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File:Jpeg2000 2-level wavelet transform-lichtenstein.png

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Summary

DescriptionJpeg2000 2-level wavelet transform-lichtenstein.png

The picture shows a 2 level wavelet transform. For the low-pass and high-pass filters, I have used the same of the JPEG2000, lossy transform, so that this is the wavelet that is used in the JPEG2000 format. In general, it is quite hard to show a wavelet transform properly, because of the different ranges of the different parts. I have made the following choices.

First of all, I wanted to use a picture with both flat surfaces, sharp edges and details. The standard test image Lenna is commonly used for this purpose, but I didn't want to use it because it is not in the public domain. I took a look at the big featured picture gallery con Commons and I have chosen Image:Lichtenstein img processing test.png because it has lots of details in the bottom left, while the top right has almost no details because of the sky. Moreover, the edges of the castle are really sharp, so they can be clearly seen in the wavelet transform.

I have saved the original image in a file called lichtenstein.png, to make it simpler.

Then I have used the following Matlab script:

clear all

%input must be 512x512
img=imread('lichtenstein.png');

img=im2double(img);
img=rgb2gray(img);

% filters in JPEG2000
Lo_D=[0.0267 -0.0168 -0.0782 0.2668 0.6029 0.2668 -0.0782 -0.0168 0.0267];
Hi_D=[0.0912 -0.0575 -0.5912 1.1150 -0.5912 -0.0575 0.0912];

%calculate the 2-level Wavelet transform
[scaled, vertical, horizontal, diagonal]=dwt2(img,Lo_D,Hi_D);
[scaled2, vertical2, horizontal2, diagonal2]=dwt2(scaled,Lo_D,Hi_D);

%crop pictures (for displaying purpose)
scaled2=imcrop(scaled2,[4 4 127 127]);
horizontal2=imcrop(horizontal2,[4 4 127 127]);
vertical2=imcrop(vertical2,[4 4 127 127]);
diagonal2=imcrop(diagonal2,[4 4 127 127]);
horizontal=imcrop(horizontal,[4 4 255 255]);
vertical=imcrop(vertical,[4 4 255 255]);
diagonal=imcrop(diagonal,[4 4 255 255]);

%increase contrast (for displaying purpose)
vertical=imadjust(abs(vertical));
horizontal=imadjust(abs(horizontal));
diagonal=imadjust(abs(diagonal));
vertical2=imadjust(abs(vertical2));
horizontal2=imadjust(abs(horizontal2));
diagonal2=imadjust(abs(diagonal2));

%put them all together
transform=zeros(512,512);

transform(1:128,1:128)=scaled2;
transform(1:128,129:256)=horizontal2;
transform(129:256,1:128)=vertical2;
transform(129:256,129:256)=diagonal2;
transform(1:256,257:512)=horizontal;
transform(257:512,1:256)=vertical;
transform(257:512,257:512)=diagonal;

imwrite(transform,'Jpeg2000_2-level_wavelet_transform-lichtenstein.png');

If I had placed all the transforms one next to the other, I would have got a picture wider than 512x512, with some odd behaviour close to the edges. In order to avoid those problems, I have cropped the transforms so that I can easily put one next to the other to get a 512x512 picture. In general, it is not a good idea to discard information, but here I just wanted to create a simple and clear example.

In order to show the derivatives, I have taken the absolute values (I want to show the magnitude of the derivative), then I have improved the contrast to make them look clearer.
Date
Source Own work
Author Alessio Damato
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Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
You are free:
  • to share - to copy, distribute and transmit the work
  • to remix - to adapt the work
Under the following conditions:
  • attribution - You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike - If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
This licensing tag was added to this file as part of the GFDL licensing update.http://creativecommons.org/licenses/by-sa/3.0/CC BY-SA 3.0Creative Commons Attribution-Share Alike 3.0truetrue

This file is licensed under the Creative Commons Attribution-Share Alike 2.5 Generic, 2.0 Generic and 1.0 Generic license.
You are free:
  • to share - to copy, distribute and transmit the work
  • to remix - to adapt the work
Under the following conditions:
  • attribution - You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike - If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
https://creativecommons.org/licenses/by-sa/2.5CC BY-SA 2.5 Creative Commons Attribution-Share Alike 2.5 truetrue
You may select the license of your choice.

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17 May 2007

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Date/TimeThumbnailDimensionsUserComment
current08:27, 6 July 2019512 x 512 (189 KB)Alexis JazzReverted to version as of 12:28, 17 May 2007 (UTC) Can't be sure if metadata, color profiles or browser compatiblity was lost. These overwrites are not needed, MediaWiki handles this.
23:28, 5 July 2019512 x 512 (183 KB)C 1File size reduced by 7KB using OptiPNG, a lossless optimizer/recompressor for PNG images. No changes made to image or metadata.
12:28, 17 May 2007512 x 512 (189 KB)Alejo2083{{Information |Description= The picture shows a 2 level wavelet transform. For the low-pass and high-pass filters, I have used the same of the JPEG2000, lossy transform, so that this is the wavelet that is used in the JPEG2000 format. In general, it is qu

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File change date and time09:39, 17 May 2007