Fun With Filters

Finite Difference Operator

Gradient Magnitude image formula:
Dx = $\begin{bmatrix} 1 & -1 \end{bmatrix}$
Dy = $\begin{bmatrix} 1\\ -1 \end{bmatrix}$
Dx_im = Dx * image
Dy_im = Dy * image
Gradient Magnitude = $\sqrt{Dx\_im ^2 + Dy\_im ^2}$
To calculate the partial derivative in x and y, I convolved the image with finite difference operators Dx and Dy. The Dx operator will do a basic slope calculation between the im[y, x] and im[y, x+1]. The Dy operator will do the same but in the y-axis. Important to note that the Dx will find vertical edges and the Dy will find horizontal edges. The gradient magnitude represents how we want the average of these changes therefore we just do the L_2 Norm of Dx_im and Dy_in. There are many edges in the image, but we want just the prominent ones and not the noisy ones. To find the real edges without noise, I binarized the gradient magnitude by choosing an appropriate threshold.

Derivative of Gaussian (DoG) Filter

Differences between 1.1 and 1.2 (Gaussian blurring): In the blurred image, the edges are more prominent and there is less noise in the image. Since the high frequencies have been removed, the threshold value can be reduced since only the most prominent edges remain. In this case it went from 0.25 to 0.03.

The graphs "Blur then Derivative Convolve" and "Single Convolution" look basically the same with the prominent edges showing up in both of them.

Fun with Frequencies!

Image "Sharpening"

Hybrid Images

Failure: I had to bump the cat high frequencies up quite a bit for it to appear on the dog and it made the border appear. I did not know how to get rid of the border.

Gaussian and Laplacian Stacks

Multiresolution Blending

Reflection

This was really cool. I’ve played around with photoshop before, but I never knew it was using a bunch of math. The hard part was finding images that would align correctly.