ACTIVITY 7 Enhancement in the Frequency Domain

7.A Convolution Theorem

Figure 1 shows a set of patterns, which illustrates the Fourier transform of the convolution of two images. The last three images can also be thought of as the convolution of the patterns (square, circle, Gaussian) with the two dots in the first image. Recall that the Fourier transform of a convolution is just the the product of the Fourier transforms of the images. Recalling the Fourier transform of a square and a circle, which are shown in the previous activity, Figure 1, indeed, illustrates that the Fourier transform of convolution of two images is just the product of their Fourier transform. Figures 2, 3 and 4 show the variation in the Fourier transform when the sizes of the patterns are varied. It can be seen that the size of the pattern in the Fourier transform, most obvious with the brightest spot, generally decreases with increasing size of the image pattern. Inverting the grayscale value of the image pattern yields a similar Fourier transform pattern but of inverted grayscale value also (see Figure 6).

Figure 1. Fourier transform of different image patterns.

Figure 2. Fourier transform of square dots of different sizes.

Figure 3. Fourier transform of circular dots of different sizes.

Figure 4. Fourier transform of Gaussian dots of different sizes.

Figure 5. Comparison of the real and imaginary parts of the Fourier transforms of the Gaussian dots and the inverted Gaussian dots.

7.B Fingerprints: Ridge Enhancement

Image enhancement can be done either in the spatial domain. Histogram equalization to improve the contrast of the image, which we did in activity 4 is an example of enhancing in the spatial domain. In this activity, the enhancement of the image is done in the frequency domain by blocking all unnecessary frequency (bias and noise) so that the Fourier transform contains only the information (frequencies) regarding the actual image. Basically, the remaining parts of this activity is about filtering in the frequency domain. Filter masks were created depending on the Fourier transform of the image and the frequency that has to be blocked. It is important that the masks are of inverted grayscale value (background-1, foreground/pattern-0). Basically, the masks just follow the shape of the frequency patterns that will be blocked. The concept behind linear filtering in the frequency domain is the convolution theorem. Thus, the inverse Fourier transform of the product of the filter mask and the Fourier transform of the image is the convoluted image of the original image and the noise pattern, which is of inverted grayscale value. This results in an image without the noise patterns.

In the case of fingerprint ridges enhancement, the filter mask created is not of inverted grayscale value because it makes filtering much easier. This is because the actual frequencies of the fingerprint are much more obvious than the unnecessary frequencies. A circular filter mask is used for the actual frequencies of the fingerprint while a Gaussian dot is used for the cutoff in the central frequency, which is the needed mask (controlling factor) for ridge enhancement. The filter used leaves all components inside the circle and Gaussian unchanged but cuts off all components outside these regions. The results are shown in Figures 6 and 8. A binarized form of the original and enhanced images are shown in Figure 7 to clearly illustrate the enhancement of the image due to the filtering process. Blotches are removed and the lines or ridges are more defined.

Figure 6. Clockwise from top left: grayscale image of a fingerprint; its Fourier transform; filter mask used; enhanced grayscale image of the fingerprint.

Figure 7. Clockwise from top left: grayscale image of the fingerprint; equivalent binary image; enhanced grayscale image; equivalent enhanced binary image.

Figure 8. Clockwise from top left: paler grayscale image of another fingerprint; its Fourier transform; filter mask used; enhanced grayscale image of the fingerprint.

7.C Lunar Landing Scanned Pictures: Line Removal

Recall that the Fourier transform of equally spaced lines is approximately a line but perpendicular to the direction of the lines in the image. So, if the horizontal (vetical) lines have to be removed, the vertical (horizontal) line in the Fourier domain has to be filtered. Figure 8 shows the resulting images when only the horizontal lines are removed and when both horizontal and vertical lines are removed.

Figure 9. First column from top: original grayscale image; vertical lines removed in the image; both horizontal and vertical lines removed in the image. Second column from top: Fourier transform of the original original image; filter mask used to remove vertical lines; filter mask used to remove both horizontal and vertical lines.

7.D Canvas Weave Modeling and Removal

Figure 10 shows the Fourier transform of the image, filter mask used to remove the canvas weave pattern and the resulting enhanced image. Figure 11 shows the inverse Fourier transform of the inverted filter mask. It is more or less similar to the canvas weave pattern in the image. It is even shown, also in Figure 11, that when the pattern is added to the enhanced image, the original image having the canvas weave pattern is restored.

Figure 10. Clockwise from top left: original grayscale image; its Fourier transform; filter mask used; enhanced grayscale image without the canvas weave pattern.

Figure 11. Clockwise from top left: inverted grayscale image of the filter mask used; reconstructed canvas weave pattern; resultant image of adding the reconstructed canvas weave pattern and the enhanced grayscale image; original image.

I created the filter masks using SciLab only because it is easier for me than creating them using Paint. There is a risk, however, for this procedure that it will also block the needed frequencies and not totally filter the undesired frequencies. But since the images are perfectly enhanced, the filter masks that were created avoided the risks mentioned above.

I would like to thank Raffy for explaining to me the process of filtering especially the correct usafe of the fftshift() function. I would also like to thank Thirdy and all others who have helped me finish this acitivity. A grade of 10 for this activity because the images were perfectly enhanced.

References:
1. Fingerprint: http://www.staffordsheriff.com/content/childsafety/Image/Fingerprint picture 1.jpg
2. Digital Image Processing using MATLAB by R. Gonzales, R. Woods and S. Eddins