Project 2: Filters and Frequencies

CS 194-26: Intro. to Computer Vision & Computational Photography, Fa21

Doyoung Kim

Overview

In this project, we have discovered how filters and frequency plays a role in image processing. Starting from finite difference operator and gaussian filter, we discover how we can create a edge filter in part 1. Then, we discover how we can use the gaussian filter as a low-band filter to create image sharpening filter followed by blending two different images seamlessly using Laplacian and Gaussian stack.

Part.1 Gaussian filters and finite difference operators

Part. 1.1 Finite Difference Operator

Using the image above, we apply each of finite difference operator, dx, dy on the cameraman image. Then, I computed the gradient magnitude of the two images filtered by dx and dy. At last, binarization is done on the gradient magnitude, which turns out to be a edge detecting image.

Original

dx filter

result

Original

dy filter

result

Gradient Magnitude

Binarized Gradient Magnitude

Part. 1.2 Derivative of Gaussian(DoG) Filters

As the results of purely applying the difference operator are quite noisy, we try to apply the gaussian filter on the image first, and then do the same thing as 1.1 to see if the result is less noisy.
On top of that, we want to discover if convolving the gaussian filter with difference operator reserves the property of the gaussian and gives us the same result as the early results where we applied guassian filter on the image first, and then applied difference operator.

Following images uses the Gaussian filter and dx, dy operators separately on the image.

Original
Cameraman

Gaussian
Filtered

dx operator
on Gaussian

dy operator
on Gaussian

Grad. Mag.
on Gaussian

Binarized Grad.
Mag. on Gaussian

Following images uses the Gaussian filter that are convolved with dx, dy operators on the image.

Gaussian Filter
Convolved with dx

Gaussian Filter
Convolved with dy

DoG Convolved
dx

DoG Convolved
dy

Dog Convolved
Grad. Mag.

DoG Convolved
Binarized Grad. Mag.

Part.2 Fun with Frequencies!

Part. 2.1 Image "Sharpening"

Using the Gaussian filter, there is a way to sharpen the image, which makes the higher frequency part of the image more clear. This is doen by extracting the higher frequency part of the image which is done by: orignal image - gaussian filtered image. Then, we add this back to the original image stack up the higher frequency.
After sharpening the image with separate steps, we can also construct an unsharpen-filter, which sharpens the image with single step. And this filter is constructed as: f+α(f−f∗g)
The following are the images that sharpening are done separately.

Original
Image

Low Frequency
Part of Image

High Frequency
Part of Image

Sharpened
Image

The following image is sharpened with the unsharpen-filter in single convolution.

Original
Image

Sharpened
Image

Part. 2.2 Hybrid Images

In this part of the project, we can discover how human eyes are more sensitive to the high-frequency part of the image when seeing at close, but low-frequency part of the image at distance. We first get two images. And take the only higher frequnecy part of one image, and get lower frequency part of the other image and add them together. This way, we can create an image that shows two different images differeing by the viewer's distance from the image.

One process necessary before adding the images is to align two images so they look alike.

Following are some process of creating a hybrid image:

Original
nutmeg

Original
Derek

Aligned
nutmeg

Aligned
Derek

Hybrid
Image

The hybrid image above is created with the high frequency part of the cat and low frequency part of the person.

Following are some more examples:

Original
Mars

Original
Apple

Aligned
Mars

Aligned
Apple

Hybrid
Image

In the following examples, frequency analysis is done. Take a look on how the frequency domain changes.

Original
Cartoon

Original
Tahoe

Aligned
Cartoon

Aligned
Tahoe

Hybrid
Image

Cartoon
Frequency

Tahoe
Frequency

High-band
Cartoon Freq.

Low-band
Tahoe Freq.

Hybrid
Freq.

Part. 2.3 Gaussian and Laplacian Stack

For part 2.3 and 2.4, we will discover a method to blend two different images seamlessly. As cutting two images into half and putting them together would create a seam between two images, we approach with gaussian and laplacian stacks. Gaussian stack is a stack of gaussian filtered single image, but as we go down the stack, the image is convolved with the gaussian filter again and again. So the 4th image of the stack would have been convolved with the gaussian filter 4 times.

The idea is similar to the Laplacian stack, however, for Laplacian stack, each image is a subtracion between each stacks from gaussian stack. For example, Laplacian_stack[0] = gaussian_stack[0] - gaussian_stack[1]. This way, Laplancian_stack[0] only contains the image of higher frequency from guassian_stack[0], which is not contained in gaussian_stack[1].

We also use a mask that allows us to distinguish the regions of two images that are supposed to be blended to each other. And as we are applying gaussian filter to the original images, in order to remove the discrepancy, we also apply the gaussian filter to the masks being used as well for each depth of the stack being created.

At the end, by adding up all the images from the Laplacian stack creates a well blended image of both images.

Following are the two images we are trying to blend: