Fun with Filters and Frequencies!

Part 1: Fun with Filters

Part 1.1: Finite Difference Operator

In this part we start off by computing the gradient of an image. First, we get a differene operator:

after that, we convolute the operator over the gray scale image. In order to find the gradient magnitude, we take the square root of sum of the squared partial derivatives. After that, we binarize the gradient magnitude with a self-defining threshold.

Part 1.2: Derivative of Gaussian (DoG) Filter

In this part, we do a very similar thing as we did in 1.1. What we are doing here is a Gaussian filter. As we can see, there is still some noise in the images. We want to reduce the noise of the edge images by first blurring our image by running it through a gaussian filter, and we do the same operations as before. After that, we can see that the final image becomes less noisy. Also, We can further improve this process doing one convolution instead of two by using the derivative of Gaussian filters, and the results should be the same.

Part 1.3: Image Straightening

This section is more mathematically heavy. It is all about image straightening algorithm. The algorithm is: Firstly, rotate the image by some angle "ang". Then, compute the gradient angle of all the edges in the rotated image by using arctan function. After that, we get a histogram of these angles from above operations. Finally, return the image with the highest number of gradient angles equal to -180 degree, -90 degree, 0, 90 degree, or 180 degree, which are the "straightest" ones.

Facade

Pisa Tower

KFC

Failure Case: Kabab

Part 2: Fun with Frequencies

Part 2.1: Image "Sharpening"

In this part, we did image sharpening, which is done by adding higher frequencies from an image to the image itself. In order to get the high frequencies, we subtract the blurred image from itself and convolve the image with a Gaussian from the original picture. After that, we add these high frequencies to the original image. Notice that we are not sharpening an image by actually getting a higher resolution, so it is more like a "fake" one.

Part 2.2: Hybrid Image

We are doing hybrid imaging in this part. First, we align two images. After that, we take the low frequency from image one and take the high frenquency from image two. The basic idea is that high frequency tends to dominate perception when it is available, but, at a distance, only the low frequency (smooth) part of the signal can be seen. By blending the high frequency portion of one image with the low-frequency portion of another, you get a hybrid image that leads to different interpretations at different distances.