This chapter has described in overview the background useful in order to under- stand the proposed method and it has focused mainly on the concepts related to shape and its analysis.
Shape is an important feature of the human visual system (HVS) as it is one of the basic features used to describe perceived objects, both natural and man-made. Dryden and Mardia discuss shape [45] as the geometrical information that stays when location, scale and rotational effects are filtered out from a particular object. Small complements this idea by further adding that the remaining information should be invariant to transformations [145]. Another important concept is shape descriptor that, in general, can be regarded as a set of numbers produced to represent a given shape feature. Such descriptors attempt to quantify the shape in ways that agree with human concepts of shape or with task-specific requirements [107]. Shapes can be described by a series of finite points locating points on the sample to be analysed, called landmarks. Shape space can be considered as the collection of all possible shapes [77]. The fundamental space for this work is Kendall’s shape space, since it provides a complete geometric setting useful for the analysis of Procrustes distances among sets of landmarks [44]. Another impotant concept isshape signature, that can be defined as any 1D function that represents 2D areas or boundaries [181]. Shape signatures are able to capture the perceptual feature of the shapes, and therefore are often used as a preprocessing to other feature extraction algorithms, such as Fourier descriptors.
Morphometry can be defined as the measurement of shape. Morphometry allows the transformation of visual information into its mathematical representation. As previously stated, Morphometry and Shape analysis but, since Morphometry can be regarded as an antecedent to Shape Analysis, the difference relies on the fact that Shape Analysis is related to the study of the geometrical properties of ob- jects. Statistical Shape Analysis (SSA) is a geometrical analysis from a set of shapes in which statistics describe the geometrical properties from similar shapes [45]. Two important aspects of SSA are introduced as well, Procrustes methods
Chapter 2. Shape, Shape Analysis and Statistical Shape Analysis 55 and PCA. Procrustes analysis is a form of statistical shape analysis used to anal- yse the distribution of a set of shapes. The name Procrustes refers to a bandit from Greek mythology who made his victims fit his bed either by stretching their limbs or cutting them off. Principal Components Analysis is a multivariate statis- tical technique that provides means to identify patterns in data, and to highlight similarities and differences in them [146]. The next section of the chapter was dedicated toPoint Distribution Models that are models for representing the mean geometry of a shape and some statistical modes of geometric variation inferred from a training set of shapes. In this section as well, descriptions of how the con- tours are extracted from images, how registration analysis is carried out, how the statistical shape variation is captured, and how the shapes are reconstructed, was given. To obtain the shape contours, first the image they come from is binarised and denoised, then the shape is traced and the boundary coordinates are obtained. In practice, preprocessing is done using standard Matlab functions from the Image Processing Data Toolbox. Given a set of shapes, the registration analysis consists of estimating an affine warp by transforming any shape to some prototype shape. The warps are searched by determining rigid body transformation parameters for each shape from the set, and the parameters are kept to reconstruct the shapes accordingly. Under correspondence free conditions, after all the shapes are reg- istered, the set is ready to capture the statistics of the set [32]. The modes of variation, or the ways in which the points of the shape tend to move together, can be found applying PCA to the deviations from the mean. Finally, any shape of the set can be approximated using the mean shape and a sum of the modes. The next two chapters are devoted to the study of the best way to find partitions out of contours. Chapter three illustrates the first attempt to generate parti- tions through the use of Fractal Dimension in a Bayesian framework (MCMC). Chapter four presents a more stable solution to the problems presented by the Markov Chain Monte Carlo algorithm and shows an alternative solution using the Curvature Scale Space method.
Chapter 3
Fractal Analysis and Markov
Chain Monte Carlo Simulation
for Contour Localisation
3.1
Introduction
Being able to partition a contour into a set of meaningful local parts has many ap- plications in machine vision, image retrieval, terrain classification or handwriting recognition. As presented in Chapter 1, several methods for contour partitioning have been developed but none of them take into account the fractal nature of the objects such as brain contours, since none of them consider types of these curves. This section presents a method of modelling fractal curves, such as the boundary of brain white matter, and partitioning them into segments having equal fractal dimension. This might lead to a better set of partitions for SSA, one of the aims of this work (refer to section 1.4). Since the solution space, for a given number of contour points and a required set of partitions is very large, a Bayesian framework of reversible-jump Markov Chain Monte Carlo (MCMC) is used together with a sampler based on the Metropolis-Hastings test. Details of the algorithm are pre- sented as well as the theoretical concepts behind it. Results on simple contours
Chapter 3. Fractal Analysis and MCMC Simulation for Contour Localisation 57 (animal silhouettes) and space-filling brain contours are shown, along with the convergence characteristics of the method. Limitations and the contributions of the method are discussed at the end of the chapter.