Abstract
Shape is an inherent feature in everything around us and one that we, as humans, can process very efficiently. The question of how to analyse shape in an objective manner, however, is a more complex one. With continually improving and more readily available imaging technologies, there are an increasing number of fields in which it is of interest to have a more quantitative means of analysing the resulting images. Shape analysis is a rapidly growing branch of statistics that aims to meet these needs.
In contrast to the early methods of shape analysis that were based on distances or angles from an object, shape is now generally assessed in terms of the full geometry of the object. This does not necessarily mean an analysis of the object in its entirety however; a simplified representation of the surface is more often used. This representation has traditionally been in the form of a set of landmarks  points of anatomical or mathematical interest on an object that act as key descriptors of its shape. A key feature of these points is therefore that they are in positions that correspond across all images. Clearly this makes them a very powerful tool for analysis, particularly useful for comparison across shapes, and as such many methods have been presented for their analysis.
However, due to the fact that they are based on significant anatomical points and are more often than not manually placed on an image, landmark points tend to be fairly small in number. A major disadvantage to this type of approach is therefore that they give a very sparse description of the object of interest's shape. In order to improve on this, alternative methods have been developed that instead present the object in terms of a set of curves or, more recently, representative surface points. These representations give a richer description of shape but, provided they are created so that they also correspond across objects, can equally be analysed by means of the many existing landmarkbased techniques.
Nevertheless, although a surfacebased approach clearly utilises a greater deal of the shape information of an image and hence provides a more comprehensive analysis, it remains a far less common technique for the analysis of shape.
This thesis therefore aims to develop tools for the analysis of threedimensional surface data, with focus lying specifically in the field of medical imaging. Three distinct studies are conducted, each of which has its own questions of interest and hence necessary techniques. The first study presented is based on a cohort of unilateral mastectomy and reconstruction patients, where interest lies in evaluating the breast asymmetry that is present postsurgery. A novel method is presented for the creation of corresponding surface points, and an analysis of these is conducted based on an established approach to the study of asymmetry.
The second study then looks to investigate the `normal' patterns of facial growth that are seen in young children, specifically between the ages of 3 months and 5 years. From the multiple, longitudinal images that are available for each child, a set of corresponding surface representations are created by means of a wellestablished technique known as the Thin Plate Spline.
A principal components analysis is then applied to the surfaces in order to reduce the dimensionality of the data, and the resulting principal components scores are modelled by a linear mixed effects approach.
The third and final study presented is an investigation into the softtissue changes that are seen as a result of craniofacial surgery. A system is devised to index the location of all points on the surface, and this information is then used to model the changes taking place at various positions on the face. While most previous approaches to this problem have been based on complex finite element models, this study aims to investigate whether a simpler and more efficient statistical modelling approach can instead prove useful.
The diversity of these studies hints at the wideranging applications of shape analysis within the medical imaging field alone, as well as many of the issues that can arise in the analysis of surface data. While it is intuitive that more informed conclusions can be drawn through these surface based analyses than would be possible by a more traditional landmarkbased approach, it is also seen that the use of a surface representation allows for an improved visualisation and interpretation of the results. The techniques developed here are illustrated on specific medical applications, although it is hoped that they would prove similarly useful in a wider variety of shape analysis settings.
In contrast to the early methods of shape analysis that were based on distances or angles from an object, shape is now generally assessed in terms of the full geometry of the object. This does not necessarily mean an analysis of the object in its entirety however; a simplified representation of the surface is more often used. This representation has traditionally been in the form of a set of landmarks  points of anatomical or mathematical interest on an object that act as key descriptors of its shape. A key feature of these points is therefore that they are in positions that correspond across all images. Clearly this makes them a very powerful tool for analysis, particularly useful for comparison across shapes, and as such many methods have been presented for their analysis.
However, due to the fact that they are based on significant anatomical points and are more often than not manually placed on an image, landmark points tend to be fairly small in number. A major disadvantage to this type of approach is therefore that they give a very sparse description of the object of interest's shape. In order to improve on this, alternative methods have been developed that instead present the object in terms of a set of curves or, more recently, representative surface points. These representations give a richer description of shape but, provided they are created so that they also correspond across objects, can equally be analysed by means of the many existing landmarkbased techniques.
Nevertheless, although a surfacebased approach clearly utilises a greater deal of the shape information of an image and hence provides a more comprehensive analysis, it remains a far less common technique for the analysis of shape.
This thesis therefore aims to develop tools for the analysis of threedimensional surface data, with focus lying specifically in the field of medical imaging. Three distinct studies are conducted, each of which has its own questions of interest and hence necessary techniques. The first study presented is based on a cohort of unilateral mastectomy and reconstruction patients, where interest lies in evaluating the breast asymmetry that is present postsurgery. A novel method is presented for the creation of corresponding surface points, and an analysis of these is conducted based on an established approach to the study of asymmetry.
The second study then looks to investigate the `normal' patterns of facial growth that are seen in young children, specifically between the ages of 3 months and 5 years. From the multiple, longitudinal images that are available for each child, a set of corresponding surface representations are created by means of a wellestablished technique known as the Thin Plate Spline.
A principal components analysis is then applied to the surfaces in order to reduce the dimensionality of the data, and the resulting principal components scores are modelled by a linear mixed effects approach.
The third and final study presented is an investigation into the softtissue changes that are seen as a result of craniofacial surgery. A system is devised to index the location of all points on the surface, and this information is then used to model the changes taking place at various positions on the face. While most previous approaches to this problem have been based on complex finite element models, this study aims to investigate whether a simpler and more efficient statistical modelling approach can instead prove useful.
The diversity of these studies hints at the wideranging applications of shape analysis within the medical imaging field alone, as well as many of the issues that can arise in the analysis of surface data. While it is intuitive that more informed conclusions can be drawn through these surface based analyses than would be possible by a more traditional landmarkbased approach, it is also seen that the use of a surface representation allows for an improved visualisation and interpretation of the results. The techniques developed here are illustrated on specific medical applications, although it is hoped that they would prove similarly useful in a wider variety of shape analysis settings.
Original language  English 

Qualification  PhD 
Awarding Institution 

Supervisors/Advisors 

Place of Publication  Glasgow 
Publisher  
Publication status  Published  31 Jul 2012 
Keywords
 shape analysis
 threedimensional surface data
 medical imaging