In this project, we first tried to get acquainted with the behavior of different images and standards in the frequency domain. We got to know the characteristics of the Fourier domain, and finally, we applied the filters in the frequency domain.
At first, to prepare the images for later use and perform various processing, we convert all the given images (except the Gradient and Gaussian image) to 0 and 1 values by applying thresholding.
Next, we apply a two-dimensional fast Fourier transform (fft2) on all images, And we use the “fftshift” command to accumulate frequencies in the center. According to the requirements of the project, we compare the images in the form of groups, The categories of images are as follows:
the results are as follows:
We have boxes in all directions of the original image and we know that the Fourier transform of the box is the sinc function. In the frequency space, these sinc functions are completely regular. In the center, which can be said more clearly, is the peak of this function, and other observations are the continuation of the sinc function, which exists in all directions and is being damped. The effect of the impulse train can be seen in this image.By looking at the original figure, we find that this figure is the same as the one above, with the difference that our rim is not very thick. Again, if we look at this photo from any direction, there are boxes and we know that the Fourier transform of the box is the sinc function. Here, because the width of the boxes is small, the sinc function becomes wider and the photo obtained from the Fourier transform of this image is brighter.If we look at the original photo from any direction, we can see the said boxes. In this picture, from there As the radius of the circle is large, the width of our boxes increases, and because of this, the sinc function becomes smaller, and the image resulting from the Fourier transform becomes about one bright point, which is the peak of the sinc function.If we want to compare the initial photo of this transformation with the photo above, we will realize the difference in the radii of the two circles. Here, as above, we have boxes from all directions, but here, due to the smaller radius of the circle, the width of the boxes is smaller, and as a result, their conversion function, which is the sinc function, is wider and the resulting image is brighter than the resulting image above. The brightest part of the photo is the peak of the sinc function and other observations are the continuation of this function. In this picture, the impact of the impulse train can be seen.With a look, we find that the image resulting from the Fourier transform of this image is similar to the image above that we talked about, and the only difference between these two primary images is the displacement of our small circle, which, as can be seen, had no effect in the frequency space. Here we understand that moving the original image does not change the frequency space.It is clear that the center of the image is brighter. When we move in the horizontal direction, we see that the width of the box is too large. But this is not the case when we move in the vertical direction. We know that the Fourier transform of the box is the sinc function, and the wider the box, the more compact this function will be, and vice versa. This is why in the resulting image, our vertical component is brighter than the horizontal component.In this photo, the above statements are also true, only here, due to the rotation of the rectangle in the original photo, our components have rotated a little in the frequency domain, but they are still perpendicular to each other, and we can see that where the width of the box is smaller, it still has a brighter component than the other. Also, we can see that the rectangle has been transformed in addition to the rotation, but the effect of this transformation is not displayed in the Fourier transform, and finally it can be concluded that the Fourier transform is sensitive to rotation but not to displacement.In this photo, because the other image is square instead of rectangular, and its vertical and horizontal widths are equal, both our vertical and horizontal components are clear in the frequency space and are exactly similar.In this photo, if we look at the photo horizontally, we can see a box that is much wider than the boxes that exist in the vertical direction. Since the Fourier transform of the box is the sinc function, the resulting image is in this form because the smaller the width of the box, the wider the sinc function and the brighter the vertical component of the transformed image. The vertical line in the middle of the photo is brighter than the others. lines, and this is because of the peak of the sinc function, and other lines are the continuation of the function.If we pay close attention to the previous photo, we can see that the width of the box is less than the same box in this photo, and for this reason, the sinc function of the Fourier transform of this photo is the sum of the level of the previous photo, so that the resulting image also looks like this.This picture is exactly the opposite of the previous pictures if we move in the direction of the horizon The rectangles are less than the vertical direction, and that’s why the image has become like this It is profitable. In this image, the horizontal boxes are wider than the boxes in the previous images, and this causes the sinc function, which is converted into a box, to be smaller and the strip on the photo is less bright than other photos.This photo is the same as the photo above, except that the lines are repeated, only the impulse train effect can be seen in this photo.The difference between this image and the image above is the width of the horizontal boxes and the proximity of the lines. From the comparison of the resulting images in these two parts, we find that the closer the lines are, the greater their distance in the frequency space.Again, in this image, we have boxes from all sides, whose width is very small and their sinc function is very large, that’s why the resulting picture is like this.All the statements of the previous photo are also true in this photo, only in this photo the impulse train effect can be seen.The star has diagonal borders, so if we move in the direction perpendicular to these borders, we still have the said boxes in this image, and we know that the Fourier transform of the box is the sinc function.
In the last part of this project, we prepare the images as in the first part. In this section, the collection of images is as follows:
The second image in row two is a sine wave with noise added on it, but it is basically the same as wave number one. Considering this point, we move the second image to the frequency space and finally recover the original wave by removing the noise.
To do this part, we move all the images to the frequency space. The resulting images look like this:
picture1picture2picture3picture4
In pictures number three and four again, due to the change of colors from white to black, both vertically and horizontally, we have the sinc function in both directions in the Fourier transform. The difference between these two images is that in small chess we are faced with a drastic change in frequency several times and the impulse train is happening.
The detailed report of this project is in the following link:
