Activity 7: Enhancement in the Frequency Domain

Introduction

Fourier transform of an image gives the spatial frequencies present in it. Because of this property of the FT it has been used to remove unwanted patterns. It will be demonstrated later how this fact will help improve the quality of an image.

A. Convolution Theorem
Convolution simply means "a little bit of the other". The FT of a convolution of two functions in space is the product of the two functions' FT

FT [ f ∗g ]=FG

Below are images which display this fact.

Figure 1. Top row: Two symmetric dots and its FT. Middle row: Two symmetric circles and its FT. Bottom row: Two symmetric squares and its FT.

The two dots represent two dirac-delta functions which has a sinusoid FT. The location of the dots represents the spatial frequency. The image with two circles is the convolution of two dirac-delta and a circular aperture with a zero-order Bessel function as its FT. The FT of the two circles is just the product of the FT of the two dirac-delta(i.e a sinusoid) and the FT of the circular apperture (i.e. Bessel function). The same goes for the square only that its FT is a sinc function in two dimensions.

FT is also an anamorphic transform. If the spacing is small in the real space domain it will be bigger in the inverse space. Below are images of two Gaussian dots of increasing radius (i.e. increasing variance) and their respective FTs.

Figure 1. Two symmetric Gaussian dots and their respective FTs. Top row: Variance = 4. Middle row: Variance = 10. Bottom row: Variance = 25.

The Gaussian dots are generated by the following code

[x,y]=meshgrid(linspace(-10,10,128),linspace(-10,10,128));

mu=5;
var=20;

I=exp(-(((x+mu).^2+y.^2).^2)./var^2)+exp(-(((x-mu).^2+y.^2).^2)./var^2);

We also compare the real and imaginary part of the Gaussian dots with its inverted counterpart.

Figure 3. Comparison between the real part of two Gaussian dots ( top left) and its inverted counterpart ( top right). The bottom row corresponds to the imaginary parts.

From figure 3. We can see that inversion only affects the real part of the FT.

B. Fingerprint: Ridge Enhancement

Since FT gives the spatial frequencies present in the image, it then also possible to remove unwanted frequencies through filtering. Basically, a filter is a mask that will block out unwanted frequencies. For this part, we will use a binary image for our filter.

Figure 4. Top left: The original image. Top right: Its FT in logarithmic scale. Bottom left: The filter. Bottom right: The filter image.

The frequencies corresponding to the fingerprint those that have lighter color (figure 4 top right). Hence we design a filter that highlights that part and zero out the rest of the frequencies. The reconstructed image has continuous ridges compare to the original showing that the method works although the image appears blur since higher frequencies are removed. High frequencies are required for sharp edges.

C. Lunar Landing Scanned Pictures: Line Removal
For this part, we will also use the same method as above only that we will this time remove unwanted features in the picture, that is the vertical and horizontal lines.

Figure 5. Top left: The original image. Top right: Its FT in logarithmic scale. Middle left: Filter to remove vertical lines. Middle right: Another filter to remove both vertical and horizontal lines. Bottom left: Filtered image with vertical lines removed. Bottom right: Filtered image with both vertical and horizontal lines removed.

The filtered image (especially figure 5 bottom right) appears cleaner.

D. Canvas Weave Modeling and Removal

For this part a more complex image will be enhanced. The canvas weave pattern contains horizontal,vertical and diagonal lines

Figure 6. Top left: The original image. Top right: Its FT in logarithmic scale. Middle left: Filter to remove diagonal canvas weave. Middle right: Another filter to remove diagonal, vertical and horizontal canvas weave. Bottom left: Filtered image with diagonal canvas weave removed. Bottom right: Filtered image with diagonal, vertical and horizontal canvas weave removed.

As with the above, filtered images are cleaner although the brushstrokes are not apparent.

We also inverted the filter and take its FT to see if we can recover the canvas weave.

Figure 7. Top row: The inverse of the filter in figure 6 and its FT. Bottom row: The other filter and its FT

The canvas weave are recovered for each filter although they are not uniform. This indicates that the filter is not perfect or asymmetric.

Paint.NET
For this activity we used lots of editing in making the filter. For reasons of quality I always save my images in PNG format, however MS Paint cannot open PNG files which is troublesome since I have to export it again to a compatible format. Then I found Paint.NET while searching how to make PNG compatible with MS Paint. It is an alternative to MS Paint but has more tools. It is almost similar to Adobe Photoshop and GIMP. The good thing is that it only takes less than 4MB of disk space. Visit: http://www.getpaint.net/index.html

Note: This is available only for Windows and requires Microsoft .NET Framework 3.5 SP1 (which can be downloaded for free at the Microsoft website).

Acknowledgements
I would like to thank Luis Buno for helpful discussions especially with recovering the canvas weave. Special thanks also to my late night buddies (i.e. cramming buddies).

For this activity I'll give myself a 10 for completing the activity, though the fingerprint result is not satisfactory but at least I did my best. ^_^

Activity 6: Properties of the 2D Fourier Transform

Introduction
The following activity demonstrate the properties unique to the 2D FT. 2D FT means applying the FT twice (once along the x and another along the y).

A. Fourier Transform of Different Patterns
Below are some patterns and their corresponding FT. (Click the image for a larger view.)

Figure 1. Top row: Some arbitrary patterns. Bottom row: Their corresponding FFT.

B. Anamorphic Property of the Fourier Transform
Basically, the FT gives the frequencies present in the input image. It treats the image as a sum of sinusoids. Below are sinusoids of different frequencies and their corresponding FT.

Figure 2. Top row: Sinusoids with frequencies 1Hz, 3Hz, 5Hz and 9Hz respectively

In figure 2, the white spots represents the frequencies of the sinusoids. The center of the images at the bottom row represents 0 Hz, thus increasing the frequency will result in white spots moving away from the center. The two symmetric white spots occur because the output of FFT is always symmetric with the center.

If we add a constant signal which effectively has a frequency equal to zero, it will result in a white spot at the center. Figure 3 demonstrate this by adding a value equal to 1 to a 1 Hz sinusoid.

Figure 3. FFT of a 1 Hz sinusoid with constant bias.

Adding a non-constant bias, for example another sinusoid, will result into another white spot representing the frequency of the added noise. (See figure 4.)

Figure 4. Left: Sum of a 3 Hz and a 0.1 Hz. Right: Its FFT.

Rotating the image will result in a rotated FFT. See images below.

Figure 5. 3 Hz sinusoid rotated at 30, 45 60 and 90 degress.

FFT can also be used in noise reduction in a signal. Usually, noise are the higher frequencies in signal. By taking the FFT of a signal and removing the higher frequency, noise can be removed.

Combining two sinusoids in X and in Y will result in FTTs with white spots having coordinates given by the frequencies in X and in Y. Below are products of two sinusoids with equation sin(2*pi*fx*X)sin(2*pi*fy*Y)

Figure 6. Top row (left): Product of sinusoids with fx=4 Hz and fy=4 Hz. Top row (right): Its FFT. Bottom row (left): Product of sinusoids with fx=4 Hz and fy=2 Hz. Bottom row (right): Its FFT.

For the final part of the activity, I sum up the above sinusoids, in particular sin(2*pi*4*X)*sin(2*pi*4*Y) and sin(2*pi*3*X) rotated at 30, 45, and 90 degrees and get the FFT. As expected, the FFT will just be the sum their respective FFTs. It will be helpful to refer back to the images above and compare it with the FFT image below.

Figure 7. Left: Sum of sinusoids. Right: Its FFT.

For this activity I'll give myself a 10. ^_^