A7 - Enhancement in the Frequency Domain

Anamorphic Property of the Fourier Transform

Varying the frequency of the sinusoid changes the location of the FFT peaks from (0,0). As observed, higher frequencies would result to peaks farther from the origin.

Rotation of the image, on the other hand, corresponds to the rotation of its FFT but on the other direction.

Multiplying two sinusoids result to peaks which form the corner of a square. Adding them, however, result to a cross-like pattern, wherein the vertical dots correspond to fft peaks of the horizontal sinusoid and horizontal dots to vertical sinusoid. The effect in the Fourier space is much like adding each FFT.

Filtering in Fourier Space

a) Fingerprint Enhancement

Shown below is a fingerprint image. We want to enhance the ridges present on it by filtering in Fourier space.

We know that the ridges would correspond to high frequencies since they are repetitive patterns present in the image. Also, we can think of them as sinusoids in different directions. Therefore, we get the FFT of the fingerprint image.

We then designed a filter mask using mkfftfilter function in SciLab. Since we want to enhance the higher frequencies corresponding to the ridges, we design a high pass filter. Shown below is a high-pass exponential filter.

We multiplied this filter with the original FFT of the fingerprint image. After this, we again get its FFT to get the following enhanced image.

The FFT of the enhanced image is shown below. The lower frequencies were removed thereby resulting to enhancement of the higher frequencies.

We then tried using a high-pass binary filter shown below.

The resulting image is shown below.

b) Lunar Landing Scanned Picture: Line Removal

The two groups of irregularly shaped craters north and west of the landing site are secondaries from Sabine Crater. This view was obtained by the unmanned Lunar Orbiter V spacecraft in 1967 prior to the Apollo missions to the Moon. The black and white film was automatically developed onboard the spacecraft and subsequently digitized for transmission to Earth. The regularly spaced vertical lines are the result of combining individually digitized 'framelets' to make a composite photograph and the irregularly-shaped bright and dark spots are due to nonuniform film development. [NASA Lunar Orbiter photograph]

By filtering in the Fourier domain, we can remove the vertical lines present in this image. We first get its FFT.

Using the hints from http://www.roborealm.com/help/FFT.php, we designed a filter mask such that the peaks which lie horizontally along the center of the FFT image will be suppressed. These peaks corresponds to the strong vertical lines in the original image. The filter mask we designed is shown below.

This mask is then multiplied to the FFT of the original image to get the following enhanced FFT.

From this enhanced FFT, we obtain the following enhanced image which obviously has no vertical lines present. Clearly, the image was enhanced.

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Collaborators:
Jeric Tugaff for the FFT discussions and hints on filtering in the Fourier space.
Cole Fabros, Benj Palmares and Ed David for the FFT discussions.

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Rating:
I give myself an 8/10 because
1. I'm not satisfied with my fingerprint enhancement algorithm. SciLab's mkfftfilter seems not suitable for this job.
2. I'm happy with the results of the Lunar Image tho, we had a hard time figuring how to do it properly. I'm thankful for Jeric's resourcefulness for he found hints in the internet.