Analysis in Fourier space can further improve the quality of an image through filtering of undesired frequencies. It is in also in Fourier space that we can filter out noise due to the concentration of the its frequencies near the center. Likewise the Fourier transform of an aperture is the pattern formed by the diffraction of a a plane monochromatic light with the pattern as the aperture. Figure 1 and 2 shows the object and the Fourier transform respectively.
Figure 1: Circular Object
Figure 2: FFT of the Circulat Object
Due to the algorithm the orientation of the Fourier Transform Image is disarranged, we must operate an fftshift on the image to return it to its original structure.
Figure 3: fftshift operation on the fft of the Circle
It can be seen that the Fourier Transform of the Circle is the Airy pattern which is also the diffraction pattern produced by a pinhole by diffracting plane wave light source.
Application of FFT on the FFT of the circle will reproduce the circle. However for the Case of the letter A this becomes the result.
Figure 4: Letter A to FFT(A) to FFT(FFT(A))
This examples shows that 2 fft operations will flip the image which we did not notice on the circle since it is radially symmetric.
A lens or an apperture in mathematical form acts on the images through convolution or in Fourier space, the multiplication of their Fourier transforms and taking its inverse fourier transform. Figure 5 shows the process of convolving the VIP image with a circle that acts as an apperture of the lens.
Figure 5: Comparison of Convolved Image(left) with the original Image(right)
Figure 6 shows the effect of the aperture size with the projected image.
Figure 6: Increasing apperture size resulting Image(0.05, 0.2, 0.5)
The correlation is like finding the pattern present on the image using a template for comparison. Figure 7 shows an example text with the same font type and font size used as the letter A in the earlier discussion. Here we try to find what would be the resulting form of the image if we determine the correlation of the letter A with the Image.
Figure 7: Sample Image to find the correlation between A
Figure 8: Result of the correlation
Figure 8 shows that the highest intensities or the whitest part coincide with the parts of the text that really do contain the same letter A which is why the are perfectly correlated and the correlation shown obtained the highest value.
Finding the correlation is like template matching and using certain matrices one can detect edges from images which is important in contour mapping of land and other image processing applications.
Figure 9 shows samples of edge detected images. Note that the template to be used must have an overall element sum of 0 for the primary reason that we are trying to prevent biases in the edge detected image.
Figure 9: Samples of edge detected images of VIP
It can be seen that the template matching or edge detection depends on the gradient of the template in which the priority edge of detection is the portion of large gradients.
Self Evaluation: 9, Late again but complete and extensive results.
References:
[1] M. Soriano, Activity 6: Fourier Transform Model of Image Formatting, July 2010
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