Wavelets for image de-noising
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Description
This research has been discontinued.
Thanks to the combination of a nice theoretical foundation and the promising applications, wavelets have become a popular tool in many research domains. In fact, wavelet theory combines many existing concepts into a global framework. This new theoretical basis reveals new insights and throws a new light on several domains of applications.
One of these applications is signal- and image de-noising. We consider two important classes of methods to suppress noise using wavelets. Algorithms of the first type start with a wavelet transform. Then they manipulate the wavelet coefficients. Finally, an inverse transform yields the result. The second type of algorithms are based on a probabilistic a priori model for noisy and noisefree wavelet coefficients. These methods use the input data as an observation and apply Bayes' rule to compute the a posteriori expected value of the noisefree coefficients. Inverse transform gives the output.
The manipulation of the wavelet coefficients in a procedure of the former type is based on a classification. This classification is often binary: the coefficients are divided into two groups. The first group contains important, regular coefficients, while the other group consists of coefficients that were catalogued as ``too noisy''. These two groups are then processed in an appropriate way. Noisy coefficients are often replaced by zero. To classify wavelet coefficients, the procedure needs a criterion to distinguish noisy from regular coefficients. In our research, we simply use the absolute value of the coefficients as a measure of regularity. Coefficients below a certain threshold are thus replaced by zero.
how to choose the threshold. The objective is a result as close as possible to the unknown noisefree data. We use a generalized cross validation procedure to estimate this optimal threshold. This method is fast and does not require an estimate of the amount of noise (the variance). An original theoretical argument justifies the combination of generalized cross validation and wavelet thresholding. We examine the assumptions and conditions that were needed for this proof and try to extend the method for more general cases.
Recent developments is wavelet theory include the so called lifting scheme. In the first instance this is an alternative and faster algorithm for a classical wavelet transform. However, the structure of this lifting scheme allows to extend the classical algorithm to cases with non-regular grids and to adapt the floating point algorithm to an integer version. Wavelet transforms that map integers to integers are no longer linear and therefore, threshold assessmment policies are to be re-examined. In the case of non-regular grids, the absolute value is no measure for importance anymore. Because of the grid structure, small coefficients may have a wide impact, while large coefficients could carry a lot of noise. We investigate how to modify a threshold procedure to deal with this structure.
Wavelet thresholding relies on a binary decision: a coefficient is affected by noise or is sufficiently clean. Moreover, the classification has a local character: each coefficient is considered separately. To construct a more continuous approach, one can try to compute the probability for a coefficient of being sufficiently clean according to the criterion. Especially for 2D-structures, like images, we introduce a geometrical a priori model of clean coefficients and use the input coefficients as an observation to compute a posteriori probabilities with Bayes' rule. The design of an appropriate a priori model is another purpose of this project.
There is also collaboration with LUCIAD on the implementation of wavelet technology in a JAVA environment for image compression.
