Reconstruction

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Normal Image Reconstruction

An original image can be reconstructed from its lowest level of its Gaussian image pyramid and all higher levels of its Laplacian pyramid. This conceptually makes sense because we take the image's lowest frequencies with the deepest level of the Gaussian pyramid. Then each time we add a level of the Laplacian, we are subsequently adding back the second lowest frequencies, then the third lowest and so on until we have all of the original frequencies and thus the complete image.

Building a Hybrid Image

To build a hybrid image, we choose a certain cutoff to split spatial frequencies into two categories, either high or low. We then construct the Gaussian and Laplacian pyramids for two similar images. We choose one image to be the low frequency image, the one seen when viewing the hybrid image from a distance; and the other image to be the high frequency image, the one seen when viewing the hybrid image up close. Theoretically, the cutoff point discussed earlier is a specific frequency, but in practice, we can consider it to be a depth in the pyramid. We construct the hybrid image by rebuilding the low frequency image as normal, however, when we reach the cutoff, we stop taking images from the low frequency image's Laplacian pyramid and instead add them from high frequency image's Laplacian pyramid. In this manner, we add the high frequencies from one and the low frequencies from the other. In practice, we do not actually need to reconstruct the low frequency image from its lowest Gaussian but rather we can just pluck the image in its Gaussian pyramid right bellow the cutoff. Afterall, this image is precisely the sum of the lowest level Gaussian image and all subsequent Laplacians.

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