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IMAGE PROGRESSING FUNDAMENTALS

Part 2: Math, Math, Math

by James Antonakos

Start ý Blob Analysis ý Edge DetectionýBehind the Mask ý A Helping Hand from Fourier ý Please Sir, May I Have Some More? ý Sources and PDF

A HELPING HAND FROM FOURIER

Most engineering and technology students are familiar with the Fourier transform:

by way of its applications in signal processing. For example, the Fourier transform of a square wave yields the spectrum of odd sinewave harmonics that actually compose the square wave. The Fourier transform converts information from the time domain into the complex frequency domain.

When the Fourier transform is applied to an image, you get the same conversion and end up with a set of frequency components that represent the image. Photo 3 shows the two-dimensional Fourier transform of the original image (Photo 1a). The center of the image represents the DC level of the image, and as you move away from the center (in any direction), the pixels represent the increasing frequency components of the image (both real and imaginary). The brighter the pixel, the higher the amplitude of the associated frequency component.

Photo 3ýThe center of the image is the DC origin. The frequency components increase as you move away from the center. High-amplitude frequency components have an associated high intensity.

What happens if you throw out some of the frequency information? As indicated in Photo 4, all information outside the small circle has been eliminated. In effect, you have run the frequency information through a low-pass filter.

Photo 4ýMost of the high-frequency components have been eliminated.

Now, using the inverse Fourier transform, convert the remaining frequency components back into the time domain. The results are shown in Photo 5. Can you see how blurry the image has become? The distinct edges and other sharp details of the image have disappeared. This confirms that you can use low-pass Fourier filtering to smooth an image. This would be especially helpful if you had a grainy image or one with a small amount of noise in it.

Photo 5ýThe result of an inverse Fourier transform on the information found in Photo 3b can be seen here. Notice the blurring in the image.

Photo 6 shows the results of high-pass Fourier filtering the image. Now the low-frequency Fourier components have been thrown out, leaving only the high-frequency components. The only details that remain in the image are the edges (edges represent high-frequency components because of their step response). If you throw out too much frequency information, you may end up with an entirely black or white image, so care and experimentation must be used to find the right filtering method. It is fascinating to examine the Fourier spectrum of different images, for they typically bear little resemblance to the original image.

Photo 6ýOnly the edges in the image stand out now.

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