni and T. Hmidi, Spectral distribution of the free Jacobi process associated with one projection, Colloq. Math. 137(2) (2014) 271-296] solves a radial Löwner equation and as such, the general theory of Löwner equations implies that it is univalent in some connected region in the open unit disc. We also prove that its inverse defines the Aleksandrov-Clark measure at of some Herglotz function which is absolutely-continuous with an essentially bounded density. As a by-product, we deduce that does not belong to the continuous singular spectrum of the unitary operator whose spectral dynamics are governed by the flow. Moreover, we use a previous result due to the first author in order to derive an explicit, yet complicated, expression of the moments of both the unitary and the free Jacobi processes. The paper is closed with some remarks on the boundary behavior of the flow's inverse. [ABSTRACT FROM AUTHOR]

We introduce a 3D deformable model-based framework for the retrieval of near-isometric flattenable objects using keypoints and BoVW (Bag-of-Visual-Words). By 3D deformable model we mean a texturemapped 3D shape which may deform isometrically. We assume that such a model is available for each object in the database. We exploit the 3D deformable models at the training and the retrieval phases. For our first contribution, we exploit the possibility of generating synthetic data from the 3D deformable models to define a new BoVW model for the database object representation. Our model chooses an optimal per-object representation by maximizing each object’s mean average precision. The maximization is done over multiple candidate representations which are generated using the criteria of keypoint repeatability, weight discriminance and stability. Our second contribution is the use of SfT (Shape-from-Template) to facilitate geometric verification at the retrieval phase, for a few objects hypothesized using the new BoVW model. Existing methods use a rigid model, such as the fundamental matrix, or a simple deformable model based on semi-local constraints. SfT however is a physics-based method which uses an object’s 3D deformable model to reconstruct its isometric 3D deformation from a single input image. The output of SfT thus directly provides a geometric verification score. A byproduct of our work is to extend the scope of SfT. The proposed object retrieval framework is used to provide SfT with a few object hypotheses which may be quickly tested for the 3D deformable object selection. Performance evaluation on synthetic and real images reveals the benefits of our retrieval framework using a database with size varying between 20 and 1000 objects. The use of the new BoVW model and SfT versus the BoVW baseline and a rigid model improves the retrieval performance by 4.2% and 11.3% with p -values of 5 × 10 − 6 and 7 × 10 − 30 respectively. [ABSTRACT FROM AUTHOR]

As a generalization of slant submersions [18], semi-slant submersions [15], and slant Riemannian maps [21], we define the notion of semi-slant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds. We study the integrability of distributions, the geometry of fibers, the harmonicity of such maps, etc. We also find a condition for such maps to be totally geodesic and investigate some decomposition theorems. Moreover, we give examples. [ABSTRACT FROM AUTHOR]

Reconstructing the 3D surface of an object using only a single image is a challenging task, which has recently attracted attention. In this paper, a template-based approach is presented to reconstruct the surface of an isometric deformable object. The proposed approach brings a solution for a class of computer vision problems named shape-from-template (SfT). In SfT, the goal is to solve single-image reconstruction for an object given its 3D template model in some rest shape. To this end, corresponding keypoints between the template and the so-called deformed image are first established. Then, a very fast method is used to estimate the first-order differential flow around the extracted keypoint pairs as an affine transformation. This is done using the keypoint pairs' surrounding texture patch. In our method, we estimate this affine transformation using the keypoint pairs' closest neighbors. This is both faster and more stable. Finally, the depth of each keypoint in the deformed image is estimated from its associated affine transformation. The robustness of keypoint matching is essential to the process. Indeed, outliers defeat depth estimation dramatically. We propose two new approaches to detect and remove the possible outliers based on geometrical properties of the matched keypoints. These two geometrical outlier removal approaches are faster than existing ones and can be used with almost any image descriptor. Experimental results show that the proposed approaches are very effective and outperform existing ones. [ABSTRACT FROM AUTHOR]

The research article describes a method of isometric transformation and determining an exterior orientation of a measurement instrument. The method is based on a designation of a "virtual" translation of two relative oblique orthogonal systems to a common, known in the both systems, point. The relative angle orientation of the systems does not change as each of the systems is moved along its axis. The next step is the designation of the three rotation angles (e.g. Tait-Bryan or Euler angles), transformation of the system convoluted at the calculated angles and moving the system to the initial position where the primary coordinate system was. This way eliminates movements of the systems from the calculations and makes it possible to calculate angles of mutual rotation angles of two orthogonal systems primarily involved in the movement. The research article covers laboratory calculations for simulated data. The accuracy of the results is 10-6 m (10-3 regarding the accuracy of the input data). This confi rmed the correctness of the assumed calculation method. In the following step the method was verifi ed under fi eld conditions, where the accuracy of the method raised to 0.003 m. The proposed method enabled to make the measurements with the oblique and uncentered instrument, e.g. total station instrument set over an unknown point. This is the reason why the method was named by the authors as Total Free Station - TFS. The method may be also used for isometric transformations for photogrammetric purposes. [ABSTRACT FROM AUTHOR]

The aim of this study was to determine pre-service elementary school teachers' capabilities of deciding the viewpoint and perspective in geometrical drawing. The study examined geometrical drawings the participants did on isometric paper. This is a case study, a qualitative study method, and the study data were analyzed using written documents. The study sample included 34 senior students in the Education Faculty's Elementary School Teaching Program at a Turkish public university. They were selected from approximately 205 pre-service teachers. The students participated in the study on a voluntary basis. The study results revealed that the pre-service teachers generally had weak skills in geometrical drawing. The participants had better skills in building structures which they saw from the air using unit blocks, and forming a structure of which they had the direct views (left and right; front and back) using unit blocks. However, the participants had weaker skills in drawing the direct views of the structures they had built, recognizing the connections between the left-right views and front-back views of the structures and drawing the structures they built on isometric paper in perspective. [ABSTRACT FROM AUTHOR]

We study the third and fourth variation of area for a compact domain in a constant mean curvature surface when there is a Killing field on $$\mathbf{R}^3$$ whose normal component vanishes on the boundary. Examples are given to show that, in the presence of a zero eigenvalue, the non negativity of the second variation has no implications for the local area minimization of the surface. [ABSTRACT FROM AUTHOR]

In this paper, based on the block operator technique and operator spectral theory, general explicit descriptions for intertwining operators and direct rotations of two orthogonal projections are established. As a consequence, Kato's result (Kato, 1996 [14] ) is improved, so are J. Avron, R. Seiler and B. Simon's Theorem 2.3 (Avron et al., 1994 [6] ) and C. Davis, W.M. Kahan's result (Davis and Kahan, 1970 [11] ). [ABSTRACT FROM AUTHOR]

The generalized Fibonacci Q d ( f ) is defined as the graph obtained from the d -cube Q d by removing all vertices that contain a given binary string f as a contiguous substring. This idea was introduced by Ilić, Klavžar and Rho. A binary string f is called isometric if Q d ( f ) is an isometric subgraph of Q d for all d ≥ 1 , otherwise it is called non-isometric. In this paper, we prove that a string f is isometric if and only if f n is isometric for any n ≥ 1 . This result can help us to construct more isometric strings and significantly increase the number of isometric strings. [ABSTRACT FROM AUTHOR]

This paper presents a new preconditioning technique for large-scale geometric optimization problems, inspired by applications in mesh parameterization. Our positive (semi-)definite preconditioner acts on the gradients of optimization problems whose variables are positions of the vertices of a triangle mesh in ℝ2 or of a tetrahedral mesh in ℝ3, converting localized distortion gradients into the velocity of a globally near-rigid motion via a linear solve. We pose our preconditioning tool in terms of the Killing energy of a deformation field and provide new efficient formulas for constructing Killing operators on triangle and tetrahedral meshes. We demonstrate that our method is competitive with state-of-the-art algorithms for locally injective parameterization using a variety of optimization objectives and show applications to two- and three-dimensional mesh deformation. [ABSTRACT FROM AUTHOR]

We consider Fock representations of the Q-deformed commutation relations ∂s∂†t=Q(s, t)∂†t∂s+δ(s, t) for s, t ∈ T. Here T:=ℝd (or more generally T is a locally compact Polish space), the function Q:T2→ℂ satisfies Q(s, t) ≤ 1 and Q(s, t)=Q(t, s), and ∫T2h(s)g(t)δ(s, t) σ(ds)σ(dt):=∫Th(t)g(t) σ(dt) σ being a fixed reference measure on T. In the case, where Q(s, t) ≡1, the Q-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev. These generalized statistics contain anyon statistics as a special case (with T=ℝ2 and a special choice of the function Q). The related Q-deformed Fock space ℱ(ℋ) over ℋ:=L2(T→ℂ,σ) is constructed. An explicit form of the orthogonal projection of ℋ⊗n onto the n-particle space ℱn(ℋ) is derived. A scalar product in Fn(ℋ) is given by an operator ƿn≥0 in ℋ⊗n which is strictly positive on ℱn(ℋ). We realize the smeared operators ∂†t and ∂t as creation and annihilation operators in ℱ(ℋ), respectively. Additional Q-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form ∂†s∂†t=Q(t, s)∂†t∂†s∂s∂t=Q(t, s) ∂t∂s, valid for those s, t∈T for which Q(s, t) =1. [ABSTRACT FROM AUTHOR]

The Shape-from-Template (SfT) problem is to recover the 3D shape of a deformable object from a single image, given a 3D template and a deformation constraint. We propose Particle-SfT, a new SfT algorithm which handles isometric and non-isometric deformations. We build Particle-SfT upon a particle system guided by deformation and reprojection constraint projections. Reconstruction is achieved by evolving particles to a globally attractive equilibrium, while taking observable external forces such as gravity into account, if any. Particle-SfT may be used to refine an existing initial shape. However, in practice we simply use the template as initial guess. This is because, as opposed to the existing refining methods, Particle-SfT has an extremely wide convergence basin. Particle-SfT is also faster than the existing refining methods. This is because it moves pieces of the shape's mesh independently to achieve larger step size by optimal constraint projection. We proved its convergence to a fixed-point. We experimented it with synthetic and real data. It has the same accuracy as the best performing isometric method and consistently outperforms all existing elastic methods in almost all cases, while being much faster. [ABSTRACT FROM AUTHOR]

Shape-from-Template (SfT) reconstructs the shape of a deforming surface from a single image, a 3D template and a deformation prior. For isometric deformations, this is a well-posed problem. However, previous methods which require no initialization break down when the perspective effects are small, which happens when the object is small or viewed from larger distances. That is, they do not handle all projection geometries. We propose stable SfT methods that accurately reconstruct the 3D shape for all projection geometries. We follow the existing approach of using first-order differential constraints and obtain local analytical solutions for depth and the first-order quantities: the depth-gradient or the surface normal. Previous methods use the depth solution directly to obtain the 3D shape. We prove that the depth solution is unstable when the projection geometry tends to affine, while the solution for the first-order quantities remain stable for all projection geometries. We therefore propose to solve SfT by first estimating the first-order quantities (either depth-gradient or surface normal) and integrating them to obtain shape. We validate our approach with extensive synthetic and real-world experiments and obtain significantly more accurate results compared to previous initialization-free methods. Our approach does not require any optimization, which makes it very fast. [ABSTRACT FROM AUTHOR]

In this article, we investigate the restrictions imposed by the dominant energy condition (DEC) on the topology and conformal type of possibly non-compact marginally outer trapped surfaces (thus extending Hawking's classical theorem on the topology of black holes). We first prove that an unbounded, stable marginally outer trapped surface in an initial data set ( M, g, k) obeying the dominant energy condition is conformally diffeomorphic to either the plane $${\mathbb{C}}$$ or to the cylinder $${\mathbb{A}}$$ and in the latter case infinitesimal rigidity holds. As a corollary, when the DEC holds strictly, this rules out the existence of trapped regions with cylindrical boundary. In the second part of the article, we restrict our attention to asymptotically flat data ( M, g, k) and show that, in that setting, the existence of an unbounded, stable marginally outer trapped surface essentially never occurs unless in a very specific case, since it would force an isometric embedding of ( M, g, k) into the Minkowski spacetime as a space-like slice. [ABSTRACT FROM AUTHOR]

We systematically study the optimal linear convergence rates for several relaxed alternating projection methods and the generalized Douglas-Rachford splitting methods for finding the projection on the intersection of two subspaces. Our analysis is based on a study on the linear convergence rates of the powers of matrices. We show that the optimal linear convergence rate of powers of matrices is attained if and only if all subdominant eigenvalues of the matrix are semisimple. For the convenience of computation, a nonlinear approach to the partially relaxed alternating projection method with at least the same optimal convergence rate is also provided. Numerical experiments validate our convergence analysis [ABSTRACT FROM AUTHOR]

A series of field trials with pipefitters in North America was conducted to explore how their spatial cognition influenced their performance on a scale model assembly task. Participants were provided one of three information formats, traditional, two-dimensional (2D) isometric drawings, 2D isometric drawings supplemented with a 2D visual rendering of a three-dimensional (3D) image of the assembly, and 2D isometric drawings supplemented with a 3D physical model of the assembly. Card rotation and cube comparison tests were administered to measure spatial cognition. This paper's primary contribution centers around the discovery that individual spatial cognition significantly influenced the pipefitter's performance, but the format of information can neutralize the effect. Both the information format and spatial cognition significantly influenced performance. When only 2D information was provided, participants with relatively low spatial cognition performed significantly worse than participants with relatively higher spatial cognition. When the 2D isometric drawings were supplemented with 3D information, participants with lower spatial cognition were able to complete the model assembly as efficiently and effectively as participants with higher spatial cognition. [ABSTRACT FROM AUTHOR]