Since the advent of digital computers and the recent advances in hardware and software development, statistical computing has become increasingly important.
Image processing is the manipulation of digital values contained in an image for subsequent processing and interpretation. For more general surveys the reader is referred to Andrews (1977) and Sonka (1999) [4], [77].
A digital image is simply the digital form of an image recorded by a sensing de-vice. Examples of sensing devices are scanners, cameras, industrial radiographs, infrared sensors, multi-spectral remote sensing and radar remote sensing. A 2-dimensional plan can be partitioned in regular polygons in three ways. These partitionings are the regular square tessellation, the regular triangular tessella-tion and the regular hexagonal tessellatessella-tion. By the term image is here meant 2-dimensional data, which constitute a series of square picture elements (pixels) arranged in a regular pattern of rows and columns. Such a partitioning is also called a square tessellation and the polygons correspond to pixels.
In case the data are obtained from e.g. an optical sensor the pixel has a digital value (grey-level) representing reflected light from the area, which the pixel covers. In radar remote sensing the pixel has a complex number representing both magnitude and phase of scattered microwaves from a resolution cell. In polarimetric SAR data a pixel may represent the complex scattering matrix. In case several values at each pixel are available the term multi-variate data or multi-dimensional image is used. In such ap-dimensional image each pixel has pvalues covering the same geographical area.
The computer vision process can be separated in three levels: low, intermediate and high. Low-level processing (or early vision) deals with raw pixel data and includes e.g. restoration, segmentation, edge detection, texture analysis and optical flow. Low-level processing is invariably data driven and nothing or little is known about the objects in the scene. The intermediate level of processing is concerning grouping the output from the low-level processing into e.g. lines.
1.3 Digital image processing 7
High-level processing is object oriented and aims at extracting symbolic features such as the recognition of e.g. characters in a handwritten letter or wetlands in SAR data. Some knowledge about the objects in the scene is therefore required.
Because the test sites are relatively small compared to the pixel spacing it is crucial for this investigation that as much information as possible is preserved about the geophysical and biological properties of the test sites. Unfortunately, the price paid for moving from one level to the next is a loss of information.
Since detail preservation in this work is of major concern the image processing in this thesis is therefore restricted to the low-level domain.
1.3.1 Contextual constraints
Textural information plays an essential role in human interpretation and analysis of visual data. Although no precise definition of texture exists texture can, according to Haralick (1979), be described assomething consisting of mutually related elements [32]. Texture can therefore be perceived as a region that is spatially homogeneous in some sense. Within the context of remote sensing such texture regions could be e.g. cities, forests or grasslands.
The type of information or features that can be extracted from image data depends strongly on the scale at which the features are detected [48]. For example large object such as cities and forests are in satellite images observed at coarse scales, whereas smaller objects such as houses and trees are observed at finer scales.
This implies that at low level of detail (large scales) smaller objects are sup-pressed and likewise at high levels of detail (small scales) all information is retained. Because detail preservation is of great importance in this thesis the images are observed at the smallest scale possible corresponding to the resolu-tion of the images. Here resoluresolu-tion refers to the size of the smallest objects that can be identified.
The presence of speckle considerably reduces the interpretability of the SAR im-ages and consequently some kind of spatial filtering is used routinely to increase the signal-to-noise ratio. Some popular representatives of SAR speckle filters are the median, Lee, Frost and Kuan filters. In Donget al. (2000) these speckle filters are examined in terms of texture preservation and in Rees and Satchell (1997) the effect of median filtering on SAR images is reviewed [24], [68]. Based on the Frost filter kernel a new method for SAR speckle reduction is proposed by Zhang et al. (2002) [89]. The simplest of these techniques is the median filter, which has edge-preserving properties but is unsuited for texture
preser-vation. The Lee, Frost and Kuan filters and the method proposed by Zhanget al. are on the whole efficient in speckle reduction but they have a tendency of slightly distorting the texture and oversmoothing fine details. Because detail preservation is of major concern in this thesis these filters do not seem suitable for speckle reduction in this investigation. More sophisticated techniques for speckle reduction are therefore applied.
As early as 1962 Chow proposed a method for using contextual information in pattern recognition [15]. Here the dependence between the pixels and their neighbours was used for character recognition. Chow utilized that neighbouring pixels tend to have similar intensities and such regularities are in a probabilistic framework conveniently described by MRF [34].
Markov Random Field (MRF) is an extension of the 1-dimensional Markov process to 2-dimensions and has attracted much attention in the image process-ing community. Hassner and Sklansky (1980) first proposed MRF as a statistical spatial interaction model for digital images [34]. One reason is that MRF pro-vides a general and natural way of modeling spatially correlated image pixels.
Another reason for using MRF is due to Hammersley and Clifford (1971) who established an equivalence between the local properties of MRF and the global properties of Gibbs distributions. This MRF–Gibbs equivalence gives an explicit formula for defining the joint distribution of MRFs through clique-potentials [6].
Most vision problems can be formulated in a general framework called image labeling. Here the task is to assign a label for a pixel, which in some sense is optimal. A label can belong to several categories depending on the problem we are trying to solve and a label set may be categorized as being continuous or discrete. For edge detection, for example, the label set is discrete containing the labels edge or non-edge, for image segmentation the label set is containing classes or regions and for image restoration it is containing grey-levels. In im-age segmentation the aim is to partition an imim-age into homogeneous exclusive regions, where each region is assigned a unique label. Here the discrete label set could e.g. be grey-levels, colour or texture. For image restoration the aim is to estimate the true signal from a degraded or noise-corrupted image using knowl-edge about its nature [77]. Since the nature of the noise in SAR data is known in advance, the method to be used to search for the scene in the polarimetric EMISAR data that best describes the observed records is image restoration.
The label set here includes both discrete and continuous grey-levels.
In a Bayesian framework the most successful criterion in optimization-based MRF modeling is the Maximum A Posteriori (MAP) estimate. The MRF-MAP framework for solving vision problems was formulated by Geman and Geman (1984) and later the subject is addressed by e.g. Besag (1986), Dubes and Jain (1989), Besag (1989) and Carstensen (1992) [29], [7], [25], [8], [11]. Our