Feng Peng  , Wei Biao  , Pan Ying-Jun  , Mi De-Ling  , 
    The Analysis of Frequency Response for Pyramidal Directional Filter Banks,
    Asian Journal of Information Technology,
    Volume 7,Issue 6,
    2008,
    Pages 249-253,
    ISSN 1682-3915,
    ajit.2008.249.253,
    (https://makhillpublications.co/view-article.php?doi=ajit.2008.249.253)
    Abstract: Aiming at the frequency aliasing issue existing in Contourlet Transform, through analyzing Laplacian Pyramidal transform, making sure that neither of lowpass filter banks in Laplace pyramid transform satisfies Nyquist sampling theorem and cutoff frequency of stopband was over  /2 was obviously the basic reason to the frequency aliasing in Contourlet Transform. Based on this, a lowpass filter which satisfies Nyquist sampling theorem was designed and an Non-aliasing Contourlet Transform was proposed, namely N-Contourlet Transform. Experimental Results of hard thresholding denosing indicated that N-Contourlet Transform denoising not only give a 2 dB (  = 30) higher PSNR than that of Contourlet Transform, but also effectively suppressed scratch after the latter denoised, providing a better visual effect.
    Keywords: Contourlet transform;frequency aliasing;poly-phase representation