2D Manifold Research Articles

Non-hexagonal neural dynamics in vowel space.

Are the grid cells discovered in rodents relevant to human cognition? Following up on two seminal studies by others, we aimed to check whether an approximate 6-fold, grid-like symmetry shows up in the cortical activity of humans who “navigate” between vowels, given that vowel space can be approximated with a continuous trapezoidal 2D manifold, spanned by the first and second formant frequencies. We created 30 vowel trajectories in the assumedly flat central portion of the trapezoid. Each of these trajectories had a duration of 240 milliseconds, with a steady start and end point on the perimeter of a “wheel”. We hypothesized that if the neural representation of this “box” is similar to that of rodent grid units, there should be an at least partial hexagonal (6-fold) symmetry in the EEG response of participants who navigate it. We have not found any dominant n-fold symmetry, however, but instead, using PCAs, we find indications that the vowel representation may reflect phonetic features, as positioned on the vowel manifold. The suggestion, therefore, is that vowels are encoded in relation to their salient sensory-perceptual variables, and are not assigned to arbitrary grid-like abstract maps. Finally, we explored the relationship between the first PCA eigenvector and putative vowel attractors for native Italian speakers, who served as the subjects in our study.

Using Low-Dimensional Manifolds to Map Relationships Between Dynamic Brain Networks.

As the field of dynamic brain networks continues to expand, new methods are needed to allow for optimal handling and understanding of this explosion in data. We propose here a novel approach that embeds dynamic brain networks onto a two-dimensional (2D) manifold based on similarities and differences in network organization. Each brain network is represented as a single point on the low dimensional manifold with networks of similar topology being located in close proximity. The rich spatio-temporal information has great potential for visualization, analysis, and interpretation of dynamic brain networks. The fact that each network is represented by a single point makes it possible to switch between the low-dimensional space and the full connectivity of any given brain network. Thus, networks in a specific region of the low-dimensional space can be examined to identify network features, such as the location of brain network hubs or the interconnectivity between brain circuits. In this proof-of-concept manuscript, we show that these low dimensional manifolds contain meaningful information, as they were able to successfully discriminate between cognitive tasks and study populations. This work provides evidence that embedding dynamic brain networks onto low dimensional manifolds has the potential to help us better visualize and understand dynamic brain networks with the hope of gaining a deeper understanding of normal and abnormal brain dynamics.

Well balanced residual distribution for the ALE spherical shallow water equations on moving adaptive meshes

We consider the numerical approximation of the Shallow Water Equations (SWEs) in spherical geometry for oceanographic applications. To provide enhanced resolution of moving fronts present in the flow we consider adaptive discrete approximations on moving triangulations of the sphere. To this end, we re-state all Arbitrary Lagrangian Eulerian (ALE) transport formulas, as well as the volume transformation laws, for a 2D manifold. Using these results, we write the set of ALE-SWEs on the sphere. We then propose a Residual Distribution discrete approximation of the governing equations. Classical properties as the DGCL and the C-property (well balancedness) are reformulated in this more general context. An adaptive mesh movement strategy is proposed. The discrete framework obtained is thoroughly tested on standard benchmarks in large scale oceanography to prove their potential as well as the advantage brought by the adaptive mesh movement.

Towards Automatic Blocking of Shapes using Evolutionary Algorithm

This work focuses on the use of evolutionary algorithm to perform automatic blocking of a 2D manifold. The goal of such a blocking process is to completely partition a 2D region into a set of conforming and non-intersecting quadrilaterals to facilitate the generation of an all-quadrilateral, or more preferably an ideal quadrilateral mesh configuration covering the closed 2D region. However, depending on the input shape, the optimal blocking strategy is often unclear and can be very user-dependent. In this work, a novel approach based on evolutionary algorithm is adapted to search for a potential set of such ideal configurations. Based on a selection within a set of candidate vertices from a pre-computed pool, blocking configurations can be derived and ranked based on the collective quality of its blocks. The quality of a block is computed based on objective functions relating to its interior angles and opposite length ratios. Using multi-dimensional ranking criteria, inferior solutions can be slowly filtered away with each successive generation. Based on observations on a range of turbomachinery test cases, it is possible to derive and improve near-optimal blocking configurations by utilizing a large number of generations.

Light Field Super-Resolution Using Edge-Preserved Graph-Based Regularization

The light field information would be captured through light field cameras and in different directions regarding 3D image view recordings. In this paper, in order to increase the spatio-angular super-resolution quality and to decrease the reconstruction error regarding the light field information, we use a graph-based light field super-resolution strategy. Accordingly, in order to apply the complementary data in the light field views, we use a graph regularizer for the total recovery of the information and an edge-preserving technique that represents an isometry between curves in the 2D manifold and 5D space of the RGB image views. Moreover, the reconstruction of the light field information is based on applying the alternating direction method of multipliers (ADMM) algorithm. Accordingly, a recent enhanced ADMM model has been used in this paper which is denominated as “Plug-and-Play” and permits the user to plug an image reconstruction technique and a denoising methodology as the first and second sub-problems respectively. On that account, we would be able to resolve the light field super-resolution problem considering the graph-based light field structure as the first sub-problem and the edge-preserving technique as the denoising methodology. Consequently, by applying the proposed super-resolution strategy, the super-resolved light field outcome would be more favorable in terms of visual quality and reconstruction errors in comparison with other state-of-the-art methodologies.

GVFOM: a novel external force for active contour based image segmentation

The GVF snake is effective on converging into concave areas and being insensitive to initialization. However, it fails to converge to areas with deep and narrow concavities or preserve weak edges. In this paper, a novel external force called gradient vector flow over manifold (GVFOM) is proposed for active contours. The proposed model takes the GVF vector as a 2D manifold embedded into a 4D Euclidean space, leading to the generalization of the Laplacian operator in Euclidean Space to that over manifold, and the two components of the external force are couples with each other to improve the properties of the GVF snake. The GVFOM snake has been tested with different kinds of images, and experimental results and comparison manifest that the GVFOM snake has better performance than the GVF and other state-of-the-art snakes on object separation, deep and narrow concavity convergence, weak edge protection, noise robustness while keeping some desirable properties of the GVF model such as initialization insensitivity and large capture range.

Diagrams of States of Single Flexible-Semiflexible Multi-Block Copolymer Chains: A Flat-Histogram Monte Carlo Study.

The combination of flexibility and semiflexibility in a single molecule is a powerful design principle both in nature and in materials science. We present results on the conformational behavior of a single multiblock-copolymer chain, consisting of equal amounts of Flexible (F) and Semiflexible (S) blocks with different affinity to an implicit solvent. We consider a manifold of macrostates defined by two terms in the total energy: intermonomer interaction energy and stiffness energy. To obtain diagrams of states (pseudo-phase diagrams), we performed flat-histogram Monte Carlo simulations using the Stochastic Approximation Monte Carlo algorithm (SAMC). We have accumulated two-Dimensional Density of States (2D DoS) functions (defined on the 2D manifold of macrostates) for a SF-multiblock-copolymer chain of length with block lengths b = 4, 8, 16, and 32 in two different selective solvents. In an analysis of the canonical ensemble, we calculated the heat capacity and determined its maxima and the most probable morphologies in different regions of the state diagrams. These are rich in various, non-trivial morphologies, which are formed without any specific interactions, and depend on the block length and the type of solvent selectivity (preferring S or F blocks, respectively). We compared the diagrams with those for the non-selective solvent and reveal essential changes in some cases. Additionally, we implemented microcanonical analysis in the “conformational” microcanonical (, where U is the potential energy) and the true microcanonical (, where E is the total energy) ensembles with the aim to reveal and classify pseudo-phase transitions, occurring under the change of temperature.

Content-Sensitive Multilevel Point Cluster Construction for ALS Point Cloud Classification

Airborne laser scanning (ALS) point cloud classification is a challenge due to factors including complex scene structure, various densities, surface morphology, and the number of ground objects. A point cloud classification method is presented in this paper, based on content-sensitive multilevel objects (point clusters) in consideration of the density distribution of ground objects. The space projection method is first used to convert the three-dimensional point cloud into a two-dimensional (2D) image. The image is then mapped to the 2D manifold space, and restricted centroidal Voronoi tessellation is built for initial segmentation of content-sensitive point clusters. Thus, the segmentation results take the entity content (density distribution) into account, and the initial classification unit is adapted to the density of ground objects. The normalized cut is then used to segment the initial point clusters to construct content-sensitive multilevel point clusters. Following this, the point-based hierarchical features of each point cluster are extracted, and the multilevel point-cluster feature is constructed by sparse coding and latent Dirichlet allocation models. Finally, the hierarchical classification framework is created based on multilevel point-cluster features, and the AdaBoost classifiers in each level are trained. The recognition results of different levels are combined to effectively improve the classification accuracy of the ALS point cloud in the test process. Two scenes are used to experimentally test the method, and it is compared with three other state-of-the-art techniques.

On Learning and Learned Data Representation by Capsule Networks

In this work, we investigate the following: 1) how the routing affects the CapsNet model fitting; 2) how the representation using capsules helps discover global structures in data distribution, and; 3) how the learned data representation adapts and generalizes to new tasks. Our investigation yielded the results some of which have been mentioned in the original paper of CapsNet, they are: 1) the routing operation determines the certainty with which a layer of capsules pass information to the layer above and the appropriate level of certainty is related to the model fitness; 2) in a designed experiment using data with a known 2D structure, capsule representations enable a more meaningful 2D manifold embedding than neurons do in a standard convolutional neural network (CNN), and; 3) compared with neurons of the standard CNN, capsules of successive layers are less coupled and more adaptive to new data distribution.

Weakly well-composed cell complexes over nD pictures

In previous work we proposed a combinatorial algorithm to “locally repair” the cubical complex Q(I) that is canonically associated with a given 3D picture I. The algorithm constructs a 3D polyhedral complex P(I) which is homotopy equivalent to Q(I) and whose boundary surface is a 2D manifold. A polyhedral complex satisfying these properties is called well-composed. In the present paper we extend these results to higher dimensions. We prove that for a given n-dimensional picture the obtained cell complex is well-composed in a weaker sense but is still homotopy equivalent to the initial cubical complex.

A Dimensional Splitting Method for 3D Elastic Shell with Mixed Tensor Analysis on a 2D Manifold Embedded into a Higher Dimensional Riemannian Space

A Dimensional Splitting Method for 3D Elastic Shell with Mixed Tensor Analysis on a 2D Manifold Embedded into a Higher Dimensional Riemannian Space

Application of path-integral quantization to indistinguishable particle systems topologically confined by a magnetic field

We demonstrate an original development of path-integral quantization in the case of a multiply connected configuration space of indistinguishable charged particles on a 2D manifold and exposed to a strong perpendicular magnetic field. The system occurs to be exceptionally homotopy-rich and the structure of the homotopy essentially depends on the magnetic field strength resulting in multi loop trajectories at specific conditions. We have proved, by a generalization of the Bohr-Sommerfeld quantization rule, that the size of a magnetic field flux quantum grows for multi loop orbits like $(2k+1)\frac{h}{c}$ with the number of loops $k$. Utilizing this property for electrons on the 2D substrate jellium we have derived upon the path integration a complete FQHE hierarchy in excellent consistence with experiments. The path-integral has been next developed to a sum over configurations, displaying various patterns of trajectory homotopies (topological configurations), which in the nonstationary case of quantum kinetics reproduces some unclear formerly details in the longitudinal resistivity observed in experiments.

Smooth 2D manifold extraction from 3D image stack

Three-dimensional fluorescence microscopy followed by image processing is routinely used to study biological objects at various scales such as cells and tissue. However, maximum intensity projection, the most broadly used rendering tool, extracts a discontinuous layer of voxels, obliviously creating important artifacts and possibly misleading interpretation. Here we propose smooth manifold extraction, an algorithm that produces a continuous focused 2D extraction from a 3D volume, hence preserving local spatial relationships. We demonstrate the usefulness of our approach by applying it to various biological applications using confocal and wide-field microscopy 3D image stacks. We provide a parameter-free ImageJ/Fiji plugin that allows 2D visualization and interpretation of 3D image stacks with maximum accuracy.

PreMosa: extracting 2D surfaces from 3D microscopy mosaics.

A significant focus of biological research is to understand the development, organization and function of tissues. A particularly productive area of study is on single layer epithelial tissues in which the adherence junctions of cells form a 2D manifold that is fluorescently labeled. Given the size of the tissue, a microscope must collect a mosaic of overlapping 3D stacks encompassing the stained surface. Downstream interpretation is greatly simplified by preprocessing such a dataset as follows: (i) extracting and mapping the stained manifold in each stack into a single 2D projection plane, (ii) correcting uneven illumination artifacts, (iii) stitching the mosaic planes into a single, large 2D image and (iv) adjusting the contrast. We have developed PreMosa, an efficient, fully automatic pipeline to perform the four preprocessing tasks above resulting in a single 2D image of the stained manifold across which contrast is optimized and illumination is even. Notable features are as follows. First, the 2D projection step employs a specially developed algorithm that actually finds the manifold in the stack based on maximizing contrast, intensity and smoothness. Second, the projection step comes first, implying all subsequent tasks are more rapidly solved in 2D. And last, the mosaic melding employs an algorithm that globally adjusts contrasts amongst the 2D tiles so as to produce a seamless, high-contrast image. We conclude with an evaluation using ground-truth datasets and present results on datasets from Drosophila melanogaster wings and Schmidtae mediterranea ciliary components. PreMosa is available under https://cblasse.github.io/premosa. blasse@mpi-cbg.de or myers@mpi-cbg.de. Supplementary data are available at Bioinformatics online.

Efficiently Storing Well-Composed Polyhedral Complexes Computed Over 3D Binary Images

A 3D binary image I can be naturally represented by a combinatorial-algebraic structure called cubical complex and denoted by Q(I), whose basic building blocks are vertices, edges, square faces and cubes. In Gonzalez-Diaz et al. (Discret Appl Math 183:59–77, 2015), we presented a method to “locally repair” Q(I) to obtain a polyhedral complex P(I) (whose basic building blocks are vertices, edges, specific polygons and polyhedra), homotopy equivalent to Q(I), satisfying that its boundary surface is a 2D manifold. P(I) is called a well-composed polyhedral complex over the picture I. Besides, we developed a new codification system for P(I), encoding geometric information of the cells of P(I) under the form of a 3D grayscale image, and the boundary face relations of the cells of P(I) under the form of a set of structuring elements. In this paper, we build upon (Gonzalez-Diaz et al. 2015) and prove that, to retrieve topological and geometric information of P(I), it is enough to store just one 3D point per polyhedron and hence neither grayscale image nor set of structuring elements are needed. From this “minimal” codification of P(I), we finally present a method to compute the 2-cells in the boundary surface of P(I).

Onset of anomalous diffusion in colloids confined to quasimonolayers.

It has been recently shown that a colloidal monolayer, e.g., formed at a fluid interface or by means of a suitable confining potential, exhibits anomalous collective diffusion. This is a consequence of the hydrodynamic interactions mediated by the three-dimensional (3D) ambient fluid when the particles are confined to reside on a two-dimensional (2D) manifold. We study theoretically and with numerical simulations the crossover from normal to anomalous diffusion as the particles are, in real systems, confined by a 3D external potential and thus have the possibility to fluctuate out of the 2D manifold, thus forming a quasimonolayer.

Localization of a mobile laser scanner via dimensional reduction

We extend the concept of intrinsic localization from a theoretical one-dimensional (1D) solution onto a 2D manifold that is embedded in a 3D space, and then recover the full six degrees of freedom for a mobile laser scanner with a simultaneous localization and mapping algorithm (SLAM). By intrinsic localization, we mean that no reference coordinate system, such as global navigation satellite system (GNSS), nor inertial measurement unit (IMU) are used. Experiments are conducted with a 2D laser scanner mounted on a rolling prototype platform, VILMA. The concept offers potential in being extendable to other wheeled platforms.

A new rotation-free isogeometric thin shell formulation and a corresponding continuity constraint for patch boundaries

This paper presents a general non-linear computational formulation for rotation-free thin shells based on isogeometric finite elements. It is a displacement-based formulation that admits general material models. The formulation allows for a wide range of constitutive laws, including both shell models that are extracted from existing 3D continua using numerical integration and those that are directly formulated in 2D manifold form, like the Koiter, Canham and Helfrich models. Further, a unified approach to enforce the G1-continuity between patches, fix the angle between surface folds, enforce symmetry conditions and prescribe rotational Dirichlet boundary conditions, is presented using penalty and Lagrange multiplier methods. The formulation is fully described in the natural curvilinear coordinate system of the finite element description, which facilitates an efficient computational implementation. It contains existing isogeometric thin shell formulations as special cases. Several classical numerical benchmark examples are considered to demonstrate the robustness and accuracy of the proposed formulation. The presented constitutive models, in particular the simple mixed Koiter model that does not require any thickness integration, show excellent performance, even for large deformations.

3D well-composed polyhedral complexes

A binary three-dimensional (3D) image I is well-composed if the boundary surface of its continuous analog is a 2D manifold. In this paper, we present a method to locally “repair” the cubical complex Q(I) (embedded in R3) associated to I to obtain a polyhedral complex P(I) homotopy equivalent to Q(I) such that the boundary surface of P(I) is a 2D manifold (and, hence, P(I) is a well-composed polyhedral complex). For this aim, we develop a new codification system for a complex K, called ExtendedCubeMap (ECM) representation of K, that codifies: (1) the information of the cells of K (including geometric information), under the form of a 3D grayscale image gP; and (2) the boundary face relations between the cells of K, under the form of a set BP of structuring elements that can be stored as indexes in a look-up table. We describe a procedure to locally modify the ECM representation EQ of the cubical complex Q(I) to obtain an ECM representation of a well-composed polyhedral complex P(I) that is homotopy equivalent to Q(I). The construction of the polyhedral complex P(I) is accomplished for proving the results though it is not necessary to be done in practice, since the image gP (obtained by the repairing process on EQ) together with the set BP codify all the geometric and topological information of P(I).

Visualization of Shape Motions in Shape Space

Analysis of dynamic object deformations such as cardiac motion is of great importance, especially when there is a necessity to visualize and compare the deformation behavior across subjects. However, there is a lack of effective techniques for comparative visualization and assessment of a collection of motion data due to its 4-dimensional nature, i.e., timely varying three-dimensional shapes. From the geometric point of view, the motion change can be considered as a function defined on the 2D manifold of the surface. This paper presents a novel classification and visualization method based on a medial surface shape space, in which two novel shape descriptors are defined, for discriminating normal and abnormal human heart deformations as well as localizing the abnormal motion regions. In our medial surface shape space, the geodesic distance connecting two points in the space measures the similarity between their corresponding medial surfaces, which can quantify the similarity and disparity of the 3D heart motions. Furthermore, the novel descriptors can effectively localize the inconsistently deforming myopathic regions on the left ventricle. An easy visualization of heart motion sequences on the projected space allows users to distinguish the deformation differences. Our experimental results on both synthetic and real imaging data show that this method can automatically classify the healthy and myopathic subjects and accurately detect myopathic regions on the left ventricle, which outperforms other conventional cardiac diagnostic methods.