Simplicial Volume Research Articles

Maximum simplex volume: an efficient unsupervised band selection method for hyperspectral image

Hyperspectral imaging makes it possible to obtain object information with fine spectral resolution as well as spatial resolution, which is beneficial to a wide array of applications. However, there is a high correlation among the bands in a hyperspectral image (HSI). Band selection (BS), selecting only some representative bands to describe well the original image, is an appropriate approach to tackle this problem. In this study, the authors propose an efficient greedy‐based unsupervised BS method, namely the maximum simplex volume by orthogonal‐projection BS method. The main contributions are two‐fold: (i) an information‐lossless compressed descriptor in the Euclidean sense that can reduce the amount of redundant information in the band analysis and (ii) an orthogonal‐projection‐based algorithm to find the band points forming the simplex of maximum volume. The experimental results on four real HSIs demonstrate that the proposed method can achieve satisfying pixel classification performances and is computationally fast.

Quantitative assessment of disease severity and rating of barley cultivars based on hyperspectral imaging in a non-invasive, automated phenotyping platform

BackgroundPhenotyping is a bottleneck for the development of new plant cultivars. This study introduces a new hyperspectral phenotyping system, which combines the high throughput of canopy scale measurements with the advantages of high spatial resolution and a controlled measurement environment. Furthermore, the measured barley canopies were grown in large containers (called Mini-Plots), which allow plants to develop field-like phenotypes in greenhouse experiments, without being hindered by pot size.ResultsSix barley cultivars have been investigated via hyperspectral imaging up to 30 days after inoculation with powdery mildew. With a high spatial resolution and stable measurement conditions, it was possible to automatically quantify powdery mildew symptoms through a combination of Simplex Volume Maximization and Support Vector Machines. Detection was feasible as soon as the first symptoms were visible for the human eye during manual rating. An accurate assessment of the disease severity for all cultivars at each measurement day over the course of the experiment was realized. Furthermore, powdery mildew resistance based necrosis of one cultivar was detected as well.ConclusionThe hyperspectral phenotyping system combines the advantages of field based canopy level measurement systems (high throughput, automatization, low manual workload) with those of laboratory based leaf level measurement systems (high spatial resolution, controlled environment, stable conditions for time series measurements). This allows an accurate and objective disease severity assessment without the need for trained experts, who perform visual rating, as well as detection of disease symptoms in early stages. Therefore, it is a promising tool for plant resistance breeding.

Fundamental groups of aspherical manifolds and maps of non-zero degree

We define a new class of irreducible groups, called groups not infinite-indexpresentable by products or not IIPP . We prove that certain aspherical manifolds with fundamental groups not IIPP do not admit maps of non-zero degree from direct products. This extends previous results of Kotschick and Löh, providing new classes of aspherical manifolds – beyond those non-positively curved ones which were predicted by Gromov – that do not admit maps of non-zero degree from direct products. A sample application is that an aspherical geometric 4-manifold admits a map of non-zero degree from a direct product if and only if it is a virtual product itself. This completes a characterization of the product geometries due to Hillman. Along the way we prove that for certain groups the property IIPP is a criterion for reducibility. This especially implies the vanishing of the simplicial volume of the corresponding aspherical manifolds. It is shown that aspherical manifolds with reducible fundamental groups do always admit maps of non-zero degree from direct products.

Archetypal Analysis for Sparse Representation-Based Hyperspectral Sub-pixel Quantification

The estimation of land cover fractions from remote sensing images is a frequently used indicator of the environmental quality. This paper focuses on the quantification of land cover fractions in an urban area of Berlin, Germany, using simulated hyperspectral <small>EnMAP</small> data with a spatial resolution of 30 m × 30 m. We use constrained sparse representation, where each pixel with unknown surface characteristics is expressed by a weighted linear combination of elementary spectra with known land cover class. We automatically determine the elementary spectra from image reference data using archetypal analysis by simplex volume maximization, and combine it with reversible jump Markov chain Monte Carlo method. In our experiments, the estimation of the automatically derived elementary spectra is compared to the estimation obtained by a manually designed spectral library by means of reconstruction error, mean absolute error of the fraction estimates, sum of fractions, R2, and the number of used elementary spectra. The experiments show that a collection of archetypes can be an adequate and efficient alternative to the manually designed spectral library with respect to the mentioned criteria.

Stable presentation length of 3–manifold groups

We introduce the stable presentation length of a finitely presented group. The stable presentation length of the fundamental group of a 3-manifold can be considered as an analogue of the simplicial volume. We show that the stable presentation length have some additive properties like the simplicial volume, and the simplicial volume of a closed 3-manifold is bounded from above and below by constant multiples of the stable presentation length of its fundamental group.

Simplicial volume of links from link diagrams

AbstractThe hyperbolic volume of a link complement is known to be unchanged when a half-twist is added to a link diagram, and a suitable 3-punctured sphere is present in the complement. We generalise this to the simplicial volume of link complements by analysing the corresponding toroidal decompositions. We then use it to prove a refined upper bound for the volume in terms of twists of various lengths for links.

Robust Minimum Volume Simplex Analysis for Hyperspectral Unmixing

Most blind hyperspectral unmixing methods exploit convex geometry properties of hyperspectral data. The minimum volume simplex analysis (MVSA) is one of such methods, which, as many others, estimates the minimum volume (MV) simplex where the measured vectors live. MVSA was conceived to circumvent the matrix factorization step often implemented by MV-based algorithms and also to cope with outliers, which compromise the results produced by MV algorithms. Inspired by the recently proposed robust MV enclosing simplex (RMVES) algorithm, we herein introduce the robust MVSA (RMVSA), which is a version of MVSA robust to noise. As in RMVES, the robustness is achieved by employing chance constraints, which control the volume of the resulting simplex. RMVSA differs, however, substantially from RMVES in the way optimization is carried out. In this paper, we develop a linearization relaxation of the nonlinear chance constraints, which can greatly lighten the computational complex of chance constraint problems. The effectiveness of RMVSA is illustrated by comparing its performance with the state of the art.

Vanishing simplicial volume for certain affine manifolds

We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. As a consequence, these manifolds have zero Euler characteristic, satisfying the Chern Conjecture. Along the way, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups of π 1 \pi _1 to have vanishing simplicial volume. This answers a special case of a question due to Lück.

Positive simplicial volume implies virtually positive Seifert volume for 3–manifolds

In this paper, it is shown that for any closed orientable $3$-manifold with positive simplicial volume, the growth of the Seifert volume of its finite covers is faster than the linear rate. In particular, each closed orientable $3$-manifold with positive simplicial volume has virtually positive Seifert volume. The result reveals certain fundamental difference between the representation volume of the hyperbolic type and the Seifert type. The proof is based on developments and reactions of recent results on virtual domination and on virtual representation volumes of $3$-manifolds.

Random Approximation of Convex Bodies: Monotonicity of the Volumes of Random Tetrahedra

Choose uniform random points $$X_1, \dots , X_n$$ in a given convex set and let $${\text { conv}}[X_1, \dots , X_n]$$ be their convex hull. It is shown that in dimension three the expected volume of this convex hull is in general not monotone with respect to set inclusion. This answers a question by Meckes in the negative. The given counterexample is formed by uniformly distributed points in the three-dimensional tetrahedron together with a small perturbation of it. As side result we obtain an explicit formula for all even moments of the volume of a random simplex which is the convex hull of three uniform random points in the tetrahedron and the center of one facet.

Rank gradient versus stable integral simplicial volume

We observe that stable integral simplicial volume of closed manifolds gives an upper bound for the rank gradient of the corresponding fundamental groups.

ARCHETYPAL ANALYSIS FOR SPARSE REPRESENTATION-BASED HYPERSPECTRAL SUB-PIXEL QUANTIFICATION

Abstract. This paper focuses on the quantification of land cover fractions in an urban area of Berlin, Germany, using simulated hyperspectral EnMAP data with a spatial resolution of 30m×30m. For this, sparse representation is applied, where each pixel with unknown surface characteristics is expressed by a weighted linear combination of elementary spectra with known land cover class. The elementary spectra are determined from image reference data using simplex volume maximization, which is a fast heuristic technique for archetypal analysis. In the experiments, the estimation of class fractions based on the archetypal spectral library is compared to the estimation obtained by a manually designed spectral library by means of reconstruction error, mean absolute error of the fraction estimates, sum of fractions and the number of used elementary spectra. We will show, that a collection of archetypes can be an adequate and efficient alternative to the spectral library with respect to mentioned criteria.

A Novel Endmember Extraction Method for Hyperspectral Imagery Based on Quantum-Behaved Particle Swarm Optimization

Endmember extraction is one of the most important issues in hyperspectral image analysis. Under the linear mixing model and pure pixel assumption, a number of convex-geometry-based methods have been developed in the past decades. However, these traditional methods generally produce unsatisfactory results since they require the hyperspectral image to have a convex structure and this is not exactly the case with the real image scene. The particle swarm optimization (PSO) algorithm has recently been employed to address the endmember extraction problem, but its performance is limited by the premature convergence and lower precision of the standard PSO, and much effort is required to enhance the optimization result. To address these problems, in this study, a novel quantum-behaved particle swarm optimization (QPSO) algorithm is proposed for hyperspectral endmember extraction. The notable advantages of the proposed method include: 1) a row–column coding approach for the particles is designed to accelerate the optimization process; 2) a cooperative approach is employed to update the particles’ individual and global best positions, in order to help the particles’ optimization behavior in the multidimensional search space; and 3) two kinds of objective functions, namely, maximizing the simplex volume formed by the endmember combination, and minimizing the root-mean-square error between the original image and its remixed image, are incorporated in a sequential optimization approach for the endmember extraction problem, which makes the algorithm robust to outliers at an acceptable time cost. The extensive experimental results prove that QPSO is able to find the optimal endmember combination.

FPGA implementation of a maximum simplex volume algorithm for endmember extraction from remotely sensed hyperspectral images

Spectral unmixing is a very important technique for remotely sensed hyperspectral unmixing. Since more hyperspectral applications now require real or near real-time processing capabilities, fast spectral unmixing using field-programmable gate arrays (FPGAs) has received considerable interest in recent years. FPGAs can provide onboard, high computing performance at low power consumption. Another important characteristic of FPGA-based systems is reconfigurability, which makes them more flexible to process different kind of scenes. Pure signature (endmember) extraction is a fundamental step in spectral unmixing, which has been tackled using the maximum volume principle by several algorithms, most notably N-FINDR and simplex growing algorithm (SGA). These algorithms find out the simplex with maximum volume as a mechanism to extract endmembers. However, a previous dimensionality reduction step is generally required, which introduces information loss and additional computational burden. To address these issues, in this work we introduce a new volume calculation formula and further develop a new real-time implementation of a maximum simplex volume algorithm (called RT-MSVA). The proposed RT-MSVA does not need dimensionality reduction, so all spectral bands can be used without losing any information to ensure robust endmember extraction accuracy. Experiments with synthetic and real hyperspectral images have been conducted to evaluate the accuracy and computational performance of our proposed method. Our experimental results indicate that proposed FPGA-based implementation significantly outperforms the corresponding software version and achieves real-time processing performance in the considered problem. It also exhibits better endmember extraction accuracy and comparable performance to other available techniques, such as a real-time implementation of a simplex growing algorithm (RT-FSGA).

Simplex Volume Maximization (SiVM): A matrix factorization algorithm with non-negative constrains and low computing demands for the interpretation of full spectral X-ray fluorescence imaging data

Technological progress allows for an ever-faster acquisition of hyperspectral data, challenging the users to keep up with interpreting the recorded data. Matrix factorization, the representation of data sets by bases (or loads) and coefficient (or score) images is long used to support the interpretation of complex data sets. We propose in this publication Simplex Volume Maximization (SiVM) for the analysis of X-ray fluorescence (XRF) imaging data sets. SiVM selects archetypical data points that represents the data set and thus provides easily understandable bases, preserves the non-negative character of XRF data sets and has low demands concerning computing resources. We apply SiVM on an XRF data set of Hans Memling's Portrait of a man from the Lespinette family from the collection of the Mauritshuis (The Hague, NL) and discuss capabilities and shortcomings of SiVM.

A Novel Spectral-Unmixing-Based Green Algae Area Estimation Method for GOCI Data

Geostationary Ocean Color Imager (GOCI) data have been widely used in the detection and area estimation of green algae blooms. However, due to the low spatial resolution of GOCI data, pixels in an image are usually “mixed,” which means that the region a pixel covers may include many different materials. Traditional index-based methods can detect whether there are green algal blooms in each pixel, whereas it is still challenging to determine the proportion that green algae blooms occupy in a pixel. In this paper, we propose a novel subpixel-level area estimation method for green algae blooms based on spectral unmixing, which can not only detect the existence of green algae but also determine their proportion in each pixel. A fast endmember extraction method is proposed to automatically calculate the endmember spectral matrix, and the abundance map of green algae which could be regarded as the area estimation is obtained by nonnegatively constrained least squares. This new fast endmembers extraction technique outperforms the classical N-FINDR method by applying two models: candidates location and distance-based vertices determination. In the first model, we propose a medium-distance-based candidates location strategy, which could reduce the searching space during vertices selection. In the second model, we replace the simplex volume measure with a more simple distance measure, thus complex matrix determinant calculation is avoided. We have theoretically proven the equivalency of volume and distance measure. Experiments on GOCI data and synthetic data demonstrate the superiority of the proposed method compared with some state-of-art approaches.

On the volume growth of the hyperbolic regular $$\varvec{n}$$ n -simplex

In this paper we give lower and upper bounds for the volume growth of a regular hyperbolic simplex, namely for the ratio of the n-dimensional volume of a regular simplex and the \((n-1)\)-dimensional volume of its facets. In addition to the methods of U. Haagerup and M. Munkholm we use a third volume form based on the hyperbolic orthogonal coordinates of a body. In the case of the ideal, regular simplex our upper bound gives the best known upper bound. On the other hand, also in the ideal case our general lower bound, improved the best known one for \(n=3\).

Piecewise straightening and Lipschitz simplicial volume

We study the Lipschitz simplicial volume, which is a metric version of the simplicial volume. We introduce the piecewise straightening procedure for singular chains, which allows us to generalize the proportionality principle and the product inequality to the case of complete Riemannian manifolds of finite volume with sectional curvature bounded from above.

Network gravity

We introduce the construction of a new framework for probing discrete emergent geometry and boundary-boundary observables based on a fundamentally a-dimensional underlying network structure. Using a gravitationally motivated action with Forman weighted combinatorial curvatures and simplicial volumes relying on a decomposition of an abstract simplicial complex into realized embeddings of proper skeletons, we demonstrate properties such as a minimal volume-scale cutoff, the necessity of a positive-definite cosmological constant-like term as a regulator for non-degenerate geometries, and naturally emergent simplicial structures from Metropolis network evolution simulations with no restrictions on attachment rules or regular building blocks. We see emergent properties which echo results from both the spinfoam formalism and causal dynamical triangulations in quantum gravity, and provide analytical and numerical results to support the analogy. We conclude with a summary of open questions and intent for future work in developing the program.

Recursive Geometric Simplex Growing Analysis for Finding Endmembers in Hyperspectral Imagery

Simplex growing algorithm (SGA) is an endmember finding algorithm which searches for endmembers one after another by growing simplexes one vertex at a time via maximizing simplex volume (SV). Unfortunately, several issues arise in calculating SV. One is the use of dimensionality reduction (DR) because the dimensionality of a simplex is usually much smaller than data dimensionality. Second, calculating SV requires calculating the determinant of an ill-ranked matrix in which case singular value decomposition (SVD) is generally required to perform DR. Both approaches generally do not produce true SV. Finally, the computing time becomes excessive and numerically instable as the number of endmembers grows. This paper develops a new theory, called geometric simplex growing analysis (GSGA), to resolve these issues. It can be considered as an alternative to SGA from a rather different point of view. More specifically, GSGA looks into the geometric structures of a simplex whose volume can be actually calculated by multiplication of its base and height. As a result, it converts calculating maximal SV to finding maximal orthogonal projection as its maximal height becomes perpendicular to its base. To facilitate GSGA in practical applications, GSGA is further used to extend SGA to recursive geometric simplex growing algorithm (RGSGA) which allows GSGA to be implemented recursively in a similar manner that a Kalman filter does. Consequently, RGSGA can be very easily implemented with significant saving of computing time. Best of all, RGSGA is also shown to be most efficient and effective among all SGA-based variants.