Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Laplacian”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS2004

Improved Discrete Approximation of Laplacian of Gaussian

An improved method of computing a discrete approximation of the Laplacian of a Gaussian convolution of an image has been devised. The primary advantage of the method is that without substantially degrading the accuracy of the end result, it reduces the amount of information that must be processed and thus reduces the amount of circuitry needed to perform the Laplacian-of- Gaussian (LOG) operation. Some background information is necessary to place the method in context. The method is intended for application to the LOG part of a process of real-time digital filtering of digitized video data that represent brightnesses in pixels in a square array. The particular filtering process of interest is one that converts pixel brightnesses to binary form, thereby reducing the amount of information that must be performed in subsequent correlation processing (e.g., correlations between images in a stereoscopic pair for determining distances or correlations between successive frames of the same image for detecting motions). The Laplacian is often included in the filtering process because it emphasizes edges and textures, while the Gaussian is often included because it smooths out noise that might not be consistent between left and right images or between successive frames of the same image.

Shuler, Robert L., Jr.↗
NASA NTRS1980

Enhanced orbital geochemical images by the Laplacian subtraction method

Orbital geochemical studies of the moon based on X-ray fluorescence and gamma-ray spectroscopy have been successful. There are, however, difficulties if geologic units or other features to be investigated are too small. Investigations have been conducted with the objective to improve with the aid of image enhancement methods both spatial resolution and contrast for the geochemical experiments. A description is provided of the Laplacian subtraction method of enhancement. It was found that the application of a least-squares version of this method to the Mg/Al ratios obtained by orbital X-ray fluorescence improved geochemical images considerably. The spatial resolution increased by about a factor of 2 and the contrast also increased. The Laplacian of an image can also be used to detect very small 'hot' or 'cold' spots in an image.

Schonfeld, E.↗
NASA NTRS1990

Galerkin finite difference Laplacian operators on isolated unstructured triangular meshes by linear combinations

The Galerkin weighted residual technique using linear triangular weight functions is employed to develop finite difference formulae in Cartesian coordinates for the Laplacian operator on isolated unstructured triangular grids. The weighted residual coefficients associated with the weak formulation of the Laplacian operator along with linear combinations of the residual equations are used to develop the algorithm. The algorithm was tested for a wide variety of unstructured meshes and found to give satisfactory results.

Baumeister, Kenneth J.↗
DOE OSTI2024

Stealth dark matter spectrum using Laplacian Heaviside smearing and irreducible representations

We present nonperturbative lattice calculations in the quenched approximation of the low-lying meson and baryon spectrum of the SU(4) gauge theory with fundamental fermion constituents. This theory is one instance of stealth dark matter, a class of strongly coupled theories, where the lowest mass stable baryon is the dark matter candidate. This work constitutes the first milestone in the program to study stealth dark matter self-interactions. Here, we focus on reducing excited state contamination in the single-baryon channel by applying the Laplacian Heaviside method, as well as projecting our baryon operators onto the irreducible representations of the octahedral group. We compare our resulting spectrum to previous work involving Gaussian smeared nonprojected operators and find good agreement with reduced statistical uncertainties. We also present the spectrum of the low-lying odd-parity baryons for the first time.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS1971

Eigenvalues of the Laplacian of a graph

Let G be a finite undirected graph with no loops or multiple edges. The Laplacian matrix of G, Delta(G), is defined by Delta sub ii = degree of vertex i and Delta sub ij = -1 if there is an edge between vertex i and vertex j. The structure of the graph G is related to the eigenvalues of Delta(G); in particular, it is proved that all the eigenvalues of Delta(G) are nonnegative, less than or equal to the number of vertices, and less than or equal to twice the maximum vertex degree. Precise conditions for equality are given.

Anderson, W. N., Jr.↗
NASA NTRS1981

Orbital inclination of Iapetus and the rotation of the Laplacian plane

This paper explores the possibility that the orbit of Iapetus, with its relatively large inclination but small eccentricity, was generated by a rapid dispersal of a gaseous circumplanetary disk, assumed to be the progenitor of the satellite system. The orientation of the local Laplacian plane is shown to be a sensitive function of the disk's structure. Modification of the disk on a time scale comparable to the precession of the orbit's nodal line can produce a large inclination from one that is initially zero, while leaving the eccentricity unchanged. This time scale is of the same order of magnitude as the viscous evolution time scale for a fully turbulent disk. Hence, Iapetus need not be a captured satellite to account for its curious orbital signature.

Ward, W. R.↗
NASA NTRS1982

The computation of Laplacian smoothing splines with examples

Laplacian smoothing splines (LSS) are presented as generalizations of graduation, cubic and thin plate splines. The method of generalized cross validation (GCV) to choose the smoothing parameter is described. The GCV is used in the algorithm for the computation of LSS's. An outline of a computer program which implements this algorithm is presented along with a description of the use of the program. Examples in one, two and three dimensions demonstrate how to obtain estimates of function values with confidence intervals and estimates of first and second derivatives. Probability plots are used as a diagnostic tool to check for model inadequacy.

Wendelberger, J. G.↗
NASA NTRS1984

The Laplacian resonance and tidal dissipation

The Laplacian resonance amongst Io, Europa, and Ganymede was examined, taking into account tidal dissipation in Jupiter, Io and Europa. In equilibrium, it is not possible to neglect dissipation in Europa nor the torques of Jupiter on Europa and Ganymede. A formal calculation was made on the assumption that the tidal torques and the torques in orbit-orbit resonance reached an equilibrium such that the rates of decrease of the mean motions of the satellites are in the ratios 4:2:1. Q(j)/k(j)=167 Q(1)/k(1), Q(2)/k(2) = 0.44 Q(1)/K(1), where Q denotes the quality at the body's frequency of rotation, k its second degree Love number and the subscripts J, 1 and 2 denote Jupiter, Io, and Europa. The Q(J) found is at Jupiter's frequency of rotation. The rate of tidal dissipation in Europa comes to 1/5 that of Io. Such a low value of Q(2)/K(2) is plausible if dissipation in both Io and Europa is due to tidal friction in fluid against underlying and overlying solid layers.

Cook, A. F.↗
NASA NTRS1984

Permutation codes for the Laplacian source

Permutation codes for the Laplacian source are developed. The performance of these codes is evaluated and compared with other quantizers and the rate-distortion function. It is shown that there is a bit-rate region in which the permutation codes outperform certain single-sample quantizers.

Townes, S. A.↗
NASA NTRS1989

Structural stereo matching of Laplacian-of-Gaussian contour segments for 3D perception

The stereo correspondence problem is solved using Laplacian-of-Gaussian zero-crossing contours as a source of primitives for structural stereopsis, as opposed to traditional point-based algorithms. Up to 74 percent matching of candidate zero crossing points are being achieved on 240 x 246 images at small scales and large ranges of disparity, without coarse-to-fine tracking and without precise knowledge of the epipolar geometry. This approach should prove particularly useful in recovering the epipolar geometry automatically for stereo pairs for which it is unavailable a priori. Such situations occur in the extraction of terrain models from stereo aerial photographs.

Boyer, K. L.↗
NASA NTRS1997

The Path Resistance Method for Bounding the Smallest Nontrivial Eigenvalue of a Laplacian

We introduce the path resistance method for lower bounds on the smallest nontrivial eigenvalue of the Laplacian matrix of a graph. The method is based on viewing the graph in terms of electrical circuits; it uses clique embeddings to produce lower bounds on lambda(sub 2) and star embeddings to produce lower bounds on the smallest Rayleigh quotient when there is a zero Dirichlet boundary condition. The method assigns priorities to the paths in the embedding; we show that, for an unweighted tree T, using uniform priorities for a clique embedding produces a lower bound on lambda(sub 2) that is off by at most an 0(log diameter(T)) factor. We show that the best bounds this method can produce for clique embeddings are the same as for a related method that uses clique embeddings and edge lengths to produce bounds.

Guattery, Stephen↗
NASA NTRS1986

Analysis of the SOR iteration for the 9-point Laplacian

The SOR iteration for solving linear systems of equations depends upon an overrelaxation factor omega. A theory for determining omega was given by Young (1950) for consistently ordered matrices. Here the optimal omega is determined for the 9-point stencil for the model problem of Laplace's equation on a square. Several orderings of the equations are considered, including the natural rowwise and multicolor orderings, all of which lead to non-consistently ordered matrices, and two equivalence classes of orderings are found with different convergence behavior and optimal omega's. The results for the natural rowwise ordering are compared to those of Garabedian (1956) and it is explained why both results are, in a sense, correct, even though they differ. Also analyzed is a pseudo SOR method for the model problem and it is shown that it is not as effective as the SOR methods. Finally, the point SOR methods are compared to known results for line SOR methods for this problem.

Loyce M. Adams↗
DOE OSTI2024

Unifying Combinatorial and Graphical Methods in Artificial Intelligence

Recently, a new graph Laplacian, called the inner product Laplacian, was introduced which generalizes many existing Laplacians, including the normalized and combinatorial Laplacian and their weighted variants. The key observation behind the inner product Laplacian is that by defining appropriate inner product spaces on the vertices and edges, the standard Laplacians can be recovered as Hodge Laplacians over the simplicial complex formed by the edges and vertices. These inner product spaces form a natural way to incorporate non-combinatorial information into the definition of a domain-specific Laplacian. In particular, in contrast to current domain-specific weighting schemes which rely solely on edge weights, information regarding the similarity of non-adjacent vertices and arbitrary pairs of edges can be effectively incorporated into the Laplacian. In order to illustrate this approach we consider the problem of calculating the potential energy of an atomistic configuration using Graph Neural Networks. In comparison with start-of-the-art approaches, such as SchNet, our approach replaces a learned (via auto-encoder) representation of the atom types with an inner product space on atoms based on scientific knowledge (e.g., electronegativity). We will illustrate how this approach captures key chemical properties of the molecules and compare the energy calculations with state-of-the-art neural network approaches. However, to compute the resulting Laplacian involves a mixture of sparse and dense matrix computation and yields a dense matrix as the basis for the graph convolution. This dense convolutional kernel necessitates moving away from the standard message passing framework for graph neural networks and increases the computational cost of applying the kernel. In order to mitigate these costs we investigate means of leveraging the mixed sparse and dense computations to reduce the overall computational cost and how these approaches can be automatically transferred to energy efficient hardware (e.g., field programmable gate arrays (FPGAs)).

97 MATHEMATICS AND COMPUTING↗