Random fields and image modeling

Matlab program files for random fields and images

General

instructions contains very simple Matlab instructions
allfiles.tar contains all Matlab and image files currently in this directory. You should read the directions for downloading. It is a good idea to download this each time you use these files, since some of the programs here use other programs not listed here.
In the recent versions of Matlab, you can rotate 3d graphs (and 2d graphs) by entering the command "rotate3d on" and then clicking and dragging on the graph. You can see how the axes will change, and when you let up on the mouse button, the graph moves to the new orientation. In Matlab 6, pull down the Tools menu on the graph window and select Move Camera. Then, when you click and drag on a graph, you see the graph itself rotate.

Loading images and making simple modifications

Note to BGSU users: the version of Matlab available in the Scientific Computing Lab does not have the image processing toolbox installed, and so cannot be used to run the image processing example program below.

Here are some basic directions for scanning things in Room 401 and the Scientific Computing Lab, plus information on image formats like GIF, JPG, PNG, BMP, and more.
Here is a brief description of the types of images in Matlab
Matlab has built-in demonstrations; run ipexindex
trigimage.m displays an intensity plot of a trig field as an image, as a way of understanding the differences between matrices and images. Then it changes the colormaps.
simpleRGBimage.m manually specifies intensities in an image
simpleRGBimage2.m manually specifies intensities as uint8
party.m reads a file, displays it and its red, green, and blue components
trigfield2A.m turns an evolving trig field into a movie
clown_treesBW.m loads, converts to grayscale (intensity image) and displays two Matlab images. It shows how to do simple modifications such as brightening, reverse video, superimposition, flipping, and blurring. It also makes a graph of the gray-scale intensity as a surface and contour graph. Page 1.
thresh.m converts an image to black and white using threshholds
edges.m illustrates Matlab edge detection methods

Image modeling

isingminimum.m tries to minimize the Ising energy pixel by pixel
isingminmac.m tries to minimize the Ising energy pixel by pixel; for the Macintosh.
grapple1.m loads and displays grapple.gif, then performs median filtering. Output
grapple2.m applies median filters until no further change occurs
grapple3.m applies median filters with succesively larger windows
party_gray.m reads a grayscale version of party_72.png and displays it. It is appropriate for Matlab on the Macintosh's at BGSU.

Generating realizations of random fields

iidfield1.m generates and graphs a matrix of iid variables. Output
iidfield2.m generates, smooths, and graphs a matrix of iid variables. Out1 Out2 Out3 Out4 Out5
trigfield1.m generates wave numbers and amplitudes and graphs a random field. Output
trigfield2.m generates wave numbers and amplitudes which evolve over time
trigfield3.m generates wave numbers with a preferred direction. Output
trigfield4.m makes it easy to zoom in on part of a field. To use it, first set keep = 0, run trigfield4, then set keep = 1 and gridrange=[xmin xmax ymin ymax]. Then run trigfield4 again.
trigfield5.m investigates the distribution of F(0). Output
trigfield6.m investigates the joint distribution of F(0,0) and F(x,y). Output
trigfield7.m allows you to explicitly choose wavenumbers. Output
trigfield8.m generates field values at random locations. Output
trigfield9.m computes the covariance associated with a given way of generating random wave numbers. Output
trigfield10.m computes the covariogram for a given way of generating the norms of isotropic two-dimensional wave numbers. Output
trigfield11.m displays a trig field using several different scales
covariogram1.m uses data to produce a sample covariogram.
covariogram2.m generates field values, computes a sample covariogram, and compares to the actual covariogram
ou1.m simulates Ornstein-Uhlenbeck by inverting the covariance matrix
kriging1.m generates field values at various locations, then predicts values at other locations using the known covariance
kriging2.m generates field values at various locations, predicts values at other locations using the known covariance, and compares to Matlab's interpolation routine griddata
cells.m generates Poisson patches of different colors
bumps.m generates a field with bumps centered at Poisson points

A gray-scale image is just a special graph of a function of two variables that takes values between 0 and 1. The graph is white where the function equals 1, it is black where the function is 0, and is various shades of gray for intermediate values. It can be manipulated like any other function, and you can add or average two different images, as you would add or average two functions. In practice, the image data is stored in a matrix whose entries are numbers between 0 and 1. For the examples loaded in the file images.m below, the matrices are 200 by 320. Color images are not much different. They simply have 3 functions (or matrices), one showing the red intensity, one for blue, and one for green.