Fun with Filters and Frequencies!

Part 1.1: Finite Difference Operator

To begin, the cameraman image is convolved with both finite difference operators D_x and D_y. The gradient magnitude is calculated by taking the magnitude, or norm, of these gradient components: grad_mag = np.sqrt(dx**2 + dy**2). This gradient magnitude image is binarized with a selected threshold, where values above the threshold qualify and are displayed as edges.

Part 1.2: Derivative of Gaussian (DoG) Filter

Seeing that the edges detected above are rather noisy, the cameraman image can be blurred before performing the process, as shown below. The differences that we see include the edges being thicker and smoother, as well as less noise on the man, camera, and background.

Blurring and computing the gradient magnitude can also be completed in a single convolution: by convolving the blurring/gaussian filter with D_x and D_y, the resulting DoG filters can be directly applied to the original cameraman image.

Part 2.1: Image "Sharpening"

Images can be "sharpened" using the unsharp masking technique. This can be done in two ways:

• Subtract the blurred image from the original and add alpha times of the remaining high frequencies back to the original
• Combine the above strategy into a single convolution operation called the unsharp mask filter

Below are results for different values of alpha, showing the progression of the original to sharpened images.

The following progression tries first blurring, then sharpening an image. The sharpened image clearly loses a lot of the original's detail, is noisy, and contains artifacts.

Part 2.2: Hybrid Images

Hybrid images (are static images! that) change with the viewer's distance to the image. From the blending of high frequencies of one image with low frequencies of another, the viewer sees the first image when they are up close, and the second when they are far.

This is an example of a failure. Since the shapes of Eren's normal face and Eren's Titan face are different sizes, it was difficult to align the two images (the final attempt tries to align the noses), resulting in a less convincing hybrid image.

The following also demonstrates a Fourier analysis. The two filtered versions of the input images keep the respective high and low frequencies, which are then combined for the final hybrid image.

Part 2.3: Gaussian and Laplacian Stacks

Before multi-resolution blending, Gaussian and Laplacian stacks are created. Gaussian stacks involve repeatedly applying the Gaussian filter at each level of the stack, and Laplacian stacks involve subtracting a level of the Gaussian stack from the previous level. However, the final level of the Laplacian matches the final level of the Gaussian so that the original image can be reconstructed by collapsing the Laplacian stack.

Stacks of an apple:

Stacks of an orange:

Part 2.4: Finite Difference Operator

To perform multi-resolution blending, each level of the Laplacian stacks across two images are blended together. This blending works best with similar Gaussian blurring of a mask. The combined layers are finally compressed into the blended image.

The following blending example involves an irregular mask: