Directional Multiscale Statistical Modeling of Images

Duncan D.Y. Po, Minh N. Do

Research output: Contribution to journalConference articlepeer-review


The contourlet transform is a new extension to the wavelet transform in two dimensions using nonseparable and directional filter banks. The contourlet expansion is composed of basis images oriented at varying directions in multiple scales, with flexible aspect ratios. With this rich set of basis images, the contourlet transform can effectively capture the smooth contours, which are the dominant features in natural images, with only a small number of coefficients. We begin with a detail study of the statistics of the contourlet coefficients of natural images, using histogram estimates of the marginal and joint distributions, and mutual information measurements to characterize the dependencies between coefficients. The study reveals the non-Gaussian marginal statistics and strong intra-subband, cross-scale, and cross-orientation dependencies of contourlet coefficients. It is also found that conditioned on the magnitudes of their generalized neighborhood coefficients, contourlet coefficients can approximately be modeled as Gaussian variables with variances directly related to the generalized neighborhood magnitudes. Based on these statistics, we model contourlet coefficients using a hidden Markov tree (HMT) model that can capture all of their inter-scale, inter-orientation, and intra-subband dependencies. We experiment this model in the image denoising and texture retrieval applications where the results are very promising. In denoising, contourlet HMT outperforms wavelet HMT and other classical methods in terms of both peak signal-to-noise ratio (PSNR) and visual quality. In texture retrieval, it shows improvements in performance over wavelet methods for various oriented textures.

Original languageEnglish (US)
Pages (from-to)69-79
Number of pages11
JournalProceedings of SPIE - The International Society for Optical Engineering
Issue number1
StatePublished - 2003
EventWavelets: Applications in Signal and Image Processing X - San Diego, CA, United States
Duration: Aug 4 2003Aug 8 2003

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics
  • Computer Science Applications
  • Applied Mathematics
  • Electrical and Electronic Engineering


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