Arnolds cat map like scrambling

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george korris
george korris 2021-10-30
编辑： yanqi liu 2021-11-9
Hello to everybody!
Does anyone know how can you use any dynamical system to scramble an image just like you would do with arnolds cat map?
For example how would you do it for the standard map:
  2 个评论
Image Analyst
Image Analyst 2021-10-31
I've posted my Arnolds cat map here before, which you've probably found. For your formula, I'm not sure what p, K, and theta represent. Do you have an image with 2-dimensions? Your formulas here seem to refer to 1-D signals.
george korris
george korris 2021-10-31
Hello and thanks for the response!
The standard map is an area-preserving chaotic map from a square with side 2*pi onto itself. It is constructed by a Poincaré's surface of section of the kicked rotator, and is defined by the formulas i posted in the original question. where θ and p are taken modulo 2*pi . Some basic pictures of the phase diagram can be found
https://en.m.wikipedia.org/wiki/Standard_map
I hope i answered your question. I have seen your beautiful code but when i tried to alter the lines that changed the equation i got errors.

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Image Analyst
Image Analyst 2021-11-6
编辑：Image Analyst 2021-11-6
George, I've tried to use your sin() formula to apply it to an image and this is what I get. I have no idea what K should be though. For every pixel in the input image, I compute p as the radius from the center of the image, and theta as the angle. Then I put them into your formula to get the next p and theta, and I send the input pixel to that location in the output image.
% Demo by Image Analyst.
clc; % Clear the command window.
close all; % Close all figures (except those of imtool.)
clearvars;
workspace; % Make sure the workspace panel is showing.
format long g;
format compact;
fontSize = 16;
numberOfIterations = 1200;
K = 1.3;
p = ones(numberOfIterations, 1);
theta = zeros(numberOfIterations, 1);
plotColors = jet(numberOfIterations);
originalImage = imread('peppers.png');
outputImage = zeros(size(originalImage), 'uint8');
[rows, columns, numberOfColorchannels] = size(originalImage)
for col = 1 : columns
fprintf('Column %d of %d.\n', col, columns)
for row = 1 : rows
% Find initial p and theta for this pixel.
p1 = sqrt((row-rows/2).^2 + (col-columns/2).^2);
theta1 = atan((row-rows/2)/(col-columns/2));
% Find next p and theta for this pixel.
% Where are we going to send this pixel to?
% Compute new p according to the formula.
p2 = p1 + K * sin(theta1);
% Compute new theta according to the formula.
theta2 = theta1 + p1;
% Make them mod 2 * pi
p2 = mod(p2, 2*pi);
theta2 = mod(theta2, 2*pi);
% Convert from polar to cartesian coordinates.
row2 = round(p2 * sin(theta2) + rows/2);
col2 = round(p2 * cos(theta2) + columns/2);
if isnan(row2) || isnan(col2)
continue;
end
% Skip it if it's outside the image.
if row2 <= 1 || row2 > rows || col2 <= 1 || col2 > columns
continue
end
outputImage(row2, col2, :) = originalImage(row, col, :);
end
end
% Display output image
imshow(outputImage)
title('Complete Chaos!', 'FontSize', fontSize);
axis('on', 'image');
impixelinfo
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Image Analyst
Image Analyst 2021-11-8
Yes. I just used the formula. Not sure what you're using for K. It looks like you'll need to adapt the code if you want to have the output image be larger. Probably need a scaling factor in there somewhere, but I didn't have time and I figured you could research that since it's your problem.

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更多回答（4 个）

Image Analyst
Image Analyst 2021-10-31
@george korris I don't really know since chaos is not my field. I know that some of the mini-hack image entries are using chaos so you might check out the code there:
https://www.mathworks.com/matlabcentral/communitycontests/contests/4/entries?sort=views_count+desc
yanqi liu
yanqi liu 2021-11-1
编辑：yanqi liu 2021-11-9
sir,may be use some logistics or chaos to generate some location change
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Image Analyst
Image Analyst 2021-11-1
OK, but do you have anything that does the chaos thing with the sin() function like he asked in the original post?
george korris
george korris 2021-11-3
@yanqi liu thank you for your answer firstly when i run your code i get this error
Unrecognized function or variable 'num'.
Error in Untitled (line 9)
for i=1 : num+1e3
and secondly what does your code do?
i cant read the comments

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Image Analyst
Image Analyst 2021-11-2
George, try this:
% Demo by Image Analyst.
clc; % Clear the command window.
close all; % Close all figures (except those of imtool.)
clearvars;
workspace; % Make sure the workspace panel is showing.
format long g;
format compact;
fontSize = 16;
numberOfIterations = 1200;
K = 1.3;
p = ones(numberOfIterations, 1);
theta = zeros(numberOfIterations, 1);
plotColors = jet(numberOfIterations);
for n = 1 : numberOfIterations-1
% Compute new p according to the formula.
p(n+1) = p(n) + K * sin(theta(n));
% Compute new theta according to the formula.
theta(n+1) = theta(n) + p(n+1);
% Make them mod 2 * pi
p(n+1) = mod(p(n+1), 2*pi);
theta(n+1) = mod(theta(n+1), 2*pi);
% Plot it
plot([p(n), p(n+1)], [theta(n), theta(n+1)], '-', 'Color',plotColors(n, :));
hold on;
end
% Plot it.
% plot(theta, p, 'b-');
grid on;
title('Complete Chaos!', 'FontSize', fontSize);
ylabel('p', 'FontSize', fontSize);
xlabel('theta', 'FontSize', fontSize);
xticks(0 : pi/4 : 2*pi)
xlim([0, 2*pi])
ylim([0, 2*pi])
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Image Analyst
Image Analyst 2021-11-3
@george korris do you mean you want to burn those lines into a digital image instead of being lines on a graph? If so, I'd need more info, like what resolution of image do you want. Do you want the lines to be colored? If so, how? And how many lines should be drawn etc. Just imagine I asked you to do it. What questions would you have for me? Well I have the same questions for you.
george korris
george korris 2021-11-3
@Image Analyst No i mean lets say i have an image of a cat (in parallel with arnolds cat map) there i use the arnolds cat map transformation in order to take the pixels of my image (the cat) and scramble them with the iteration. Can i somehow do the same thong here ?

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yanqi liu
yanqi liu 2021-11-4
编辑：yanqi liu 2021-11-9
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yanqi liu
yanqi liu 2021-11-6
i use it in watermark、encrypt、compress，so it must can transfer and inverse transfer

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