Introduction

In this project, we explore different ways of using frequencies to process and combine images interestingly. For example, similar to our phones, we can sharpen images by emphasizing high frequencies. We can also extract edges using finite difference operators. Hybrid images that look one way close up and another far away can also be created by combining the high frequencies of one image with the low frequencies of another. Lastly, images can be blended at various frequencies with the help of Gaussian and Laplacian stacks.

Fun with Filters

Finite Difference Operator

Finite Difference Operators help detect edges by highlighting regions where pixel intensity changes rapidly, which usually corresponds to edges or boundaries in the image. In two-dimensional images, the finite difference operator computes the derivative in horizontal (x) and vertical (y) directions. These derivatives are calculated using simple convolution filters, such as:

• Dx = [1, -1]: for detecting horizontal changes (vertical edges).
• Dy = [[1], [-1]]: for detecting vertical changes (horizontal edges).

When these operators are applied to an image, they produce two gradient images:

• The gradient in the x-direction highlights vertical edges.
• The gradient in the y-direction highlights horizontal edges.

By combining these gradients, we can produce an edge-detection image that reveals significant changes in intensity.

Results

Derivative of Gaussian (DoG) Filter

Blurred Finite Difference

A Gaussian filter is applied to the image to reduce noise and small details. I used a kernel of size 5 with sigma = 1 created using cv2.getGaussianKernel. I created a blurred version of the original image by convolving with the Gaussian and repeating the procedure in the previous part.

There are some very noticeable differences between the two methods. The binarized edges are much thicker and more prominent than before. Furthermore, some of the unnecessary details Details near the bottom of the pictures have been removed, and the fine details from the camera have also been removed.

Derivative of Gaussian (DoG)

Another method is to do the same thing with a single convolution instead of two by creating a derivative of Gaussian filters. the Gaussian kernels used to blur the image are instead convolved beforehand using the dx and dy kernels to produce dx_gaussian and dy_gaussian. We then convolute the original image with the new filters.

Comparison

As can be seen, the two images are identical, apart from a few slight differences in length and shape of minor edges.

Image "Sharpening"

Unsharp Masking Technique

We can sharpen images by adding more high frequencies to the image. We do this using the unsharp masking technique. We first use a low pass filter (Gaussian) to retain only the low frequencies. Then, we subtract the low frequencies from the original image to get only the high frequencies. These details are then emphasized in the final image through result = target + alpha * details where alpha is a sharpening factor.

Taj Mahal

For the Taj Mahal image, I used a sharpening factor of 2.0. As seen below, the edges of the building and the structures of the trees were emphasized.

Bee + Flower

For the Bee on Flower image, I used a sharpening factor of 5.0. As can be seen below, the edges of the flower, as well as the outline of the bee, were emphasized. The sharpened image is much more clear than the original image.

Resharpening a Blurred Image

For this example, I took a sharp image of a cat, blurred it, and then sharpened it again. As can be seen, the resharpened image looks sharper than the original image on the edges such as whiskers and eyes. But we still lose some detail in the fur of the cat.