# A0-SeanArietta

## Hybrid Images

Hybrid images are images that are comprised of two images in a special way that allows the visual appearance of the resulting image to change as the scale at which that image is viewed is changed. More simply, hybrid images are two (maybe more) images that are stacked on top of one another with the unique property that if you are close the hybrid image you see one image, but if you are far away you see the other.

Here is an example of two images and their corresponding hybrid image:

Notice that the second image shows up only once the scale of the hybrid image has been decreased. We can see why this effect is possible by examining the Laplacian pyramid.

### Visualizing the Hybrid Image via the Laplacian Pyramid

The higher levels in the pyramid correspond to the higher-frequency content in the image, while the lower levels in the pyramid contain the lower frequency content. Notice that the first image shows up in the higher levels of the pyramid while the second image comprises the lower levels. Since moving away from an image (or equivalently downsampling it) has the effect of discarding high-frequency content, we see the first image when we're close up and the second image when we are far away.

So why does the Laplacian pyramid reveal this structure? Recall that the Laplacian pyramid is built recursively filtering the original image by a Gaussian filter, subtracting the original image, and then downsampling the filtered image. The reason this results in the decomposition seen above is obvious if we inspect the Fourier transform of the relevant images:

First note that the Fourier transform of a Gaussian is also a Gaussian, and that convolution is just multiplication in the domain that we are visualizing above. Now notice the differences between the original image, the filtered image, and the Laplacian level image. The filtered image has support only near the center of the Fourier plot. This area corresponds to the low-frequency content in the image, which is why convolving with a Gaussian has a blurring effect. The Laplacian level has some support near the center of the plot, but it now lacks some information in that region (this is a bit tricky to visualize but you should be able to see the darkened area to some degree). Thus, the Laplacian level retains the high-frequency content at each level and is the reason why we are able to see exactly why the hybrid image behaves the way it does.