congruency irrespective of the angle at which congruency occurs atand, up to a point, irrespective of the rate at which the amplitudespectrum decays with frequency.  A gradient based edge detectorproduces a double response for all features that have congruence ofphase at angles other than zero (towards the bottom of the testimage).  The phase congruency detector marks features with a singleresponse.  The colour coded image was generated by <ahref=PhaseCongruency/dispfeat.m>dispfeat.m</a></ul><a name="noisecomp"></a><hr><h4>Image denoising</h4><center><img align=top src=WWWImages/srvb0019gamma-tn.jpg> &nbsp;<img align=top src=WWWImages/gclean0019a-tn.jpg> </center><br><ul><li type=square><a href=PhaseCongruency/noisecomp.m>noisecomp.m</a>Code for denoising images.  This code differs from standard waveletdenoising techniques in that it uses non-orthogonal wavelets, andunlike existing techniques, ensures that phase information ispreserved in the image. Phase information is of crucial importance tohuman visual perception.  Also, this code <i>does</i> have aneffective way of determining threshold levels automatically.<p>See the example below, under grey scale transformation andenhancement, for an example of the use of this function.</ul><p><b>Reference:</b><ul><li> Peter Kovesi,      <a href="http://www.csse.uwa.edu.au/~pk/research/pkpapers/denoise.pdf">     "Phase Preserving Denoising of Images". </a>     <i> The Australian Pattern Recognition Society Conference:     DICTA'99. </i> December 1999. Perth WA. pp 212-217.</ul><a name="shapelet"></a><hr><h4>Surface normals to Surfaces</h4><center><table><tr><td><img align=top src=WWWImages/rampneedlesm.png><br><center> Surface Normals </center></td><td>&nbsp; &nbsp; &nbsp;</td><td> <img align=top src=WWWImages/rampreconsm.png><br><center>Surface Reconstruction</center></td></tr></table></center><ul><li type=square><a href=Shapelet/shapeletsurf.m>shapeletsurf.m</a>Function reconstructs an estimate of a surface from its surfacenormals by correlating the surface normals with that those of a bankof shapelet basis functions. The correlation results are summed toproduce the reconstruction.  The sumation of shapelet basis functionsresults in an implicit integration of the surface while enforcingsurface continuity.<p>Note that the reconstruction is only valid up to a scale factor(which can be corrected for). However the reconstruction process isvery robust to noise and to missing data values.  Reconstructions (upto positive/negative shape ambiguity) are possible where there is anambiguity of pi in tilt values.  Low quality reconstructions are alsopossible with just slant, or just tilt data alone.  However, if youhave full gradient information you are better off with the FrankotChellappa algorithm below.<li type=square><ahref=Shapelet/frankotchellappa.m>frankotchellappa.m</a> Animplementation of Frankot and Chellappa's algorithm for constructingan integrable surface from gradient information.  If you have fullgradient information in x and y this is probably the best algorithm touse.  It is very simple, very fast and highly robust to noise.  If youhave surface normal information in the form of slant and tilt, and youhave an ambiguity of pi in your tilt data, or only have slant, thentry using shapeltsurf.m above.<li type=square><a href=Shapelet/grad2slanttilt.m>grad2slanttilt.m</a>Converts gradient values over a surface to slant and tilt angles.<li type=square><a href=Shapelet/slanttilt2grad.m>slanttilt2grad.m</a>Converts slant and tilt angles over a surface to gradients.<li type=square><a href=Shapelet/needleplotgrad.m>needleplotgrad.m</a>Generates a needle plot given surface gradients over a surface.<li type=square><a href=Shapelet/needleplotst.m>needleplotst.m</a> Generates aneedle plot given slant and tilt values over a surface.<li type=square><a href=Shapelet/testp.m>testp.m</a> Generates asynthetic test surface along with its surface normals for testingshapeletsurf.</ul><p><b>Reference:</b><ul><li> Peter Kovesi, <a href="http://www.csse.uwa.edu.au/~pk/research/pkpapers/shapeletsICCV.pdf">"Shapelets Correlated with Surface Normals Produce Surfaces". </a>10th IEEE International Conference on Computer Vision.Beijing. pp 994-1001. 2005<li><a href="http://www.csse.uwa.edu.au/~pk/Research/ShapeFromContour/index.html">An example</a> of how much 3D shape you can get from very minimalsurface normal information.</ul><a name="scalogram"></a><hr><h4>Scalogram calculation</h4> <center><img align=top src=WWWImages/dphase2.jpg></center> <br><ul><li type=circle><a href=FrequencyFilt/scalogram.m>scalogram.m</a>Function to calculate the phase and amplitude scalograms of a 1Dsignal. The analysis is done using quadrature pairs of log Gaborwavelets.</ul><a name="anisodiff"></a><hr> <h4>Anisotropic diffusion</h4><center> <img align=top src=WWWImages/Mona_LisaGs.jpg> &nbsp;<img align=top src=WWWImages/Mona_diffuses.jpg> </center>  <br><ul><li type=square><a href=Spatial/anisodiff.m>anisodiff.m</a> Functionto perform anisotropic diffusion of an image following Perona andMalik's algorithm.  This process smoothes regions while preserving,and enhancing, the contrast at sharp intensity gradients.</ul><a name="greytrans"></a><hr><h4>Grey scale transformation and enhancement</h4><ul><li type=square><a href=GreyTrans/extractfields.m>extractfields.m</a>separates fields from a video frame, and optionally interpolatesintermediate lines.<li type=square><a href=GreyTrans/interpfields.m>interpfields.m</a>interpolates lines on a field extracted from a video frame.<li type=square><a href=GreyTrans/normalise.m>normalise.m</a> rescalesimage values to 0-1.<li type=square><a href=GreyTrans/adjcontrast.m>adjcontrast.m</a>adjusts image contrast using sigmoid function.<li type=square><a href=GreyTrans/adjgamma.m>adjgamma.m</a> adjustsimage gamma.<li type=circle><a href=GreyTrans/greytrans.m>greytrans.m</a> allowsyou to interactively remap intensity values in a colour or greyscaleimage via a mapping function defined by a series of spline points.  Afeeble attempt at replicating <i>xv's</i> intensity mapping tool.  Itis not as fast but it does operate on floating point images allowingyou to better preserve image fidelity. (Needs remapim.m).<li><a href=GreyTrans/remapim.m>remapim.m</a> is a non-interactiveversion of greytrans that allows you to apply an intensity mapping toa colour or greyscale image using a mapping function determinedexperimentally with greytrans.  Useful if you want to apply the samemapping function to a sequence of images. <li type=square><a href=GreyTrans/histtruncate.m>histtruncate.m</a>truncates ends of an image histogram.  Useful for enhancing imageswith outlying values.<!--<li><a href=GreyTrans/histtruncatex.m>histtruncatex.m</a> truncatesends of an image histogram.--><li> <a href=GreyTrans/example/index.html>Example</a> of using some ofthese functions above to enhance a video image.</ul><a name="freqfilt"></a><hr><h4>Frequency domain transformations</h4><ul><li type=square><ahref=FrequencyFilt/lowpassfilter.m>lowpassfilter.m</a> constructslow-pass Butterworth filter.<li type=square><ahref=FrequencyFilt/highpassfilter.m>highpassfilter.m</a> constructshigh-pass Butterworth filter.<li type=square><ahref=FrequencyFilt/highboostfilter.m>highboostfilter.m</a> constructshigh-boost Butterworth filter.<li type=square><ahref=FrequencyFilt/bandpassfilter.m>bandpassfilter.m</a> constructsband-pass Butterworth filter.<li type=square><a href=FrequencyFilt/homomorphic.m>homomorphic.m</a>performs homomorphic filtering on an image.  One of my favourite imageenhancement techniques. (needs histtruncate.m and normalise.m)<!--<li><a href=FrequencyFilt/homomorphicGUI.m>homomorphicGUI.m</a> GUIinterface to homomorphic. (needs homomorphic.m) This code is a bitflakey under version 6.0 R12, I think it is OK under 6.1--><li type=square><a href=FrequencyFilt/psf.m>psf.m</a> generates avariety of point-spread functions.  This function can be useful whenmanually specifying point-spread functions for Wiener filtering orwith deconvolution functions such as the Richardson-Lucy algorithm(see the MATLAB image toolbox function deconvlucy.m).<li type=square><a href=FrequencyFilt/psf2.m>psf2.m</a> is identicalto psf, it just has a different way of specifying the function shapewhich may be more convenient for some applications.<li type=square><a href=FrequencyFilt/imspect.m>imspect.m</a> plotsimage amplitude spectrum averaged over all orientations.<li type=circle><a href=FrequencyFilt/freqcomp.m>freqcomp.m</a>demonstrates image reconstruction from its Fourier components.</ul><a name="projective"></a><hr><h4> Functions supporting projective geometry</h4><center><table><tr><td><img align=top src=WWWImages/trigg.jpg><br><center>image of beach</center></td><td><img align=top src=WWWImages/recttrigg.jpg><br><center>rectified beach</center></td></tr></table></center><ul><li type=square> <a href=Projective/homography1d.m>homography1d.m</a>computes the 2x2 1D homography of 3 or more points along a line.<li type=square> <a href=Projective/homography2d.m>homography2d.m</a>computes the 3x3 2D homography of 4 or more points in a plane.  Thiscode follows the normalised direct linear transformation algorithmgiven by Hartley and Zisserman.<li type=square> <a href=Projective/fundmatrix.m>fundmatrix.m</a>computes the fundamental matrix from 8 or more matching points in astereo pair of images using the normalised 8 point algorithm.<li type=square> <ahref=Projective/affinefundmatrix.m>affinefundmatrix.m</a> computes theaffine fundamental matrix from 4 or more matching points in a stereopair of images.<li type=square> <ahref=Projective/normalise2dpts.m>normalise2dpts.m</a> translates andnormalises a set of 2D homogeneous points so that their centroid is atthe origin and their mean distance from the origin is sqrt(2).  Thisis used to improve the conditioning of any equations used to solvehomographies, fundamental matrices etc.<li type=square> <a href=Projective/hnormalise.m>hnormalise.m</a>normalises an array of homogeneous coordinates so that their scaleparameter is 1. Points at infinity are unchanged.<li> <a href=Projective/imTrans.m>imTrans.m</a> applies a homogeneoustransform to an image.<li type=circle> <a href=Projective/digiplane.m>digiplane.m</a> allowsyou to digitise and transform points within a planar region in animage.<li> <ahref=Projective/equalAngleConstraint.m>equalAngleConstraint.m</a> Affinetransform constraints given two equal angles.<li> <ahref=Projective/knownAngleConstraint.m>knownAngleConstraint.m</a> Affinetransform constraints given a known angle.<li> <ahref=Projective/lengthRatioConstraint.m>lengthRatioConstraint</a> Affinetransform constraints given a length ratio.<li> <a href=Projective/circleintersect.m>circleintersect.m</a> Findsintersection of two circles. Use this function to solve for affinetransformation constraints.<li type=square> <a href=Projective/circle.m>circle.m</a> Draws acircle.<li type=square> <a href=Projective/hcross.m>hcross.m</a> Homogeneouscross product, result normalised to s = 1.<li type=circle> <a href=Projective/hline.m>hline.m</a> Plot 2D linesdefined in homogeneous coordinates.<li type=square> <a href=Projective/homoTrans.m>homoTrans</a> 2D homogeneoustransformation of points/lines.<li type=square> <a href=Projective/plotPoint.m>plotPoint.m</a> Plotspoint with specified mark and optional text label.<p><li> If you are using these functions above you should look at AndrewZisserman's <a href="http://www.robots.ox.ac.uk/~vgg/hzbook/code/">MATLAB Functions for Multiple View Geometry</a>