An overview of the self-organizing map algorithm, on which the papers in this issue are based, is presented in this article.

The Self-Organizing Map

Dependence of biodiesel fuel properties on the structure of fatty acid alkyl esters

IEEE Transactions on Pattern Analysis and Machine Intelligence

Production of liquid biofuels from renewable resources

Progress and recent trends in biodiesel fuels

Integral invariants obtained from Principal Component Analysis on a small kernel domain of a submanifold encode important geometric information classically defined in differential-geometric terms. We generalize to hypersurfaces in any dimension major results known for surfaces in space, which in turn yield a method to estimate the extrinsic and intrinsic curvature of an embedded Riemannian submanifold of general codimension. In particular, integral invariants are defined by the volume, barycenter, and the EVD of the covariance matrix of the domain. We obtain the asymptotic expansion of such invariants for a spherical volume component delimited by a hypersurface and for the hypersurface patch created by ball intersetions, showing that the eigenvalues and eigenvectors can be used as multi-scale estimators of the principal curvatures and principal directions. This approach may be interpreted as performing statistical analysis on the underlying point-set of a submanifold in order to obtain geometric descriptors at scale with potential applications to Manifold Learning and Geometry Processing of point clouds.

Manifold Curvature Descriptors from Hypersurface Integral Invariants.

The existence of characteristic structure, or shape, in complex data sets has been recognized as increasingly important for mathematical data analysis. This realization has motivated the development of new tools such as persistent homology for exploring topological invariants, or features, in large data sets. In this paper, we apply persistent homology to the characterization of gas plumes in time dependent sequences of hyperspectral cubes, i.e. the analysis of 4-way arrays. We investigate hyperspectral movies of Long-Wavelength Infrared data monitoring an experimental release of chemical simulant into the air. Our approach models regions of interest within the hyperspectral data cubes as points on the real Grassmann manifold Gk,i¾?n whose points parameterize the k-dimensional subspaces of $$\mathbb {R}^n$$Rn, contrasting our approach with the more standard framework in Euclidean space. An advantage of this approach is that it allows a sequence of time slices in a hyperspectral movie to be collapsed to a sequence of points in such a way that some of the key structure within and between the slices is encoded by the points on the Grassmann manifold. This motivates the search for topological features, associated with the evolution of the frames of a hyperspectral movie, within the corresponding points on the Grassmann manifold. The proposed mathematical model affords the processing of large data sets while retaining valuable discriminatory information. In this paper, we discuss how embedding our data in the Grassmann manifold, together with topological data analysis, captures dynamical events that occur as the chemical plume is released and evolves.

Persistent Homology on Grassmann Manifolds for Analysis of Hyperspectral Movies

Biodiesel, which consists of the fatty acid esters of simple alcohols, is a potential replacement for a portion of thediesel fuel used in transportation. It is produced from both used (oil that has been utilized for frying and discarded) and newvegetable and animal fats and oils. It has several advantages. Among these advantages are its classification as a renewableresource, its ability to reduce HC, CO, and CO2 exhaust emissions, its nontoxic character, and its biodegradability. One ofthe keys to making biodiesel a viable and profitable energy source is the use of a continuous flow transesterification processto reduce time and cost, thereby increasing production and profit. A continuous flow esterification process for producing biodiesel from rapeseed oil and ethanol was investigated. Theequipment consisted of an oil metering pump, centrifugal mixing pump, ethanol and catalyst metering pump, static mixers,laddertype retention reactor, water injection system, and continuous flow centrifugal separating system. The oil feed ratewas 0.38 L/min that yields about three times the weekly production of the existing batch type transesterification system thatproduces 945 L (250 gal) per batch. It was anticipated that if methanol were used instead of ethanol that the flow rate couldbe considerably increased. The system in its present configuration has met ASTM PS12199 standard for free and total glycerol. The centrifugalseparation resulted in release of excessive alcohol vapors. A vapor recovery and condensing system should be installed. Thislatter system could help further reduce costs of the biodiesel produced by recycling some of the alcohol used.

Continuous flow biodiesel production.

One of the fundamental assumptions of compressive sensing (CS) is that a signal can be reconstructed from a small number of samples by solving an optimization problem with the appropriate regularization term. Two standard regularization terms are the L1 norm and the total variation (TV) norm. We present a comparison of CS reconstruction results based on these two approaches in the context of chemical detection, and we demonstrate that optimization based on the L1 norm outperforms optimization based on the TV norm. Our comparison is driven by CS sampling, reconstruction, and chemical detection in two real-world datasets: the Physical Sciences Inc. Fabry-Perot interferometer sensor multispectral dataset and the Johns Hopkins Applied Physics Lab FTIR-based longwave infrared sensor hyperspectral dataset. Both datasets contain the release of a chemical simulant such as glacial acetic acid, triethyl phosphate, and sulfur hexafluoride. For chemical detection we use the adaptive coherence estimator (ACE) and bulk coherence, and we propose algorithmic ACE thresholds to define the presence or absence of a chemical of interest in both un-compressed data cubes and reconstructed data cubes. The un-compressed data cubes provide an approximate ground truth. We demonstrate that optimization based on either the L1 norm or TV norm results in successful chemical detection at a compression rate of 90%, but we show that L1 optimization is preferable. We present quantitative comparisons of chemical detection on reconstructions from the two methods, with an emphasis on the number of pixels with an ACE value above the threshold.

Total variation vs L1 regularization: a comparison of compressive sensing optimization methods for chemical detection

A flag is a nested sequence of vector spaces. The type of the flag is determined by the sequence of dimensions of the vector spaces making up the flag. A flag manifold is a manifold whose points parameterize all flags of a particular type in a fixed vector space. This paper provides the mathematical framework necessary for implementing self-organizing mappings on flag manifolds. Flags arise implicitly in many data analysis techniques for instance in wavelet, Fourier, and singular value decompositions. The proposed geometric framework in this paper enables the computation of distances between flags, the computation of geodesics between flags, and the ability to move one flag a prescribed distance in the direction of another flag. Using these operations as building blocks, we implement the SOM algorithm on a flag manifold. The basic algorithm is applied to the problem of parameterizing a set of flags of a fixed type.

Chris Peterson

Papers

Manifold Curvature Descriptors from Hypersurface Integral Invariants.

Persistent Homology on Grassmann Manifolds for Analysis of Hyperspectral Movies

Continuous flow biodiesel production.

Total variation vs L1 regularization: a comparison of compressive sensing optimization methods for chemical detection

Self-Organizing Mappings on the Flag Manifold