The multiresolution Fourier transform : a general purpose tool for image analysis
The extraction of meaningful features from an image forms an important area of image analysis. It enables the task of understanding visual information to be implemented in a coherent and well defined manner. However, although many of the traditional approaches to feature extraction have proved to be successful in specific areas, recent work has suggested that they do not provide sufficient generality when dealing with complex analysis problems such as those presented by natural images. This thesis considers the problem of deriving an image description which could form the basis of a more general approach to feature extraction. It is argued that an essential property of such a description is that it should have locality in both the spatial domain and in some classification space over a range of scales. Using the 2-d Fourier domain as a classification space, a number of image transforms that might provide the required description are investigated. These include combined representations such as a 2-d version of the short-time Fourier transform (STFT), and multiscale or pyramid representations such as the wavelet transform. However, it is shown that these are limited in their ability to provide sufficient locality in both domains and as such do not fulfill the requirement for generality. To overcome this limitation, an alternative approach is proposed in the form of the multiresolution Fourier transform (MFT). This has a hierarchical structure in which the outermost levels are the image and its discrete Fourier transform (DFT), whilst the intermediate levels are combined representations in space and spatial frequency. These levels are defined to be optimal in terms of locality and their resolution is such that within the transform as a whole there is a uniform variation in resolution between the spatial domain and the spatial frequency domain. This ensures that locality is provided in both domains over a range of scales. The MFT is also invertible and amenable to efficient computation via familiar signal processing techniques. Examples and experiments illustrating its properties are presented. The problem of extracting local image features such as lines and edges is then considered. A multiresolution image model based on these features is defined and it is shown that the MET provides an effective tool for estimating its parameters.. The model is also suitable for representing curves and a curve extraction algorithm is described. The results presented for synthetic and natural images compare favourably with existing methods. Furthermore, when coupled with the previous work in this area, they demonstrate that the MFT has the potential to provide a basis for the solution of general image analysis problems.