| This dissertation develops the fuzzy transformation (FZT) theory and investigates its applications in image processing. The theory introduces a set of new concepts, i.e., the crisp spatial-rank (SR) space and fuzzy SR space and their components. The fuzzy transformation refers to the mapping that takes the components of the crisp SR space to the fuzzy SR space. These new concepts construct a compact framework to explicitly characterize the data based on the spatial, rank and spread information. A collection of deterministic and statistic properties of these new concepts is developed. Among them, the element invariant, order invariant and clustering properties are the most important, which show that the spread information embedded in the data can be explicitly expressed without distorting the spacial and rank information. These properties establish the principles of utilizing the FZT in developing new signal processing methods.; The new methods developed in this dissertation include the fuzzy generalizations of the conventional identity, weighted median (WM), and lower-upper-middle (LUM) filters, as well as a clustering method for one dimensional data. The filter generalizations jointly utilize the spatial, rank and/or spread information of the samples. Their properties and optimization are investigated, which show their advantages in noise removal, detail preservation, and robustness to various types of interference. The new clustering method effectively utilizes the change of sample spread that is caused by the clustering effects of the FZT. It detects the number of clusters and identifies the clusters in the data simultaneously, without relying on any prior knowledge or assumption about the number of clusters.; The generalized and newly developed methods are applied to a wide range of image processing applications, such as coding artifacts removal, noise smoothing, zooming, enhancing, and microarray image segmentation. The experimental results demonstrate the superiority of the FZT-based over their crisp fuzzy counterparts and/or state-of-art techniques reported in the literature.; In conclusion, FZT theory provides a systematical methodology to develop effective image processing tools based on the joint spatial-rank-spread information. The theory has great potential to be exploited in other signal processing applications, and entails interesting open questions that deserve future research efforts. |
