Multiscale wavelet analysis of edges : issues of uniqueness & reconstruction

Signal analysis has traditionally been the domain of Fourier-based techniques. Although it is very powerful, Fourier analysis suffers from the fundamental lack of time locali-sation and therefore is not really suited to the analysis of signals containing localised features such as transients and edges. A recent technique, known as Wavelet analysis, has attracted a remarkable amount of activity and interest. Researchers in diverse fields, ranging from applied mathematics, signal and image processing to seismic analysis have begun to consider Wavelet-based methods. This is due to the fact that Wavelet analysis possesses dual localisations in both time and frequency domains. It thus provides a multiresolution approach to signal analysis such that a signal with localised features either in time or in frequency domains can be analysed effectively. In addition, Wavelet analysis allows the flexibility of choosing a particular wavelet and a particular scheme to suit a particular application. A common type of wavelet scheme is the orthonormal or biorthonormal wavelet trans-form typically used in coding and compression. This scheme can be implemented with a fast algorithm and it allows a non-redundant decomposition of a signal. Unfortunately, the scheme suffers a fundamental problem of not being translation invariance. This means that a shifted copy of the same signal will produce totally different transformed coefficients. For applications such as transient analysis and object recognition, this behaviour is clearly undesirable. The aim of this study is to find a wavelet scheme that overcomes this lack of translation invariance. Besides having the invariant property, the scheme is also required to give an efficient and meaningful representation of a signal. During the early nineties, a group led by Mallat introduced the Wavelet Transform Modulus Maxima Representation. Not only is the representation invariant to translation, it is also meaningful in the sense that it corresponds to the multiscale edges in a signal. Mallat presented extensive empirical evidence to show that the representation is complete and that the original signal can be reconstructed from the representation to a high degree of accuracy. Nevertheless, a theoretical analysis of the uniqueness of the representation is lacking. In this thesis, the uniqueness issue is first thoroughly investigated in term of the irregular sampling theory and the frame theory. The relationship between the uniqueness of the representation and the locations of the modulus maxima of the continuous dyadic Wavelet Transform of a signal is discussed. It was shown that the representation is unique if and only if the set of wavelets that produces the representation forms a frame in the signal subspace. Three methods that can be used to reconstruct a one dimensional signal from its Wavelet Transform Modulus Maxima Representation are described. They are the projection onto constraint spaces method, the frame based method and the Singular Value Decomposition based direct reconstruction method. All three methods will give an exact reconstruction if the representation is unique. The frame based method and the Singular Value Decomposition based method will give a least square optimal solution when the representation is not unique. For the projection based method and the Singular Value Decomposition based method, it is possible to utilise the a priori information in the representation to further improve the least square solution. Simulation results show that all three reconstruction methods are capable of giving accurate reconstruction. The studies on the uniqueness of the Wavelet Transform Modulus Maxima Representation and the reconstruction of a signal from this representation is extended to deal with two dimensional signals. Similar to the one dimensional case, the two dimensional representation is unique if and only if the set of wavelets that produces the representation forms a frame in the two dimensional image subspace. A two dimensional projection based reconstruction algorithm that is capable of providing accurate reconstruction is described. Finally, as an application, a contour based image coding scheme based on the Wavelet Transform Modulus Maxima Representation is proposed and studied. Coding experiments indicate that it is possible to achieve a compression ratio of more than 70:1 while still giving an acceptable reconstruction.