Archive for the 'mathematica' Category

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The algorithms below are ready to be downloaded. Biomedical Imaging Group. EPFL

Available Algorithms

The algorithms below are ready to be downloaded. They are generally written in JAVA or in ANSI-C, either by students or by the members of the Biomedical Imaging Group.Please contact the author of the algorithms if you have a specific question.
JAVA: Plug-ins for ImageJ
JAVA classes are usually meant to be integrated into the public-domain software ImageJ.
bullet Drop Shape Analysis. New method based on B-spline snakes (active contours) for measuring high-accuracy contact angles of sessile drops.
bullet Extended Depth of Focus. The extended depth of focus is a image-processing method to obtain in focus microscopic images of 3D objects and organisms. We freely provide a software as a plugin of ImageJ to produce this in-focus image and the corresponding height map of z-stack images.
bullet Fractional spline wavelet transform. This JAVA package computes the fractional spline wavelet transform of a signal or an image and its inverse.
bullet Image Differentials. This JAVA class for ImageJ implements 6 operations based on the spatial differentiation of an image. It computes the pixel-wise gradient, Laplacian, and Hessian. The class exports public methods for horizontal and vertical gradient and Hessian operations (for those programmers who wish to use them in their own code).
bullet MosaicJ. This JAVA class for ImageJ performs the assembly of a mosaic of overlapping individual images, or tiles. It provides a semi-automated solution where the initial rough positioning of the tiles must be performed by the user, and where the final delicate adjustments are performed by the plugin.
bullet NeuronJ. This Java class for ImageJ was developed to facilitate the tracing and quantification of neurites in two-dimensional (2D) fluorescence microscopy images. The tracing is done interactively based on the specification of end points; the optimal path is determined on the fly from the optimization of a cost function using Dijkstra’s shortest-path algorithm. The procedure also takes advantage of an improved ridge detector implemented by means of a steerable filterbank.
bullet PixFRET. The ImageJ plug-in PixFRET allows to visualize the FRET between two partners in a cell or in a cell population by computing pixel by pixel the images of a sample acquired in three channels.
bullet Point Picker. This JAVA class for ImageJ allows the user to pick some points in an image and to save the list of pixel coordinates as a text file. It is also possible to read back the text file so as to restore the display of the coordinates.
bullet Resize. This ImageJ plugin changes the size of an image to any dimension using either interpolation, or least-squares approximation.
bullet SheppLogan. The purpose of this ImageJ plugin is to generate sampled versions of the Shepp-Logan phantom. Their size can be tuned.
bullet Snakuscule. The purpose of this ImageJ plugin is to detect circular bright blobs in images and to quantify them. It allows one to keep record of their location and size.
bullet SpotTracker Single particle tracking over noisy images sequence. SpotTracker is a robust and fast computational procedure for tracking fluorescent markers in time-lapse microscopy. The algorithm is optimized for finding the time-trajectory of single particles in very noisy image sequences. The optimal trajectory of the particle is extracted by applying a dynamic programming optimization procedure.
bullet StackReg. This JAVA class for ImageJ performs the recursive registration (alignment) of a stack of images, so that each slice acts as template for the next one. This plugin requires that TurboReg is installed.
bullet Steerable feature detectors. This ImageJ plugin implements a series of optimized contour and ridge detectors. The filters are steerable and are based on the optimization of a Canny-like criterion. They have a better orientation selectivity than the classical gradient or Hessian-based detectors.
bullet TurboReg. This JAVA class for ImageJ performs the registration (alignment) of two images. The registration criterion is least-squares. The geometric deformation model can be translational, conformal, affine, and bilinear.
bullet UnwarpJ. This JAVA class for ImageJ performs the elastic registration (alignment) of two images. The registration criterion includes a vector-spline regularization term to constrain the deformation to be physically realistic. The deformation model is made of cubic splines, which ensures smoothness and versatility.
Most often, the ANSI-C pieces of code are not a complete program, but rather an element in a library of routines.
bullet Affine transformation. This ANSI-C routine performs an affine transformation on an image or a volume. It proceeds by resampling a continuous spline model.
bullet Registration. This ANSI-C routine performs the registration (alignment) of two images or two volumes. The criterion is least-squares. The geometric deformation model can be translational, rotational, and affine.
bullet Shifted linear interpolation. This ANSI-C program illustrates how to perform shifted linear interpolation.
bullet Spline interpolation. This ANSI-C program illustrates how to perform spline interpolation, including the computation of the so-called spline coefficients.
bullet Spline pyramids. This software package implements the basic REDUCE and EXPAND operators for the reduction and enlargement of signals and images by factors of two based on polynomial spline representation of the signal.
bullet E-splines. A Mathematica package is made available for the symbolic computation of exponential spline related quantities: B-splines, Gram sequence, Green function, and localization filter.
bullet Fractional spline wavelet transform. A MATLAB package is available for computing the fractional spline wavelet transform of a signal or an image and its inverse.
bullet Fractional spline and fractals. A MATLAB package is available for computing the fractional smoothing spline estimator of a signal and for generating fBms (fractional Brownian motion). This spline estimator provides the minimum mean squares error reconstruction of a fBm (or 1/f-type signal) corrupted by additive noise.
bullet Hex-splines : a novel spline family for hexagonal lattices. A Maple 7.0 worksheet is available for obtaining the analytical formula of any hex-spline (any order, regular, non-regular, derivatives, and so on).
bullet MLTL deconvolution : This Matlab package implements the MultiLevel Thresholded Landweber (MLTL) algorithm, an accelerated version of the TL algorithm that was specifically developped for deconvolution problems with a wavelet-domain regularization.
bullet OWT SURE-LET Denoising : This Matlab package implements the interscale orthonormal wavelet thresholding algorithm based on the SURE-LET (Stein’s Unbiased Risk Estimate/Linear Expansion of Thresholds) principle.
bullet WSPM : Wavelet-based statistical parametric mapping, a toolbox for SPM that incorporates powerful wavelet processing and spatial domain statistical testing for the analysis of fMRI data.

FEM, finite element method, PDE, partial differential equation;mathematica
This package allows to solve second order elliptic differential equations in two variables:

div(a*grad u) – b*u = f in the domain domain u = gD Dirichlet boundary conditions on first part of boundary a*du/dn = gN Neumann condition on the other part of the boundary

If the functions a, b f, gD and gN are given, then a numerical approximation is computed, using the method of finite elements. To generate meshes the programm EasyMesh can be used.

a general finite element environment;mathematica

The Mathematica application package AceFEM is a general finite element environment designed to solve multi-physics and multi-field problems. Using Mathematica, the program provides both extensive symbolic capabilities and the numeric efficiency of a commercial finite element environment.

AceFEM consists of two components. The main module performs tasks such as processing user-input data, mesh generation, control of solution procedures, and graphic post-processing of results. A fully integrated second module handles numerically intensive operations such as evaluation and assembly of finite element quantities, solution of linear systems of equations, contact search procedures, and more.


The upcoming release of Mathematica (2005) establishes the next level of sophistication in scientific function and data visualization. All the plotting functions have been rewritten to incorporate state-of-the-art algorithms in numeric and algebraic analysis, mesh generation, and computational geometry. Arbitrary-precision function evaluation, region, mesh overlays, piecewise functions, and singularities handling are just part of the new functionality. All of this makes Mathematica a unique system for function and data visualization with features not found in any other algebraic computational system. In this presentation we will give a general description and present several examples of the new capabilities.

mesh generation, literate programming, matlab; mathematica; persson

This Mathematica notebook is an effort to transcribe the MATLAB code of a 2-D mesh generation algorithm as described explicitly in Persson and Strang’s paper [1]. The goal is to make the algorithm executable in Mathematica so that its users can also experiment with the algorithm.

Since the algorithm was expressed very clearly from their original paper [1] including the MATLAB code, which is a perfect example of literate programming in MATLAB, it is pretty easy to translate the MATLAB code « literally » into Mathematica. Such translation is virtually always possible in either direction even without human interference. And such a Rosetta Stone kind of translation might be useful if one species of people coding in either MATLAB or Mathematica were to disappear, future generations would still be able to rediscover one programming language by reading its interpretation in the other one.

However, it is so tempting to present the literate programming capability of Mathematica by following its general principles; that is, (a) documentation mingles with code and both get pretty-printed; (b) shuffle code pieces for human readability. I decided to transcribe the code manually.

The original MATLAB code was documented as 8 steps (sections) in sequential order, which is easy to follow because the ideas behind the code were explained beforehand in early parts of the paper. So it is recommended that you read part 1 and 2 of the original paper. Instead of following the MATLAB code literally in 8 steps, this notebook breaks the code pieces apart and examines each of them separately.

finite volume method

The finite volume method is a numerical method for solving partial differential equations that calculates the values of the conserved variables averaged across the volume. One advantage of the finite volume method over finite difference methods is that it does not require a structured mesh (although a structured mesh can also be used). Furthermore, the finite volume method is preferable to other methods as a result of the fact that boundary conditions can be applied noninvasively. This is true because the values of the conserved variables are located within the volume element, and not at nodes or surfaces. Finite volume methods are especially powerful on coarse nonuniform grids and in calculations where the mesh moves to track interfaces or shocks.

Hyman et al. (1992) have derived local, accurate, reliable, and efficient finite volume methods that mimic symmetry, conservation, stability, and the duality relationships between the gradient, curl, and divergence operators on nonuniform rectangular and cuboid grids.

SEE ALSO: Finite Element Method


Hyman, J. M.; Knapp, R.; and Scovel, J. C. « High Order Finite Volume Approximations of Differential Operators on Nonuniform Grids. » Physica D 60, 112-138, 1992.

Rübenkönig, O. « The Finite Volume Method (FVM). »

Versteeg, H. K. and Malalasekera, W. An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Reading, MA: Addison-Wesley, 1995.


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