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Topic: Laplacian

Related Topics

Vector fields in cylindrical and spherical coordinates

Nabla in cylindrical and spherical coordinates

Curvilinear coordinates

Dynamical system

Divergence

Curvature of Riemannian manifolds

Christoffel symbols

Ricci curvature

Edge detection

Intrinsic metric

Dirac equation

Sobel

Riemannian manifold

Nash embedding theorem

Canny

	Image Sharpening with a Laplacian Kernel
		The Laplacian operator is an example of a second order or second derivative method of enhancement.
		Thus, one application of a Laplacian operator is to restore fine detail to an image which has been smoothed to remove noise.
		The Laplacian operator is implemented in IDL as a convolution between an image and a kernel.
	www.dfanning.com /ip_tips/sharpen.html (954 words)

	Laplace operator - Wikipedia, the free encyclopedia
		In mathematics and physics, the Laplace operator or Laplacian, denoted by Δ, is a differential operator, specifically an important case of an elliptic operator (or a hyperbolic operator, when defined on pseudo-Riemannian manifolds), with many applications in mathematics and physics.
		In mathematics, functions with vanishing Laplacian are called harmonic functions; the Laplacian is at the core of Hodge theory and the results of de Rham cohomology.
		The discrete Laplace operator is an analog of the continuous Laplacian, defined on graphs and grids.
	en.wikipedia.org /wiki/Laplacian (1033 words)

	PlanetMath: Laplacian
		A coordinate independent definition of the Laplacian is
		A harmonic function is one for which the Laplacian vanishes.
		This is version 13 of Laplacian, born on 2002-06-04, modified 2006-07-11.
	planetmath.org /encyclopedia/Laplacian.html (109 words)

	Theory of Atoms in Molecules: The Laplacian of the Electron Density and the Lewis and VSEPR Models
		The Laplacian of the electron density recovers the shell structure of an atom by displaying a corresponding number of alternating shells of charge concentration and charge depletion.
		The geometry of approach of the acid and base molecules is predicted through the alignment of the corresponding "lumps" and "holes" in their Laplacian distributions, as illustrated for the approach of the non-bonded charge concentration on carbon of the CO molecule to the hole on the boron atom in BH, Figure 12.
		The use of the Laplacian of the electron density to account for the bent geometries of the hydride, halide and methylide molecules of calcium, strontium and barium, in terms of a distortion of the outer core of the electron density of the metal atom is discussed in reference 12.
	www.chemistry.mcmaster.ca /faculty/bader/aim/aim_5.html (984 words)

	Laplacian of Gaussian Filter
		Laplacian filters are derivative filters used to find areas of rapid change (edges) in images.
		Since derivative filters are very sensitive to noise, it is common to smooth the image (e.g., using a Gaussian filter) before applying the Laplacian.
		This is called a negative Laplacian because the central peak is negative.
	www.marquette.edu /~matthysd/web226/Lab02.htm (407 words)

	How Unsharp Masking and Laplacian Sharpening Work (Site not responding. Last check: 2007-10-27)
		Laplacian sharpening blurs an image at several different scales and produces a mask at each level (or layer).
		The Laplacian transform is a true wavelet transform, which is why this method of sharpening is sometimes generally referred to as a wavelet sharpening, but that's too general since there are countless wavelet transforms, many of which can be used for image enhancement in various ways.
		Laplacian sharpening gives you control over the scaling factor for each of the masks, just the way unsharp masking gave you control over the scaling factor for the single blurred mask that was generated.
	www.unm.edu /~keithw/astroPhotography/imageSharpening.html (1061 words)

	The Laplacian of the Electron Density (Site not responding. Last check: 2007-10-27)
		The laplacian of the electron density is the trace of the Hessian matrix.
		In regions where the Laplacian is negative, the potential energy is dominant and the negative charge is concentrated.
		In regions where the Laplacian is positive, the kinetic energy density dominates and a depletion of negative charge occurs.
	www.cmbi.kun.nl /%7Eschaft/molden/laplacian.html (358 words)

	Charge coupled device incorporating Laplacian thresholding with TDI array - Patent 4264930
		The Laplacian for each pel is calculated by measuring the average brightness within an area surrounding the pel, and subtracting the average brightness from the brightness measured at the pel.
		The Laplacian for 3P3 may then be approximated by subtracting a charge proportional to 1/25 of the area sum from the charge 3P3.
		The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the claims rather than by the foregoing description, and all changes which come within the meaning and range of the equivalents of the claims are therefore intended to be embraced therein.
	www.freepatentsonline.com /4264930.html (8508 words)

	FeatureJ: Laplacian (Site not responding. Last check: 2007-10-27)
		This plugin allows you to compute the Laplacian of an image and detect its zero-crossings - features that have been shown by psychophysical and neurophysiological research to play a key role in human vision as well [1,2].
		The smoothing scale is equal to the standard deviation of the Gaussian derivative kernel used in computing the second-order derivatives constituting the Laplacian and must be larger than or equal to zero.
		If you have indicated in the Options dialog that use should be made of specific voxel sizes, in each dimension the scale is divided by the voxel size in that dimension.
	imagescience.bigr.nl /meijering/software/featurej/laplacian.html (239 words)

	Gaussian and Laplacian Pyramids
		The Laplacian is then computed as the difference between the original image and the low pass filtered image.
		The Laplacian pyramid can thus be used to represent images as a series of band-pass filtered images, each sampled at successively sparser densities.
		We were able to extract the text from the background in an image by using 3 levels of the Laplacian pyramid.
	www.cs.utah.edu /~arul/report/node12.html (419 words)

	IRAF help : images laplace (Dec85)
		The Laplacian filters are a set of four three by three kernels which approximate the Laplacian operator, where a Laplacian operator is defined as the sum of the partial second derivatives in x and y.
		The elements of the central column and row of a 3 by 3 image subraster are combined to estimate the Laplacian at the position of the central pixel.
		Another characteristic of Laplacian operators is that a single grey level transition may produce two distinct peaks one positive and one negative in the Laplacian which may be offset from the gradient location.
	www.astro.uni-bonn.de /~webstw/cm/lfa_html/iraf/images.laplace.html (490 words)

	Laplacian (Site not responding. Last check: 2007-10-27)
		The divergence of the gradient of a scalar function is called the Laplacian.
		The Laplacian finds application in the Schrodinger equation in quantum mechanics.
		In electrostatics, it is a part of LaPlace's equation and Poisson's equation for relating electric potential to charge density.
	hyperphysics.phy-astr.gsu.edu /hbase/lapl.html (43 words)

	Laplacian Surface Editing (Site not responding. Last check: 2007-10-27)
		We provide such a representation of a surface, based on the Laplacian of the mesh, by encoding each vertex relative to its neighborhood.
		The Laplacian of the mesh is enhanced to be invariant to locally linearized rigid transformations and scaling.
		Based on this Laplacian representation, we develop useful editing operations: interactive free-form deformation in a region of interest based on the transformation of a handle, transfer and mixing of geometric details between two surfaces, and transplanting of a partial surface mesh onto another surface.
	www.cs.tau.ac.il /~sorkine/ProjectPages/Editing/lse.html (430 words)

	Spatial Filters - Laplacian/Laplacian of Gaussian
		The Laplacian is a 2-D isotropic measure of the 2nd spatial derivative of an image.
		The Laplacian of an image highlights regions of rapid intensity change and is therefore often used for
		The Laplacian is often applied to an image that has first been smoothed with something approximating a Gaussian smoothing filter in order to reduce its sensitivity to noise, and hence the two variants will be described together here.
	homepages.inf.ed.ac.uk /rbf/HIPR2/log.htm (1331 words)

	IngentaConnect A Combinatorial Laplacian with Vertex Weights (Site not responding. Last check: 2007-10-27)
		One of the classical results in graph theory is the matrix-tree theorem which asserts that the determinant of a cofactor of the combinatorial Laplacian is equal to the number of spanning trees in a graph (see [1, 17, 11, 15]).
		Namely, a Laplacian ℒ for G is a matrix with rows and columns indexed by the vertex set V of G, and the (u, v)-entry of ℒ, for u, v in G, u≠ v, is associated with the edge-weight of the edge (u, v).
		In this note, we consider a vertex weighted Laplacian which is motivated by a problem arising in the study of algebro-geometric aspects of the Bethe Ansatz [18].
	www.ingentaconnect.com /content/ap/ta/1996/00000075/00000002/art00080 (309 words)

	CVM 1.1 (VW): Eigenfunctions of the Laplacian
		Applying the Laplacian operator to a function usually results in a new function totally different from the original.
		This is quite similar to what happens when we look for eigenfunctions of the Laplacian that are periodic with respect to a lattice.
		(y) is an eigenfunction of the planar Laplacian with eigenvalue
	www.geom.uiuc.edu /~dpvc/CVM/1998/01/vw/article/constructing/efl.html (258 words)

	Scientific Page Introduction to Laplacian Montages Article 1 (Site not responding. Last check: 2007-10-27)
		The Laplacian operator can be thought of as the second spatial derivative of the potential field at a specific electrode with respect to each surrounding electrode.
		The Laplacian montage attempts to circumvent this problem by looking at only a subset of the electrodes in constructing the reference, those closest to the electrode of interest.
		Overall, the Laplacian montage is most useful when the actual source topography is relatively focal and of high amplitude.
	www.wset.org /sci04/jan/scijan1.htm (1437 words)

	Laplacian Effect
		The Laplacian filter is used for detection of edges in an image.
		The Laplacian filter is a standard Laplacian of Gaussian convolution.
		Because the Laplacian of Gaussian produces a fairly wide convolution for a small radius this filter can become quite computationally expensive as radius is increased.
	www.websupergoo.com /helpie/source/2-effects/laplacian.htm (144 words)

	Atlas Aware Laplacian Smoothing (Site not responding. Last check: 2007-10-27)
		Laplacian smoothing is the building block for defining linear filters, which can be described in terms of polynomials of the Laplacian operator.
		We focus on extending basic Laplacian smoothing to continuous vector valued functions irregularly sampled on multi-chart parameterized surfaces.
		We present an algorithm for Laplacian smoothing over a texture atlas which takes into account the discontinuities imposed by the charts.
	www.cs.brown.edu /people/pgs/atlasawarelaplacian (174 words)

	Variational Subdivision for Laplacian Splines
		The result will be a local stationary subdivision scheme for bounded rectangular grids which produces a limit surface that is close to the minimizer of the original variational problem.
		Laplacian Splines are defined as the minimizer to the variational problem derived from Laplace's equation
		The energy matrix for Laplacian splines is derived from Laplace's equation.
	www.cs.rice.edu /~jwarren/Laplace-tr (1204 words)

	[No title] (Site not responding. Last check: 2007-10-27)
		Subject: Re: Solving Laplacian on Sphere Date: Sun, 20 Feb 2000 11:11:04 -0000 Newsgroups: sci.math.research,sci.math Stephen I think I'm right in saying that the function K you seek is the Green's function for your problem.
		As has been said above, the Laplacian is the simplest differential operator that is invariant under "rigid motions" (translation, rotation, etc.) and so shows up in all forms of physics.
		As for the second part of your query, I think you are confusing "the Laplacian" with the "Laplace Transform"- a completely different thing.
	www.math.niu.edu /~rusin/known-math/00_incoming/laplacian (594 words)

	Source Coding Tools (Site not responding. Last check: 2007-10-27)
		--/ / declare the Laplacian parameters / it_function_args(laplacian_pdf) laplacian_args; / set the Laplacian parameter lambda to 1/sqrt(2) (variance equal to 1) / laplacian_args.lambda = 1/sqrt(2.0); / generate laplacian source of 20 samples, assuming the distribution is / / null outside the range [-10,10] */ double s = it_randpdf(-10, 10, laplacian_pdf, &laplacian_args);
		The pdf must be centered on its maximum value and bounds must be provided where the pdf may be assumed to be zero.
		/* declare the Laplacian parameters / it_function_args(laplacian_pdf) laplacian_args; / set the Laplacian parameter lambda to 1/sqrt(2) (variance equal to 1) / laplacian_args.lambda = 1/sqrt(2.0); / generate laplacian source of 20 samples, assuming the distribution is / / null outside the range [-10,10] */ source = source_pdf(20, -10, 10, laplacian_pdf, &laplacian_args);
	libit.sourceforge.net /doc/source.html (780 words)

	laplacian-guide.html
		Description: Generate the discrete laplacian matrix for a graph G. This is a symmetric matrix of dimension (# of vertices).
		Warning: laplacian was perhaps a bad name for this function since the `linalg' package already has a function named laplacian.
		We define this as (1/2)Qv + delta_i, where Q is the laplacian matrix, v is the effective resistance vector based around the vertex i and delta_i is the zero vector except for a 1 in the i'th position.
	www.math.uga.edu /~mbaker/REU/maple/laplacian-guide1.html (1618 words)

	Laplacian vector field - Wikipedia, the free encyclopedia
		In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible.
		Therefore, the potential of a Laplacian field satisfies Laplace's equation.
		This page was last modified 17:09, 20 September 2005.
	en.wikipedia.org /wiki/Laplacian_vector_field (117 words)

	CiteULike: Tag laplacian (Site not responding. Last check: 2007-10-27)
		Graph Laplacians and their convergence on random neighborhood graphs
		Automated 3-D extraction and evaluation of the inner and outer cortical surfaces using a Laplacian map and partial volume effect classification.
		Laplacian operators and Radon transforms on Grassmann graphs
	www.citeulike.org /tag/laplacian (506 words)

	Computer Vision - Homework #2 by Benjamin Berger
		The brightness of the Laplacian pyramid images shown is increased for better visibility.
		Negative values in the Laplacian pictures would not be shown, so there is 50% gray added (i.e.
		The second picture is achieved by blending (averaging) each level of the Laplacian pyramid using one transistion line, whereas the third picture uses three transistion lines (with 1/4 and 3/4 weight) according to the method described in the paper (p.226 bottom, "apple and orange method").
	www.eecis.udel.edu /~qili/ta/cis489/2 (603 words)

	The Laplacian Spectrum of a Graph II
		The Laplacian Spectrum of a Graph II SIAM Journal on Discrete Mathematics
		The Laplacian Spectrum of a Graph II Robert Grone and Russell Merris
		The first section of this paper is devoted to properties of Laplacian integral graphs, those for which the Laplacian spectrum consists entirely of integers.
	epubs.siam.org /sam-bin/dbq/article/22265 (141 words)

	Laplacian of Gaussian Filter
		Laplacian filters are derivative filters used to find areas of rapid change (edges) in images.
		Since derivative filters are very sensitive to noise, it is common to smooth the image (e.g., using a Gaussian filter) before applying the Laplacian.
		This is called a negative Laplacian because the central peak is negative.
	academic.mu.edu /phys/matthysd/web226/Lab02.htm (407 words)

	Graph Embeddings and Laplacian Eigenvalues
		For an n × n Laplacian, these embedding methods can be characterized as follows: The lower bound is based on a clique embedding into the underlying graph of the Laplacian.
		The embedding that produces these correspondences has a simple description in electrical terms if the underlying graph of the Laplacian is viewed as a resistive circuit.
		Laplacian matrices, graph eigenvalues and eigenvectors, graph embeddings
	epubs.siam.org /sam-bin/dbq/article/32982 (325 words)

	Laplacian (Site not responding. Last check: 2007-10-27)
		The Laplacian of a smooth function on this mesh is also smooth.
		Although the Laplacian has much better accuracy using this method, there is some loss of numerical stability, because the effective stencil of the Laplacian is now larger.
		In this approach, the Laplacian does not appear at all on the right hand side of the equations, only on the left hand sides of the two Poisson equations for the potentials.
	www.math.nyu.edu /research/strauss/fem7pl/node6.html (543 words)

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