Activity 5: Fourier Transform Model of Image Formation

Objective: To introduce the use of the Fourier Transform or FT as the standard model for imaging applications highlighting the varied uses of this method for image processing.

Tools: Scilab with SIP toolbox.

Procedure:The activity is divided into 5 parts each with its own focus.

1. Familiarization with the discrete FFT:
This section focuses on the use of the FFT as the mathematical equivalent of a lens. For this section we use two shapes, a circle and a square as our object and essentially image it with a lens of infinite aperture via the 2-D FFT function:

The corresponding FTs for these are as follows:

These image on the left represents the Airy disk of a point like image and the left image represnts the sinc(x)*sinc(y) function from a square aperture. This is how the patterns will appear on the transform plane on the back focal plane of a lens. Incidentally, these two images represent the far field or Fraunhofer diffraction pattern of the two aperture illuminated by a plane wave. When the FT of these images are taken we get the FT of the FT which are just the original images. On the other hand, if we use as our image a capital letter "A" as we have below:

We get a Fourier transform like the one we have below:
This indicates a center of symmetry even if the original image is only laterally symmetric. We also note that this patten may be thought of as a superposition of many point spread functions interfering with each other thus yielding this complex pattern. If we now take the FT of this image, we get an inverted "A" which can be thought of as the image of the object as seen through a lens system:

2. Convolution-Simulation of an Imaging Device.

The Convolution theorem essentially states that when the transform of two functions are multiplied it is equal to the transform of the product of the original functions. When this is translated into imaging, the image of an object ban be thought of as the convolution of the aperture function and the transfer function of the lens (or any other imaging system). In this case we use the the capital letters "VIP" in white against a blackbackground and a series of circular apertures simulating the aperture stop of the imaging lens. We take the FT of the aperture as imaged by a lens and then multiply it with the aperture which is in the transfrom plane. we then take the FT one more time to simulate another imaging lens.

We use the following apertures with increasing size.

The corresponding imaged convolutions are:

These correspond well to the fact that a larger aperture will produce a clearer image compared to a smaller aperture. At small apertures, the diffraction from the aperture actually limits the resolution of the imaging system.

3. Correlation-Template Matching

The section on correlation utilizes an image with certain text and a template letter. The correlatio is maximum when the section of the image matches completely with the template. This results in a large spot or spread function at the location of the match. In this case, we use the image on the left and then a template for the letter "A." The correlation is displayed on the right of these two:

The bright points indicate the regions of maximum correlation and these correspond clearly with the locations of the letter A in the image on the left.

4. Edge detection using convolution integral

In this last section we use the same concept as template matching to find the edges of an image pattern. This time we use matrices as our teplate instead of imges. Again we use the VIP image and find its edges using the following matrices:

A= -1 -1 -1

2 2 2

-1 -1 -1

B= -1 2 -1

-1 2 -1

-1 2 -1

and

C= -1 -1 -1

-1 8 -1

-1 -1 -1

With the following results:

Reference: AP 186 Activity 5 Manual

Evaluation: For the sheer amount of work expended as well as the proper results a grade of

10 is warranted.

Acknowledgements: I would like to thank the usual people for their assistance.