Fun With Filters And Frequencies

Part 1.2: Derivative of Gaussian (DoG) Filter

In this part, I used blurring to improve the noise in the edge image from part 2.1. To create the blurred image, I convolved the cameraman image with a gaussian of kernel size 5. Then, I applied the same steps as in part 2.1 to get the edge image. My results are shown below.

Applying the Dx and Dy convolutions after blurring the image helped reduce a lot of the noise in the gradient images and therefore the edge image. I also noticed that the edges appeared thicker in the smoothed image, and they also appeared thicker in the edge image produced from the smoothed image. The max magnitude of the x and y gradients was larger with the non-smoothed image, but the average gradient magnitude was higher for the smoothed image.

I also tried combining the Dx and Dy convolutions with the gaussian convolution, for which my results are shown below.

I ended up with the exact same results as I had when applying the convolutions separately.

Part 2.1: Image "Sharpening"

In this part, I convolved images with the unsharp mask filter to make them appear sharper. The way this filter works is by adding some constant alpha times the image to the image, then subtracting alpha times the image convolved with a gaussian (I used a gaussian of kernel size 5, and alpha = 1). These operations can be combined into one convolution. This essentially adds more of the high frequency parts of the image to the image (the high frequency parts are obtained by subtracting the low frequency parts from the image).

For the snowshoe image, I also tried blurring it and sharpening the blurred image to see if I would end up with the original image.

The sharpened version of the blurred image is sharper than the blurred image, but it not quite as sharp as the original image. This is probably because some information is lost in the blurring/resharpening process. When we sharpen, we are not actually getting new information and adding it to the image (all we are doing is enhancing the high frequencies), so we can't get back the original image from the blurred image by sharpening it.

Part 2.2: Hybrid Images

This was my favorite part of the project! For this part, given a pair of images, I extracted the high frequency part of one image and the low frequency part of the other. Then I averaged these together. When you look at the result from far away, you see the image that I took the low frequency components from. When you look at it from close up, you see the image I took the high frequency components from.

This example did not turn out well, because the original input images did not look similar at all.

For the last set of images, I also display the fourier transforms of the input images, the filtered images, and the hybrid image.

Part 2.3/2.4: Gaussian and Laplacian Stacks, Multiresolution Blending

In this part of the project, our goal was to blend two images together seamlessly. To do this, I first created Gaussian and Laplacian stacks for each of the two images (the Laplacian stack contains the different frequency components of the image). The stacks were of length 5, and to create them I used a Gaussian kernel size of 11. Then, I created a mask for the part of the first image I wanted to include, and created a Gaussian stack of that mask (length 5, kernel size 21). Then, I weighted the different layers of the two Laplacian stacks by the mask gaussian layer and (1 - the mask gaussian layer) respectively. Finally, I summed the results for each layer and then summed over all the layers to get the blended image. This process works because the lower frequency Laplacian layers are weighted by a larger Gaussian, so they get blended more, and the opposite is true for the highest frequency Laplacian layers.

Results for the oraple:

The first 3 rows represent the 0'th, 2'nd, and 4'th layers of the Laplacian stacks, weighted by the relevant layers of the mask gaussian stack. The last row is the sum of the weighted laplacian layers. The first column is the apple, the second column is the orange, and the third column is the sum of the first two columns. The bottom right corner is the final blended result.

Below, I show the results for three other sets of images.

For the foot and hand blending, I used the same mask as for the oraple. I think the result looks very creepy!

I like the way this turned out, but the boat does appear lighter here than it did in the original image.

I think the result turned out okay here as well, but it would have worked better if I was blending two things with more similar shapes.

Reflection

The most important thing I learned from this project is that images are actually made up of many layers of different frequencies, and that each frequency layer differently affects how we perceive an image. For example, low frequencies are what we see from far away, and we perceive high frequencies as image sharpness. Overall, I think this was a really fun project to work on.