Fun with Filters and Frequencies!

Part 1: Fun with Filters

Part 1.1: Finite Difference Operator

In this part, we find partial derivatives of the image by convolving the original image with the given operators (dx, dy). The second image is the gradient magnitude found using the convolution. By applying an appropriate threshold to the image, a binarized image consisting of edge was found. To preserve the essential contour, a relatively large threshold was used, which led to the noises shown at the bottom of the edge graph

Derivative of Gaussian (DoG) Filter

Part 1.2: Derivative of Gaussian (DoG) Filter

In the second part, a Gaussian filter is used to eliminate noises. A 2D Gaussian filter was constructed using its 1D outer-product. To remove noises, the original image is convolved with the gaussian filter first and then with the operators given (dx, dy). An alternative approach is to convolve gaussian kernel with (dx, dy) and then convolve the original image with the differentiated gaussian filter. Based on results, we could tell that these 2 approaches could significantly remove noises and they actually yield the same results

Image Straightening

The concept of gradient could also be used to straighten a tilted image. In this part, I used a scoring function based on the sum of horizontal edges and vertical edges that could be found in an image. After iterating through a set of feasible rotations, I kept the image with best score as a straightened image. However, this approach only works well for images with sharp and sensible edges. For images with no distinct edges, other metrics need to be used to find the correct rotations (Failure example: Galaxy)

Image "Sharpening"

Since the convolution between a gaussian filter and original image would yield the low-frequency part of an image, we could use it to perform a preliminary sharpening on a blurred image. By subtracting low frequency part of the image from the original image, we are left with only high frequency part of the image. Although we may lose some information during this process, the image would look sharper to our eyes.

Hybrid Images

In this part, we use gaussian and laplacian filters to generate hybrid images . After making perceptual alignment for two images, we extract low-frequency part from one of the image and the high frequency part of the other by applying corresponding filters. By combining parts with different frequencies, we would see one image at close while see another image in distance as our sensitivity to image frequency changes. This approach works well for images taken from the front and images with similar background brightness; however its effects is not ideal when images are taken from other angles or with various background brightness, of which more tweaking is neededto show a better result.

Gaussian and Laplacian Stacks

In this part, we are asked to find the gaussian and laplacian stack for images. The Gaussian stack is created by consecutively applying gaussian filter over previously created image while the laplacian stack is constructed by finding out the difference between images in the gaussian stack. In my code, a gaussian filter of size 20 and sigma 2.5 is used to create images listed below. Gaussian filter and the origin,

Multiresolution Blending

In this part, we are asked to use the gaussian stack and laplacian stack created in Part 2.3 to do the image-blending. To blend images, we need to find the laplacian stacks as well as the gaussian stacks for both images; we also need to find out the gaussian stack for the mask we use to blend images. With all information collected, we simply use the formula from Burt and Adelson's 1983 paper to compute comprehensive information needed to blend images and then wrap the summed results together with the last stack from both images' gaussian stack, which guides the contours for colors.

Conclusion

By interacting with images and techniques introduced in this project, I had a more thorough understanding towards human's visual perception: it relies heavily on the images' frequencies and orientations. By altering frequencies over different parts of the images, we would be able to change what audiences see and understand. Its applications over image compressing would also be profound.