Introduction to Fourier Analysis of Images Introduction The objective of this laboratory session will be to become familiar with the Fourier transformation of images, and gain experience in using the basic tools for the Fourier analysis of image data using IDL. Following this you will learn how to perform the inverse Fourier transform, and hence be able to perform image enhancement in the spectral domain. The images to be enhanced will possess different types of noise, in this case random and coherent, which require different approaches to remove the noise from the image. Fourier Transform of an Image If we wish to process an image in some way, in order to improve its appearance, there are two ways in which this can be done. One way is in the spatial domain, where we alter the image's pixels directly in order to change its appearance. The other way is to modify the image in the spectral, or frequency domain, which is the subject of Fourier analysis. Fourier analysis makes the assumption that any image can be constructed by adding together a large number of sinusoidal components of differing frequencies, with each component having its own amplitude. By obtaining the Fourier transform of an image (which can also be represented by an image in the frequency domain), we can get a pictorial representation of the frequency content of an image. This information can then be used, for example, to attenuate the high frequency components of an image, which has the effect of reducing the noise in an image. Exercise 1 The program lab_5a.pro reads an image file and displays its Fourier transform. Run the program to read in the stick1.bmp image. 1/ Describe briefly what the Fourier image is showing. Try to explain why it has particular features. Investigate Fourier transforms of an image with little periodicity, and an image containing some regular features. Briefly describe how the Fourier transforms of these two images relate to the original images and the Fourier transform of the stick. You will note that the command which displays the images is; TVSCL, ALOG(1 + 100*ABS(transform)) which indicates that the transform image has had a logarithmic transformation applied to it before it is displayed. Why is this done, and what would be the result if this step was not taken? (Hint: Try displaying the transform image without the use of the logarithmic transformation, for part 2 only. Part 1 Stick1. bmp Figure 1: Original image Figure 2: Fourier transform of image Figure 3: Centered fourier transform of image Train. bmp Figure 4: Original image Figure 5: Fourier transform of image Figure 6: Centered fourier transform of image (Hint / important to comment on the small white dots on the right and left edges of this image) Exercise 1 Part 2 Strick1. bmp Figure 7: Original image Figure 8: Fourier transform of image Figure 9: Centered fourier tra.

1. Introduction to Fourier Analysis of Images
Introduction
The objective of this laboratory session will be to become
familiar with the Fourier transformation of images, and gain
experience in using the basic tools for the Fourier analysis of
image data using IDL. Following this you will learn how to
perform the inverse Fourier transform, and hence be able to
perform image enhancement in the spectral domain. The images
to be enhanced will possess different types of noise, in this case
random and coherent, which require different approaches to
remove the noise from the image.
Fourier Transform of an Image
If we wish to process an image in some way, in order to
improve its appearance, there are two ways in which this can be
done. One way is in the spatial domain, where we alter the
image's pixels directly in order to change its appearance. The
other way is to modify the image in the spectral, or frequency
domain, which is the subject of Fourier analysis.
Fourier analysis makes the assumption that any image can be
constructed by adding together a large number of sinusoidal
components of differing frequencies, with each component
having its own amplitude. By obtaining the Fourier transform
of an image (which can also be represented by an image in the
frequency domain), we can get a pictorial representation of the
frequency content of an image. This information can then be
used, for example, to attenuate the high frequency components
of an image, which has the effect of reducing the noise in an

2. image.
Exercise 1
The program lab_5a.pro reads an image file and displays its
Fourier transform. Run the program to read in the stick1.bmp
image.
1/ Describe briefly what the Fourier image is showing. Try to
explain why it has particular features.
Investigate Fourier transforms of an image with little
periodicity, and an image containing some regular features.
Briefly describe how the Fourier transforms of these two images
relate to the original images and the Fourier transform of the
stick.
You will note that the command which displays the images is;
TVSCL, ALOG(1 + 100*ABS(transform))
which indicates that the transform image has had a logarithmic
transformation applied to it before it is displayed. Why is this
done, and what would be the result if this step was not taken?
(Hint: Try displaying the transform image without the use of the
logarithmic transformation, for part 2 only.
Part 1
Stick1. bmp
Figure 1: Original image Figure 2:
Fourier transform of image
Figure 3: Centered fourier transform of image

3. Train. bmp
Figure 4: Original image Figure 5: Fourier
transform of image
Figure 6: Centered fourier transform of image
(Hint / important to comment on the small white dots on the
right and left edges of this image)
Exercise 1
Part 2
Strick1. bmp
Figure 7: Original image Figure 8: Fourier
transform of image
Figure 9: Centered fourier transform of image
Exercises 2
Run the program lab_5b.pro. Select the circle image for input.
You will see three images:
(a) the original image, (b) the transform of (a), and (c) the
inverse transform of (b). Try another image. Comment on the
relationship these images (a, b and c) have to each other, with
respect to the operation of the Fourier transform. You need only
report on one set of images (circle or other).
Circle. bmp
Figure 10: Original image the transform of original the
inverse transform of transform

4. Fork. bmp
Figure 11: Original image the transform of original the
inverse transform of transform
Run the program lab_5c.pro. You should see an image of the
circle with incoherent noise, the circle with coherent noise, and
their respective transforms.
(a). Relate each image of the noisy circle with it's respective
transform, making reference to specific features seen within
each set of images.
(b). Describe qualitatively how you would construct a filter to
remove the ``spikes'' in the frequency spectrum of the circle
with coherent noise. Concentrate on the appearance of the filter,
and comment on how your filter would remove the spikes but
preserve the rest of the frequency spectrum (and why this is an
important consideration).
Circle. bmp
Figure 12: Random noisy image, and transform
Figure 13: Coherent noisy image, and transform
Run the program lab_5d.pro. You should see an image of (a) the
ideal lowpass filter, (b) the filtered spectrum of the original
circle image with incoherent noise, and (c) the inverse
transform of (b). Comment briefly on how the application of the
ideal lowpass filter has altered the spectrum of the circle with
incoherent noise, and what implications this has for the final
image (c).
Circle

5. Figure 14: Noisy original, fft spectrum
Figure 15: Lowpass filter, filtered spectrum and filtered image
Run the program lab_5e.pro. You should see an image of (a) the
butterworth filter, (b) the filtered spectrum of the original circle
image with incoherent noise, and (c) the inverse transform of
(b). Comment briefly on how the application of the butterworth
filter has altered the spectrum of the circle with incoherent
noise, and what implications this has for the final image (c).
Comment briefly on the differences between the ideal lowpass
filter and the butterworth filter, and what implications these
differences have on each filter's ability to filter incoherent noise
from an image.
Circle
Figure 16: Original, fft spectrum
Figure 17: Butterworth ilter, filtered spectrum and filtered
circle