1. Researchlab 4 Presentation
Fourier vs Wavelets
Researchlab 4 Presentation
Maurice Samulski
June 27th, 2005

2. Contents Introduction Discrete Fourier Transform
Discrete Cosine Transform
Wavelet Transform
Comparison between DCT and WT
Conclusions
Thursday, November 22, 2018
Research Lab 4 presentation 2

3. Fourier analysis Joseph Fourier 1807
Represent functions by superposing sines and cosines with different frequencies and amplitudes
s(t) = 3 sin (t) sin(4t) - 20 sin (200t)
Fourier analysis is named after its inventor Joseph Fourier, who discovered that we can represent functions by superposing sines and cosines with different frequencies and amplitudes.
For example, an arbritary signal can be described as …… where 3, 100 and 20 are the amplitudes and 1, 4, 200 are the periodic times
Thursday, November 22, 2018
Research Lab 4 presentation 3

4. Fourier analysis 4 Thursday, November 22, 2018
Here are some basic signals which can be represented as a sum of sines. The more harmonics the better the signal will look like it’s original.
Thursday, November 22, 2018
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5. Discrete Fourier Transform (DFT)
DFT of image f(x,y) with size m x n
To get a discrete Fourier transform of an image f(x,y) with size m x n, you feed it in this formula.
We can see that F(u,v) is calculated for every pixel, and that it has to do the sum over the whole image, it’s therefore quite a complex computation.
Thursday, November 22, 2018
Research Lab 4 presentation 5

6. Discrete Fourier Transform (DFT)
Inverse DFT of F(u,v)
The basic idea behind this transform is that any signal can be seen as an infinite sum of a series of sinusoidal signals, where each sine can have a different amplitude and phase
n practical terms, it means that you can feed in an arbitrary signal and get out data that represents the individual frequencies and their amplitudes that make up the original data.
Thursday, November 22, 2018
Research Lab 4 presentation 6

7. Discrete Fourier Transform
Image f(x,y) is real
Fourier transform F(u,v) is complex
F(u,v) often represented as
One of the things that should be noted is that the image function f(x,y) is real,
but the fourier transform F(u,v) is complex
x,y are image variables, location of the pixel
But u,v are frequency variables
In the literature F(u,v) is often represented as magnitude and phase rather than its real and imaginary parts.
Basically, the magnitude tells how much of a certain frequency component is present and
the phase tells where the frequency component is in the image.
Thursday, November 22, 2018
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8. Discrete Fourier Transform
To see how a FT looks like, we have to use a simple image. Because Fourier works with sine functions, the image is build up with sines. 8 to be exactly. The picture on the right is only the magnitude picture.
If frequency where higher, then the points would be further away from the middle point
Thursday, November 22, 2018
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9. Discrete Fourier Transform
This is an example of an FT from a normal image (MAGNITUDE!).
Because a real world image normally contains lots of low frequencies, there is a bright blob in the middle
You can do some basic stuff to the transform and then transform it back to the original image. If you make a circle cut out, the image will be blurred. But with this transform
you can also compress the image by letting away high frequencies which the eye can’t see and then transform it back.
Thursday, November 22, 2018
Research Lab 4 presentation 9

10. Discrete Fourier Transform
This is an example of an FT from a normal image (MAGNITUDE!).
Because a real world image normally contains lots of low frequencies, there is a bright blob in the middle
You can do some basic stuff to the transform and then transform it back to the original image. If you make a circle cut out, the image will be blurred. But with this transform
you can also compress the image by letting away high frequencies which the eye can’t see and then transform it back.
Thursday, November 22, 2018
Research Lab 4 presentation 10

11. Discrete Cosine Transform (DCT)
Very similar to the discrete Fourier transform, but
Uses only real numbers
Decomposes a function into a series of even cosine components only
Different ordering of coefficients
Computationally cheaper than DFT and therefore very commonly used in image processing, eg JPEG and MPEG
The basic idea behind this transform is that any signal can be seen as an infinite sum of a series of sinusoidal signals, where each sine can have a different
amplitude and phase
n practical terms, it means that you can feed in an arbitrary signal and get out data that represents the individual frequencies and their amplitudes that make up the original data.
Thursday, November 22, 2018
Research Lab 4 presentation 11

12. (1) Divide image into 8x8 blocks
It first divides an image in 8x8 blocks
Input image
8x8 block
Thursday, November 22, 2018
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13. (2a) 2-D DCT basis functions
Low
High
Low
Low
Here are the 64 basis functions which DCT uses.
Horizontal axis is the horizontal frequency
Vertical axis is the vertical frequency
Each 8 x 8 block can be build up with these basis functions, and can be seen as a weighted sum of these basis functions.
High
High
8x8 block
Low
High
Thursday, November 22, 2018
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14. (2b) 2-D Transform Coding
DC coefficient
(average color) +
y00
y23
y01
y12
y10
The basis function in the upperleft corner, here left, is the DC coefficient of the block..which is the average color. Then the detail is expressed with the other basis functions with their coefficient.
And if you have the information below, you can reconstruct the original block
...
AC coefficients (details)
Thursday, November 22, 2018
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15. (3) Zig-zag ordering DCT blocks
Why? To group low frequency coefficients in top of vector.
Maps 8 x 8 to a 1 x 64 vector.
Then these DCT blocks coefficients are ordered zigzag, to group the low frequency coefficients in top of vector. Low frequencies are more important for the eye than high frequency blocks.
Thursday, November 22, 2018
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16. DCT compression
Because human eye is most sensitive to low frequencies, less sensitive to high frequencies, we can truncate the coefficients which represent these high frequencies
The lower quality setting, the more coefficients are truncated
Lesser coefficients mean less detail of the block which leads to the famous blocking artifact
Thursday, November 22, 2018
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17. Wavelets
The major advantage of using wavelets is that they can be used for analyzing functions at various scales
It stores versions of an image at various resolutions, which is very similar how the human eye works.
As you zoom in at smaller and smaller scales, you can find details that you did not see before.
Looking to a forest from an airplane, it appears to be a solid green plane
If you look to a forest from a car on the ground, this solid plane resolves into individual trees
If you got out of the car, you can see branches and leaves
Thursday, November 22, 2018
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18. Haar wavelet example (1D)
Suppose we have a one-dimensional data set containing eight pixels:
[ ]
We can represent this image in the Haar basis by computing a wavelet transform, by averaging the pixels together pairwise:
[ ]
Clearly, some information has been lost in this averaging process, we need to store detail coefficients:
[ ]
To get a sense for how these wavelets work and give an impression on what is meant by resolution
and scale, we give a simple example that is based on the simplest wavelet basis: the Haar wavelet.
Suppose we have a one-dimensional data set containing eight pixels. For example, they could be the first row of an two-dimensional 8x8 pixel image:
We can represent this image in the Haar basis by computing a wavelet transform. To do this, we first average the pixels together,
pairwise to get a new, lower resolution image with four pixel values.
Thursday, November 22, 2018
Research Lab 4 presentation 18

19. Haar wavelet example (1D)
The full decomposition will look like
We will store this as follows: [ −1 1 −1 −2 1 ]
No information has been gained or lost by this process
To get a sense for how these wavelets work and give an impression on what is meant by resolution
and scale, we give a simple example that is based on the simplest wavelet basis: the Haar wavelet.
Suppose we have a one-dimensional data set containing eight pixels. For example, they could be the first row of an two-dimensional 8x8 pixel image:
We can represent this image in the Haar basis by computing a wavelet transform. To do this, we first average the pixels together,
pairwise to get a new, lower resolution image with four pixel values.
Thursday, November 22, 2018
Research Lab 4 presentation 19

20. Haar wavelet example (1D)
The full decomposition will look like
This transform will be stored as:
[ −1 1 −1 −2 1 ]
No information has been gained or lost by this process
To get a sense for how these wavelets work and give an impression on what is meant by resolution
and scale, we give a simple example that is based on the simplest wavelet basis: the Haar wavelet.
Suppose we have a one-dimensional data set containing eight pixels. For example, they could be the first row of an two-dimensional 8x8 pixel image:
We can represent this image in the Haar basis by computing a wavelet transform. To do this, we first average the pixels together,
pairwise to get a new, lower resolution image with four pixel values.
Thursday, November 22, 2018
Research Lab 4 presentation 20

21. Haar wavelet
This may look wonderful and all, but what good is compression that takes eight values and compresses it to eight values?
Pixel values are similar to their neighbors
The image can be compressed by removing small coefficients from this transform
The one-dimensional Haar Transform can be easily extended to two-dimensional
Input matrix instead of an input vector
apply the one-dimensional Haar transform on each row
apply the one-dimensional Haar transform on each column
You may think that a compression of eight values to other eight values may be useless,
but the difference is that because neighbour pixels are very similar to eachother, the detail coefficients are close to zero and can
be compressed better.
…which only introduces some small errors in the reconstructed image
Thursday, November 22, 2018
Research Lab 4 presentation 21

22. Other wavelets
The Haar wavelet uses simple basis functions (discontinuous) for scaling and determining detail coefficients
Not suitable for smooth functions
You may think that a compression of eight values to other eight values may be useless,
but the difference is that because neighbour pixels are very similar to eachother, the detail coefficients are close to zero and can
be compressed better.
…which only introduces some small errors in the reconstructed image
Thursday, November 22, 2018
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23. JPEG vs JPEG2000
Generally, there are two visible damages caused by image compression:
Blocking artifacts: artificial horizontal and vertical borders between blocks
Blur: loss of fine detail and the smearing of edges
…which only introduces some small errors in the reconstructed image
Thursday, November 22, 2018
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24. Test: Image quality Test results are subjective
With ‘normal’ compression (2+ bits/pixel), quality advantage of JPEG2000 is negligible
Real quality advantage will only become clear by using very high compression ratios (0.5 or less b/p)
At 0.25 b/p, JPEG images begin to look like a mosaic while with JPEG2000 it gets a elegant blur across the image
JPEG2000 image files tend to be 20 to 60% smaller than their JPEG counterparts for the same subjective image quality
The test results are subjective because they are based on our human visual experience,
however I think we can conclude the following:
With normal compression (above 2 bits/pixel) the quality advantage of jpeg2000 is negligible. I can’t see the difference unless I zoom in 1000%.
Real quality advantage will only become clear by using very high compression ratios (0.5 or less b/p)
At a quarter bits per pixel, JPEG images begin to look like a mosaic while with JPEG2000 it gets a elegant blur across the image
Thursday, November 22, 2018
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25. Test: Image quality (Original)
Although the test results are subjective because they are based on our human visual experience
we come to the conclusion that JPEG2000 provides a slightly better image quality than JPEG when
the compressed rates are 1-2 bits per pixel.
Lena Original (512x512x24b) Building Plan (small piece)
Thursday, November 22, 2018
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26. Results: Image quality (Lena)
Although the test results are subjective because they are based on our human visual experience
we come to the conclusion that JPEG2000 provides a slightly better image quality than JPEG when
the compressed rates are 1-2 bits per pixel.
JPEG (0.2 b/p) JPEG2000 (0.2 b/p)
Thursday, November 22, 2018
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27. Results: Image quality (Building plan)
Although the test results are subjective because they are based on our human visual experience
we come to the conclusion that JPEG2000 provides a slightly better image quality than JPEG when
the compressed rates are 1-2 bits per pixel.
JPEG (0.2 b/p) JPEG2000 (0.2 b/p)
Thursday, November 22, 2018
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28. Results: Performance
Price to pay: considerable increase in computational complexity and memory usage
For these quality improvements there’s a price to pay
In the table below we compare both algorithms by recording
the time needed to compress the selected images known for their difficulty.
Thursday, November 22, 2018
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29. Conclusions JPEG2000 works better with sharp spikes in images
Quality advantages are really visible when compressing with very high compression ratios
Only to be used with very large datasets like fingerprints, MRI scans, building plans, etc.
You can choose between different wavelet basis functions to get the optimal result for a specific application
Blur isn’t experienced as bad as blocking artifacts
Time needed to compress high resolution images takes a lot of time with JPEG2000
JPEG 2000 works better with sharp spikes in images, as can be seen in the building plan. Also fingerprints are compressed with more accuracy than JPEG.
Possibly because JPEG2000 needs to process the whole file and JPEG works per block.
Thursday, November 22, 2018
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30. Questions? 30 Thursday, November 22, 2018 Research Lab 4 presentation
JPEG 2000 works better with sharp spikes in images, as can be seen in the building plan. Also fingerprints are compressed with more accuracy than JPEG.
Possibly because JPEG2000 needs to process the whole file and JPEG works per block.
Thursday, November 22, 2018
Research Lab 4 presentation 30