Filters and Frequency

Mengyang Zhang

Finite Difference Operator

part 1.1

To calculate the partial derivatives of of D_x and D_y, I first turned these two vectors into a 2d matrix of [[1, -1]] for D_x and [[1],[-1]] for D_y. Then apply the scipy.signal.convolve2d with the original picture

To calculate the gradient, the magnitude of the gradient is (dx ** 2 + dy ** 2) ** 1/2

part 1.2

Compared with the previous ones, the DoG filters gives us more smooth edges, since guassian itself is a low-pass filter and blur the images
It also shows less noise compared with the previous one, since guassian will blur the noises

DoG filters ones

part 2.1 Image Sharpening

The process is to blur the image with the gaussian first and get low frequecies
Then substract it from the original image to get higher frequecy part.
Add the higher frequecies part onto the original image to get the sharpening effect.

part 2.2 hybrid images

In order to hybrid two images:
Blur one images with the guassian filters
Subtract the blurred image2 from the original image2 to get the higher frequecy part
Combine the two modified images

FFT analysis for the previous 3 pictures

More examples here

Part 2.3 Guassian and Laplacian Stacks

Guassian Stack production process:

First layer is the original image.

Use guassian blur the previous layer repeatedly

Laplacian Stack production process:

Get the guassian stacks for the original image first

Each layer of the laplacian stack is the result of subtract two adjacent gaussian layers

The last layer is the last layer of the gaussian stack

Mask for the orange and apple are a 2d matrix with the same shape, and left half 0, and the other half 1, or vice versa

General multiresolution process

calculate laplacian stacks for the images

calculate the guassian stacks for the masks

combine them for each level

add the laplacian stacks to get the final image

Lakers and moon