Activity 7 Enhancement in the Frequency Domain

Filtering in Fourier Space

Signal processing as the name implies is an area that deals with analyzing and processing of signals. A major concern in signal processing is the signal to noise ratio. By definition, anything that you do not want is considered noise [1]. One of the methods use in signal processing is filtering in Fourier space. In essence, if we know the characteristic frequency of a signal we can use a filter to remove unwanted frequencies and enhance the signal thereby increasing the signal to noise ratio. In this activity, we demonstrate basic use of filtering in Fourier space to remove unwanted signals.

Activity 7.A: Convolution Theorem

Before we dwell on filtering, we must first familiarize ourselves with the convolution theorem. This is a crucial step since the process of filtering in Fourier space is basically just a convolution. For example, the convolution of a Dirac delta is just equal to the function itself on the location of the Dirac delta. Below are Fourier Transform images of different patterns.

FT of two dots is a sinusoid with a distinct frequency(Dirac delta at f).

FT of circles of different radius. As we increase the radius of the circle the FT decreases in radius. This is expected. i.e., similar to the rect function since FT is anamorphic.

FT of squares of different sides. Again as we increase the length of the side, the width of the FT decreases. In fact, the limit (side -> inf) will result to a dirac delta (point).

FT of gaussian of varying variance. The unique property of a gaussian is that its FT is also a gaussian. Again similar to square and cirlces, as you increase the variance of a gaussian its FT decreases in variance.

Comparing the imaginary and real part of gaussian and inverse gaussian.

click on the image to ZOOM

Gaussian of varying variance with the (a)[leftmost] image of gaussian (b) its real part (c) imag part (d) and FT (e) [bottom] inverse gaussian image (f) real part of inv gaussian (g) imag part of inv gaussian and (h) FFT of inv gaussian.

Observe that the real part of the gaussian and inverse gaussian are the same.

Activity 7.B Fingerprints: Ridge Enhancement
One of the use of filtering in Fourier space is through ridge enhancement of fingerprints. The process can be basically summarize as follows

• the FT of the image (i.e, fingerprint - input) was obtained. In order to enhance the visibility of the frequencies, the FT was obtained in log scale.
  
• The obtained FT was analyzed and was used as a template to create a mask that will enhance the frequencies of ridge while decreasing other unwanted frequencies.
  
• It must be noted that use of binary mask is not too adviseable as this will pick only the desired frequencies but not enhance them relative to the other frequencies. Use of gradient mask or grayscale mask is more adviseable. If we know exactly the frequency of the object we want, we can maximize the pixel value of the mask at that location while blurring or decreasing the pixel value of other frequencies that we do not want.
  
• In this example, the author was having a hard time separating the background from the frequency of the ridges. As as a solution, I use a gradient mask on the frequencies of the ridges while blurring it as you go away from the signal. Similarly for the zero order (background), a gradient mask was used which is darker at the center and increasingly bright moving away from the center. This was done because complete removal of the zero-order will not result to a very good reconstruction.
  

For our purposes, this reconstruction is not the best reconstruction that could be achieve but its a start. The ridges are well defined and darkening of the background provided additional contrast. It is recommended however that further refining be done for better reconstruction.

Activity 7.C Lunar Landing Scanned Pictures: Line Removal

From our previous discussion of FT, we know that the FT of vertical lines are two points symmetric about the center along the horizontal axis. The picture of the lunar above was filled with vertical lines resulting from stitching of several digital "framelets" to obtain a composite image. We can remove this in Fourier space by filtering the image using a mask by filtering the frequencies along the horizontal direction. (see mask used)

Activity 7.D Canvass Weave Modelling and Removal

Similar to ridge enhancement we can remove the pattern created by the canvass weave using filtering in Fourier space. Observe that the canvass weave looks a lot like corrugated sinusoids, thus, we must remove frequencies along the vertical and horizontal side. Also by virtue of rotation, other frequencies might also be evident around the center. The presence of dots in the masked pattern was due to thresholding in scilab (i.e, img(img~=0)=1).

For this activity, I give myself a grade of 10 for obtaining the requirements needed to complete the activity. However, I admit that the reconstructions I presented above are not the best reconstructions. Further research can be devoted to this area to provide better algorithms for filtering and pattern recognition.

I would like to acknowledge Irene and Miguel for all the help they've given while doing this activity. (jokes, company and intellectual discussions)

References
[1] http://en.wikipedia.org/wiki/Noise
[2] Applied Physics 186 Activity 7 Manual.