CS 194-26: Fall 2020
Project 2: Fun with Filters & Frequencies!
Megan Lee

Part 1: Fun with Filters

In this part, we will build intuitions about 2D convolutions and filtering.

1.1: Finite Difference Operator

An image gradient is a directional change in the intensity or color in an image. Thus, in order to detect the edges of our image, we can utilize the gradient of our image. A useful tool to analyze the edges of our image is by using the derivatives of the image and the gradient magnitude. We began this project by using the finite difference operators above and convolving them with an image to obtain the partial derivative in x and y of our image.

In order to obtain the gradient magnitude image, we applied the formula above (sqrt(dx^2 + dy^2) to the partial derivatives we previously found to view the overall image gradient.

I binarized the gradient magnitude image by picking the appropriate threshold, attempting to suppresss the noise while still showing the edges. After playing around with it, I ended up choosing a threshold of 45, which displayed clear edges but a little noise at the bottom still.

1.2: Derivative of Gaussian (DoG) Filter

With just the difference operator, the results were pretty noisy. In order to combat this, we use our smoothing operator: the Gaussian filter. I blurred the original image by convolving the image with a Gaussian filter of sigma 2, and then repeated the process in part 1.1 to find it's edges.

Intuitively, this should address the noisiness returned with just the difference operator since the gaussian blur acts as a low pass filter, removing the high frequency fluctuations in the image and getting rid of the grainy noise.

I used a threshold of 13 to binarize the gradient magnitude image. My guess was correct: this time, the edges are much thicker and cleaner and there is a lot less noise. This makes sense as removing the high frequencies in the image would remove the noise (e.g. the grass) in the photo, creating much more defined edges. However, this came at the cost of having much thicker edges. Since the image has been blurred, the actual edges that we want to detect are no longer as sharp, causing the edge to get thicker.

1.3: Image Straightening

It is known that statistically there is a preference for vertical and horizontal edges in most images (due to gravity!). Using this insight, I was able to automate image straightening. I set my function to rotate each image between -10 degrees and 10 degrees, computed the gradient angle of the edges in the image, and created a heuristic that picked the angle rotation with the maximum number of verticle edges (defined by 90 and -90 degree gradient angles).

I ran into a little trouble here: my heuristic was picking up edges created by the rotation. To account for this, I center cropped the image before doing any scoring.

Note that it is hard to see the changes reflected in the histogram as the degree rotations where both under 10 degrees.

Successful Cases: Left to right: Original Image, Straightened Image


Angle chosen: -2 degrees


Angle chosen: -7 degrees


Angle chosen: -5 degrees

Failure Cases: Left to right: Original Image, Straightened Image

In this instance, the road was very curved, and there were a lot of trees and items in the photo. In the histograms shown, we can see that there are a lot of different angles in this photo. Since there's so much going on, and not a prominent amount of horizontal and vertical edges, my algorithm was unable to find a suitable angle.


Angle chosen: -10 degrees

Part 2: Fun with Frequencies

2.1: Image Sharpening

The "sharpen" an image, we learned that we can apply a low pass filter such as a Gaussian filter to retain only the low frequencies of an image, and subtract this blurred version from the original imagee to obtain the high frequencies. An image with higher frequency tends to look sharper, allowing us to create a sharpening effect.

Process is illustrated below:

2.2: Hybrid Images

The goal of this part of the assignment is to create hybrid images using the approach described in the SIGGRAPH 2006 paper by Oliva, Torralba, and Schyns. Here, I combined a high frequencies of one image with low frequencies of another image to product a hybrid image.

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. Upon looking more closely, the eye detects the high frequency image, but when there is a bit more of a distance the image appears more like the low frequency image.

The process is illustrated through frequency analysis. The log magnitude of the Fourier transform of the two input images, the filtered images, and the hybrid image are shown below.

Here are some more hybrids just for fun!

The hybrid of Derek and the fish was a failure. Since their faces are so different, aside from the eyes, the images do not overlap well and it is easy to see both Derek and the fish at close distances and far distances in the hybrid photo.

2.3: Gaussian and Laplacian Stacks

Lincoln: From top to bottom: Gaussian and Laplacian stacks.





Derek the Fish: From top to bottom: Gaussian and Laplacian stacks.

2.4: Multiresolution Blending

The goal of this part of the assignment is to blend two images seamlessly using a multi resolution blending as described in the 1983 paper by Burt and Adelson. An image spline is a smooth seam joining two image together by gently distorting them. Multiresolution blending computes a gentle seam between the two images seperately at each band of image frequencies, resulting in a much smoother seam.



Splined an apple and orange together to create an oraple!



Splined a basketball and orange together.



I have always wanted to go to Egypt!

Illustration of the process of creating the photo of me in Egypt:

Final Thoughts

I loved this project! I had a laugh making hybrids of my friends and different animals, and but my favorite part was splining images together. The more useful and interesting takeaway from this project was that images are made up of high frequencies and low frequencies, and upon first glance you can sort of tell if the image will be suitable for manipulation (whether it's straighten, splining, or hybriding!) and what type of manipulation just based on the amount of frequencies in the image.