2.2 | Laplacian Pyramid
The image pyramid is a canonical multi-scale representation
fundamentally established via a recursive method. This formation
ultimately leads to a pyramid of images with progressively decreasing
resolutions. Constructing the Laplacian Pyramid is contingent upon the
foundation of the Gaussian Pyramid, with the following steps:
1. Gaussian Pyramid Construction: At the heart of this step is the
down-sampling of the image, which effectively reduces its size. Firstly,
the image from the preceding layer undergoes convolution with a Gaussian
kernel. Subsequent to this convolution, down-sampling is performed with
a scale factor of 2.
2. Laplacian Pyramid Construction: Once the Gaussian pyramid is
acquired, up-sampling of its r -th layer is car- ried out,
followed by Gaussian convolution. This results in a new image whose
dimensions match those of the r-1 -th layer of the Gaussian
pyramid. By subtracting the newly procured image from the r-1 -th
layer of the Gaussian pyramid, we obtain the r-1 -th image of the
Laplacian pyramid. The aforementioned steps are iteratively executed
until images for all layers of the Laplacian pyramid are generated.
3. Reconstruction of the Laplacian Pyramid Images: The reconstruction
mechanism mirrors the procedures fro- m step 2. Post up-sampling of ther -th layer of the Gaussian pyramid and subsequent Gaussian
convolution f- iltering, the resulting image, when added to the
residuals stored in the Laplacian pyramid, yields a new image.
Replicating the above steps facilitates the reconstruction of images
from the Laplacian pyramid.