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Highlights from
2D Polynomial Data Modelling: Version 1.0

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2D Polynomial Data Modelling: Version 1.0

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This toolbox used discrete orthogonal polynomials to model 2D data, e.g. surfaces and images

Demonstration of the 2D Discrete Gram Transformation

Demonstration of the 2D Discrete Gram Transformation

(C) 2013 Matthew harker and Paul O'Leary Institute for Automation University of Leoben A-8700 Leoben Austria

email: office@harkeroleary.org

This file produces a demonstration of the 2D discrete gram polynomial transformation of an image and the corresponding inverse transformation. It used the function dgt to compute the discrete Gram transformation and the function idgt to compute the inverse transformation.

It is also demonstarted that the error in the reconstrcuted image ins negligible, i.e., approaching the numerical accuracy available in MATLAB. This demonstrates that the polynomial basis functions are free from error evant at very high degrees in this example d = 1024.

This tool requires the discrete orthogonal toolbox: dopBox:

http://www.mathworks.com/matlabcentral/fileexchange/41250

Contents

close all;
clear all;
%
% Set some defaults
%
FontSize = 12;
set(0,'DefaultaxesFontName','Times');
set(0,'DefaultaxesFontSize',FontSize);
set(0,'DefaulttextFontName','Times');
set(0,'DefaulttextFontSize',FontSize);
set(0,'DefaultTextInterpreter', 'latex');
%

Load an image and convert to a double gray data set

This is an image of an art object. The snails are modelled in modelling material, they are not real animals.

[pic, map] = imread('snails1.jpg');
%
% Make the data mean free so that the spectrum is better viwible
%
D = double( rgb2gray( pic ));
D = D - mean(D(:));

View the Test Image

View the image

fig1 = figure;
imagesc( pic );
axis image;
%
title('Photo Finish ($$\copyright$$ 2012 Paul O''Leary and Wolfgang Trettnak)');
xlabel('These are artificial snails and not real animals');
%

Compute the Gram Spectrum

Call the dgt2 function to compute the two dimensional Gram polynomial transformation. This is akin to the fft2 for the Fourier basis.

[S, Bx, By] = dgt2( D );
%
[ny,nx] = size( S );
%
% Compute the degrees in x and y directions.
%
xScale = 0:(nx-1);
yScale = 0:(ny-1);

View the 2D Polynomial Spectrum

This is the complete 2D spectrum for the complete image.

fig2 = figure;
imagesc( xScale, yScale, S );
axis image;
colorbar
title('2D Gram Spectrum (information is at lower degrees)');
xlabel('Polynomial degree in $$x$$');
ylabel('Polynomial degree in $$y$$');
%

Zoomed Section of Spectrum

Virtually all of the Information of the test image is concentrated at lower degrees. This can be shown by zooming in on the relevant portion of the spectrum.

fig3 = figure;
imagesc( xScale, yScale, S);
axis image;
colorbar
title('2D Gram Spectrum');
xlabel('Polynomial degree in $$x$$');
ylabel('Polynomial degree in $$y$$');
axis([0,20,0,20]);
%
%

Reconstruct the Image and Test for Errors

The aim here is to reconstruct the image from its spectrum and then to compute a norm for the total error. This gives a measure for the quality of the transform inverse-transform pair.

Note: the basis functions computed for the transform are passed to the inverse transform. this avoids the necessity to recompute the polynomial basis functions.

As can be seen in the final figure, the Frobeneus norm of the error is in the order of $10^{-15}$, consequently, the polynomial transformation can be regarded as free from error.

Dr = idgt2( S, Bx, By );
%
% Calculate the difference between the original gray scale image and its
% reconstruction.
%
E = D - Dr;
%
% Compute the ratio of the Frobenius of the error to the norm of the data.
%
normE = norm( E, 'fro');
normD = norm( D, 'fro');
normEtoD = normE / normD;
%

View the Reconstructed Image.

fig3 = figure;
imagesc( Dr );
axis image;
colormap gray;
axis off;
title(['Reconstructed Gray Image ($$ \epsilon = ',num2str(normEtoD),' $$)']);
xlabel('These are artificial snails and not real animals');

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