Gaussian pyramid

Gaussian or Laplacian pyramid, the Burt and Adelson pyramids or Gaussian and Laplacian pyramid called, are digital signal processing algorithms. They were 1981/83 introduced by Peter J. Burt and Edward H. Adelson in digital image processing to unify some known algorithms systematically. 1988, the basic idea of ​​this data structure by Stéphane Mallat and Yves Meyer was transferred to Functional Analysis. There it is known as multiresolution analysis ( MRA) of wavelet theory.

One of the characteristics of digital images, which are used to detect specific structures or image manipulation, including blur and sharpness. Information on these features can be found in the frequency bands. To determine the individual frequency bands, filter cores or the Fourier transform may be used, which is associated with a considerable computational effort. Alternatively, one uses a Gaussian - Laplacian pyramid.

Creating a Gaussian - Laplacian pyramid

To develop a Gaussian Laplacian pyramid, a first Gaussian pyramid to be constructed. The original image represents the lowest pyramid level G0 represents the next level G1 is calculated through a low-pass convolution () and halving the sampling points of G0. This process continues from step to step until the image reaches a size of 1 × 1 pixel. The low-pass convolution is realized by a convolution with a Gaussian bell. In practice, the image is convolved with a binomial filter. It is to be noted that the original image G0 must have a side length of pixels ( a picture can be divided into image blocks ). The at the end resulting Gaussian pyramid from the images of the various stages corresponds to a subdivision into images that represent a certain frequency component respectively. Each successor of an image has only a quarter of the pixels of its predecessor.

After a Gaussian pyramid has been constructed, a Laplacian pyramid is developed from it. A Laplacian pyramid level is achieved through the formation of the difference of two adjacent Gaussian pyramid levels. This is referred to as algorithm DoG ( Difference of Gaussian). Both stages must be the same size. Since this is not the case in the Gaussian pyramid, the smaller image to be brought by interpolation on the size of the other image. The individual Laplacian pyramid levels represent the sharpness of an image proportions. The image L0 in this case contains the highest frequency components.

After the Gaussian Laplacian pyramid level is formed and the individual layers may have been processed, the Gaussian Laplacian pyramid has to be reconstructed. Therefore the desired Laplacian pyramid levels and the highest Gaussian pyramid level are summed.

Application

The Gaussian - Laplacian pyramid is used to solve many image processing. A favorite application is the data compression. The data compression of an image high frequencies are removed, since they represent the least information content. For this, the highest Laplace pyramid levels are omitted. Furthermore, the support points and the individual quantized levels are represented by means of a quadtree. An advantage of this method is the smart decompression: the lower levels of the image pyramid are first decompressed because the low frequencies contain most of the information and require the least computational effort. The image is built up during decompression.

Another possible application is the mosaicing. Different images are fused together by the images are decomposed into image pyramids and weighted using a mask and summed. Then the image is reconstructed and possibly reprocessed. You edited the frequency bands separately, to avoid edge formation.

Gaussian - Laplacian pyramid also find application in surface or structure detection. Particularly convenient is the sharpness and blur images advantage. In this method, certain frequency bands of the image are determined, in order to subsequently process such information.