Contourlet

The contourlet transform has a fast implementation based on a Laplacian pyramid decomposition followed by directional filterbanks applied on each bandpass subband.

However, the ability of 1-D transform processing of the intrinsic geometrical structures, such as smoothness of curves, is limited in one direction, then more powerful representations are required in higher dimensions.

[2] The Laplacian pyramid (LP) decomposition only produce one bandpass image in a multidimensional signal processing, that can avoid frequency scrambling.

This double filter bank structure of combination of LP and DFB is also called as pyramid directional filter bank (PDFB), and this transform is approximate the original image by using basic contour, so it is also called discrete contourlet transform.

[4] The reason for this lies in the up-sampling and down-sampling present in both the Laplacian Pyramid and the directional filter banks.

[4] Though the contourlet and this variant are relatively new, they have been used in many different applications including synthetic aperture radar despeckling,[5] image enhancement[6] and texture classification.

According to the authors there were some properties that they desired with this transform such as: perfect reconstruction, a sharp frequency response, easy implementation and linear-phase filters.

And the second stage of the wavelet-based contourlet transform is still a directional filter bank (DFB) to provide the link of singular points.

Therefore, the HMT model, that captures the highly non-Gaussian property, is used to get the dependence on neighborhood through the links between the hidden states of the coefficients.

There are two contributing factors to the aliasing, the first is the periodicity of 2-D frequency spectra and the second is an inherent flaw in the critical sampling of the directional filter banks.

Though it fixes all of those issues, this method requires more filters than the original contourlet transform and still has both the up-sampling and down-sampling operations meaning it is not shift-invariant.

This method of image enhancement significantly outperformed the nonsubsampled wavelet transform (NSWT) both qualitatively and quantitatively.