Multidimensional Geometrical Signal Representation : Constructions and Applications

One of the key differences between one-dimensional (1-D) and N -D (N ≥ 2) signals is the geometrical information, an important and unique feature of multidimensional signals. The goal of this research is to develop a new set of theories and techniques in signal processing that can make better use of the intrinsic geometrical information in multidimensional data in a robust and efficient way. The primary technique we employ to approach this problem is multidimensional filter banks, due to their computational advantages, their design flexibilities, and very importantly, their direct connection with the theory of basis (nonredundant) and frame (redundant) decomposition of multidimensional signals. Directional information is an important geometric feature of multidimensional signals. As a result of a separable extension from 1-D bases, multidimensional wavelet transforms have very limited directionality. Furthermore, different directions are mixed in certain wavelet subbands. To solve this problem, we propose a simple Directional Extension for Wavelets (DEW) that fixes this subband mixing problem and improves the directionality. In a nutshell, the proposed directional extension provides an optional tool to efficiently enhance the directionality of multidimensional wavelet transforms. Numerical experiments show that certain wavelet-based image processing applications will benefit from this improved directionality. The contourlet transform was proposed as a directional multiresolution image representation that can efficiently capture and represent singularities along smooth object boundaries in natural images. Its efficient filter bank construction as well as low redundancy make it an attractive computational framework for various image processing applications. However, a major drawback of the original contourlet construction is that its basis images are not localized in the frequency domain. We analyze the cause of this problem, and propose a new contourlet construction as a solution. Instead of using the Laplacian pyramid, we employ a new multiscale decomposition defined in the frequency domain. The resulting basis images are sharply localized in the frequency domain and exhibit smoothness along their main ridges in the spatial domain. Numerical experiments on image denoising show that the proposed new contourlet transform can significantly outperform the original transform both in terms of the peak signal-to-noise ratio (PSNR) – by several decibels – and in visual quality, while with similar computational complexity.