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  R E S E A R C H  -  D I R E C T I O N L E T S              

 Image processing using directionlets

• Standard wavelet transform (WT) is widely used in image processing (including JPEG-2000) because of:
• Sparse representation of general images
• Separable filtering and subsampling
• Simple (one-dimensional) filter design
• Low computational complexity

• However, the WT is isotropic and built only along horizontal and vertical directions. The corresponding basis functions are thus  symmetric. Thus, the representation of elongated and oriented features in images (like contours) is inefficient because of many large coefficients.

• Several transforms with different orientations have been proposed to address this inefficiency of the WT, like curvelets (Donoho et al.), contourlets (Do et al.), wedgelets (Donoho, Baraniuk), bandelets (Mallat et al.), etc.

• However, some of them require non-separable (two-dimensional) operations, oversampling, high computational complexity, or fail to satisfy perfect reconstruction.

• Directionlets retain the basic properties of the WT:
• Critical sampling
• Separability and simple filter design
• Low computational complexity

• Additionally, directionlets provide anisotropy and multi-directionality. The basis functions are elongated and oriented in different directions. Thus, oriented features can be represented more efficiently.

• Directionlets provide an efficient tool for non-linear approximation (NLA) of images. Image is segmented and different transform directions are chosen in each segment. Asymptotic decay of mean-square error of NLA of C²/C² images is faster (~N ^-1.55), as compared to the decay provided by the WT (~N ^-1), for N retained largest coefficients. The approximation performance using directionlets is also better for natural images with strong directionality.

• Application of directionlets in image compression is straightforward because of critical sampling. Lagrange-based optimization can be used to determine the optimal choice of transform directions and tiling.

• Directionlets improve the performance of the zerotree method (Shapiro) and space-frequency quantization (Xiong et al.) when applied instead of the WT.

• Selected references:


• V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, Space-frequency quantization for image compression with directionlets, IEEE Trans. on Image Proc, to appear.
• V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, P. L. Dragotti, Low-rate reduced complexity image compression using directionlets, IEEE Int. Conf. on Image Proc. (ICIP), Atlanta, GA, October 2006.
• V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, P. L. Dragotti, Directionlets: anisotropic multi-directional representation with separable filtering, IEEE Trans. on Image Proc., July 2006.
• V. Velisavljevic, Directionlets: anisotropic multi-directional representation with separable filtering, Ph.D. Thesis no. 3358(2005), LCAV, School of Computer and Communication Sciences, EPFL, Lausanne, Switzerland, October 2005.
• V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, P. L. Dragotti, Approximation power of directionlets, IEEE Int. Conf. on Image Proc.(ICIP), Genova, Italy, September 2005.