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CS 180 Project 2: Fun with Filters and Frequencies!

+

Jaxon Zeng

+ +

+ +
+ +

Overview

+

This project explores the frequency domain of images. In the first part, I built an edge detection algorithm + using the Finite Derivative Operator and used the Gaussian filter to eliminate the effect of noises in edge + detection. In the second part, I tried two ways of image blending. The first method blends the high and low + frequencies of two images to create different impressions when looking from far and close distances. The second + method used the technology called multiresolution blending introduced in the 1983 paper by Burt and Adelson. This + method can create a seamless effect when blending two images with masks.

+ + +

Part 1: Fun with Filters!

+

1.1: Finite Derivative Operator

+

Finite derivative operator is a primitive way to detect edges. In the code, I used two simple convolution kernel + to detect edges horizontally and vertically.

+ +

I used `signal.convolve2d` from the SciPy library to convolve the image with the two kernels. The result of this + convolution is the gradient of the image.

+ +
+ + +
Camera Man Original +
+ +
+
+
+ + + + + + +
+ +
Camera Man gradient dx
+ 		+ +
Camera Man gradient dy
+
+
+

To create a better result, I binarized the gradient images to make the edges obvious. I choose a threshold of + 0.25 (in range [0, 1]) so that any pixel under this value will be black and above this value will be white.

+
+ + + + + + +
+ +
Camera Man gradient dx binarized
+ 		+ +
Camera Man gradient dy binarized
+
+
+

Combining the two edges using `np.sqrt(edge_imgX ** 2 + edge_imgY ** 2)` (l2 norm) can create the complete edge + image: +

+ + + + + + +
+ +
Camera Man edge binarized
+
+
+
+
+

1.2: Derivative of Gaussian (DoG) Filter

+ +

It's easy to notice that there is noise on the background. Applying a Gaussian filter can be a good way to get + rid of the noise. I created the Gaussian filter by first called cv2.getGaussianKernel() to create a 1D filter. + Then, I calculated the outer product of two 1d Gaussian filter to create a 2D Gaussian convolution kernel. Then I + convolved the image using the Gaussian kernel. It creates blurred images that also eliminate the noise.

+ +
+ + + + + + +
+ +
Camera Man Blurred
+
+
+

I can now take the finite derivative operator on the blurred image to achieve a better result from edge + detection. We can see from the result that the detected edges are smoother and less noise. +

+
+ + + + + + +
+ +
Camera Man Blurred gradient dx
+ 		+ +
Camera Man Blurred gradient dy
+
+

Choose 0.1 as the threshold for binarizing:

+ + + + + +
+ +
Camera Man Blurred gradient dx binarized
+ 		+ +
Camera Man Blurred gradient dy binarized
+
+

Combining the two edge images using the algorithm above creates this:

+
+ + + + + + + +
+ +
Camera Man Blurred edge
+ 		+ +
Camera Man Blurred edge binarized
+
+
+
+ +
+ + +

Since both applying the Gaussian filter and taking the finite derivative are linear operation, the order of the + operations does not matter. I could take the finite derivative on the Gaussian filter first and then apply the + derivative of Gaussian on the image.

+
+ + + + + + + + +
+ +
Gaussian Kernel
+ 		+ +
Gaussian Kernel dx
+ 		+ +
Gaussian Kernel dy
+
+
+
+
+ + + + + + + + + +
+ +
Camera Man Gaussian dx
+ 		+ +
Camera Man Gaussian dy
+
+

Still choose 0.1 as the threshold for binarizing:

+ + + + + + + + +
+ +
Camera Man Gaussian dx binarized
+ 		+ +
Camera Man Gaussian dy binarized
+
+

Combining the two edge images using the algorithm above creates this:

+ + + + + + + + +
+ +
Camera Man Gaussian dx binarized
+ 		+ +
Camera Man Gaussian dy binarized
+
+ +
+

We can see either ways create mostly identical results except for some minor differences. The two binarized + images are exactly the same under the same threshold. This is becasue finite derivative and Gaussian filter are + both linear process. Hence the order of the two manipulation does not matter.

+
+ + + + + + + +
+ +
Gaussian then Derivative
+ 		+ +
Derivative of Gaussian
+
+
+
+
+ +

Part 2: Fun with Frequencies!

+

2.1 Image Sharpening

+

The idea of image sharpening is to add more high frequency content to the image. Adding high frequency content + can make the image look like having more details. To extract the high frequency content, we first create a blurred + version of the image (represent to low frequency part). Then we subtract the original image with the blurred + image. Finally, we could add the extracted high frequency content to the original image to create a sharpen image. +

+ + +
+

High frequency of the R, G, B channels of Taj:

+ + + + + + + + +
+ +
High frequency of R channel
+ 		+ +
High frequency of G channel
+ 		+ +
High frequency of B channel
+
+
+
+
+

Adding high frequency to the original image:

+ + + + + + + + +
+ +
Taj Original
+ 		+ +
Taj Sharpen
+ 		+ +
+
+ +
+

High frequency of the R, G, B channels of Parrot:

+ + + + + + + + +
+ +
High frequency of R channel
+ 		+ +
High frequency of G channel
+ 		+ +
High frequency of B channel
+
+
+
+
+

Adding high frequency to the original image:

+ + + + + + + + +
+ +
Parrot Original
+ 		+ +
Parrot Sharpen
+
+
+
+

I then blur the image first and try sharpen the image to recover its initial status.

+
+

Blur the Parrot using Gaussian filter:

+ + + + + + + + +
+ +
Parrot Original
+ 		+ +
Parrot Blurred
+
+
+ + +
+

High frequency of the R, G, B channels of Parrot:

+ + + + + + + + +
+ +
High frequency of R channel
+ 		+ +
High frequency of G channel
+ 		+ +
High frequency of B channel
+
+
+
+
+

Adding high frequency to the original image:

+ + + + + + + + + +
+ +
Parrot Blurred
+ 		+ +
Parrot Blurred Sharpen
+ 		+ +
Parrot Original
+
+
+

The result shows that although sharpening the blurred image can make it looks sharper than the blurred one. + However, it's still pretty obvious that it lacks some details compare to the original image. The lost information + cannot be recovered by image sharpening.

+ +
+ +

2.2 Hybrid Images

+

This part of the project is to create hybrid images using the method introduced in the paper published in the + SIGGRAPH + 2006 by Oliva, Torralba, and Schyns. The key idea is to merge the high frequency of one image and the low frequency + of + another image. When looking close to the image, the high frequency part is clear, thus look like the first image. + However, when looking in a far distance, we can only see the low frequency part, so it looks like the second image. +

+

My approach is still use the Gaussian filter to get the low frequency part of an image. Then use the original image + minus the low frequency part to get the high frequency part.

+ +

In Nutmeg and Derek's example, I will create an image using the high frequency of Nutmeg and low frequency of + Derek.

+
+ + + + + + + + + + +
+ +
Nutmeg
+ 		+ +
Derek
+
+ +
Nutmeg Frequency Domain
+ 		+ +
Derek Frequency Domain
+
+
+
+
+
+ + + + + + + + + + + + + +
+ +
Nutmeg Low Frequency
with Gaussian sigma 10
+ 		+ +
Nutmeg High Frequency
_
+ 		+ +
Derek Low Frequency
with Gaussian sigma 13
+
+ +
Nutmeg Low Frequency
Frequency Domain
+ 		+ +
Nutmeg High Frequency
Frequency Domain
+ 		+ +
Derek Low Frequency
Frequency Domain
+
+
+

Diving into the frequency domain, the algorithm works successfully on extracting the high and low frequencies. From + Nutmeg's high frequency image, the high frequency part is enhanced and low frequency part is reduced. In the Derek + Low Frequency's frequency domain, only low frequencies are left in the image.

+ +
+ + + + + + +
+ +
Nutmeg Derek hybrid
+
+
+
+

In the next example, I blended me with a dog I don't know the name who trying to steal my food.

+
+ + + + + + +
+ +
Me
+ 		+ +
Dog
+
+
+
+
+
+ + + + + + + + + +
+ +
Dog Low Frequency
with Gaussian sigma 10
+ 		+ +
Dog High Frequency
_
+ 		+ +
Dog Low Frequency
with Gaussian sigma 13
+
+
+
+ + + + + + +
+ +
Dog and Me hybrid
+
+
+

In another example, I tried to blend two emoji together.

+
+ + + + + + +
+ +
Smile Face
+ 		+ +
Cool Face
+
+
+
+
+
+ + + + + + + + + +
+ +
Smile Face Low Frequency
with Gaussian sigma 10
+ 		+ +
Smile High Frequency
_
+ 		+ +
Cool Low Frequency
with Gaussian sigma 13
+
+
+
+ + + + + + +
+ +
Cool Face and Smile Face hybrid
+
+
+ + +

2.3 Gaussian and Laplacian Stack

+

A image stack is a group of images with the same resolution. However, filters are applied to different layer of + stack.

+

To create a Laplacian Stack, I need to first create a Gaussian stack. The Gaussian stack implementation is simply + taking the original image as the first layer and applying the Gaussian filter on the previous layer as the next + layer. Then for the Laplacian stack, each layer will be the same layer of the Gaussian stack minus the next layer of + the Gaussian stack. For the last layer of the Laplacian stack, it just uses the same image as the last layer of the + Gaussian stack.

+ + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + +
+ + + + +
+ + + +
+ + + +
+ + + +
+ +
Apple
+ 		+ +
Orange
+
+
+ +

2.4 Multiresolution Blending

+

The key point of the multiresolution blending is to blend different frequency with differnt masks to create a + seamless effect. The approach first crate a Gaussian stack for the mask and Laplacian stacks for two images that + are going to stack. After that, for each layer of the stack, the algorithm blends them using `blended_img = (1 - + mask) * img1 + mask * img2`. As a result, different frequencies are blended with suitable masks.

+
+ +

The oraple is blended from an apple image and an orange image from the previous part.

+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ +
+ +
+ +
+ +
+ +
+ +
Oraple
+
+
+ +
+

I find a set of images collected by Pablo Rochat interesting. I decided to blend some images from his photo collection

+
+

Fishliper: blend the image of a fish with a slipper

+
+ + + + + + + + + + + + + + + + + + + +
+ +
Slipper
+ 		+ +
Fish
+
+ +
Mask
+ 		+ +
Fishliper
+
+
+

Sauceglasses: blend the image of some sauce with a sunglasses

+
+ + + + + + + + + + + + + + + + + + + +
+ +
Sunglasses
+ 		+ +
Sauce
+
+ +
Mask
+ 		+ +
Sauceglasses
+
+
+ + + + + + + + + \ No newline at end of file