Research

Probability Models and Wavelet-Based Denoising

Non-Gaussian probability models are important for achieving high quality results in wavelet-based image and video processing. Such probability models are the starting point for the derivation of effective non-linear processing rules. The simplest method for wavelet-based denoising is a thresholding rule. More advanced approaches begin with a probability model for the wavelet coefficients and then obtain an estimator via Bayesian estimation techniques, such as the MAP or MMSE estimator. For denoising, several issues then arise: (1) What kind of probability distributions accurately represent the wavelet coefficients? (2) How do we estimate the parameters of that distribution from noisy data? (3) How do we estimate the noise-free wavelet coefficients from the noisy wavelet coefficients using this probability model?

We are developing non-Gaussian probability models and algorithms for wavelet-based denoising. While the multivariate Gaussian model is relatively easy to work with, it is usually not accurate for wavelet-domain modeling natural signals. This is because wavelet coefficients of natural imagery are generally (1) heavy-tailed, (2) approximately uncorrelated, and (3) strongly dependent on adjacent coefficients (in scale and spatially). These are characteristics the multivariate Gaussian can not achieve. We have developed fast and effective non-linear denoising algorithms based on multivariate Laplacian distributions and we are now working with mixtures of such distributions.

Based on this research, we have developed software for denoising digital X-ray images that is used in a dental X-ray sensor manufactured by AFP Inc, a company that sells X-ray machines and related imaging equipment. This software implements a fast real-time algorithm for the reduction of signal-dependent noise in digital dental X-ray images using a wavelet-based nonlinear signal processing algorithm.

Design of Wavelet Transforms

The simplest wavelet transform for multi-dimensional digital data is the critically-sampled separable wavelet transform. This transform uses a 1-D wavelet transform in each dimension and is the one that is conventionally used. However, one way to improve the performance of wavelet-based signal and image processing algorithms is to use specialized wavelet transforms in place of the conventional wavelet transform. There are several advances in the design of specific wavelet transforms that lead to substantially improved performance. For example, the undecimated wavelet transform, the steerable pyramid, and curvelet transform all give improved results in applications involving multidimensional data. At Polytechnic we have been designing several types of specialized wavelet transforms. Recently we have built upon the recently developed dual-tree transform, an oriented complex-valued wavelet transform shown to be highly beneficial for multi-dimensional signal processing. This transform has several advantages over the conventional multi-dimensional wavelet transform: (1) near shift invariance, (2) directional selectivity, and (3) improved energy compaction. We have also been developing expansive wavelet transforms (ones that transforms an N-point signal into M expansion coefficients with M > N) that possess properties not possible with a critically-sampled transform.

Design of Wavelet Frames

Design of Complex Wavelets

Design of Multiwavelets

Other Wavelet Design

Video Coding using a 3-D Motion-Selective Wavelet Transform

The transmission, storage, and related processing of video are important technical problems in today's society. Progress related to the efficient representation of video is of increasing importance for personal mobile electronics and for Internet applications. Video coding algorithms are often based on motion-compensated predictive coding. This project involves an original approach for video representation that is based on recent developments in wavelet theory: recently developed transforms designed to overcome basic problems that degrade the performance of the wavelet transform when it is applied to multidimensional data using the standard separable implementation.

The conventional separable 3-D wavelet transform is rarely used for video compression because it mixes 3-D orientations in its subbands; this artifact reduces the effectiveness of the separable transform for providing an efficient representation of video. However, the new 3-D wavelet transform is free of the mixing artifact and gives a meaningful multi-scale decomposition for video. With the new transform, it is more likely that the multiresolution framework, which has proved very effective for image compression and efficient feature extraction, can also be effectively applied to video representation. The new transform isolates motion in different directions in separate subbands, so the direction of motion can be inferred to some degree from the wavelet coefficients.

This research is in collaboration with Professor Yao Wang.

Filter Design