1. Completeness: Apart from the data structure, the completeness of a given deformable-template based shape representation depends on how the group acts on the shape space. In the strictest sense, a deformation group Ψ of a shape space can be considered complete if its action is regular. That is, for every c1 and c2 in a given shape space there exists exactly one ψi ∈ Ψ
such that ψic1 = c2. Alternatively, completeness can be framed as an
optimization problem in cases where several possible deformations exist that can map c1 to c2– the distance metric in deformation based representation
is mostly defined in terms of optimization.
2. Metric: The general distance framework for a group based shape defor- mation analysis is formalized in [You98], for both finite or infinite dimen- sional representations. Assuming the group action is transitive (there exists ψi ∈ Ψ for every c1, c2 ∈S such that ψic1 = c2), the similarity between two
shapes is measured by a deformation that minimizes an objective func- tion called effort functional – the distance between the group’s identity and the given deformation, see [You98, You10a]. Similar to the landmark and immersion based approaches variation due to scale, location and rotation are cleared from the considered shape space. Subsequently, the distance between a template shape ct and a given shape c1 is formulated as
ds(c1, ct) = min
ψ∈Ψ{dΨ(e, ψ) : c1 = ψ(ct)}, (2.27)
where dΨis the metric in the deformation space. In the infinite dimensional
setting, distance in the deformation space is closely related with distance in the original shape space. Moreover, (2.27) can further be formalized to address optimal parameter estimation of shape representations. Computa- tional cost of (2.27) is dependent on the type of deformation space, and distance metric defined on it.
2.4
Summary
In Chapter 2, we have described three distinct types of curved shape representa- tion approaches and discussed the advantages and challenges of each approach.
Despite the apparent incompleteness of the landmark based approach, there are several landmark based applications in medical image analysis problems mainly because of its computational efficiency [DM98]. Meanwhile, the immersed func- tion based representation has been used in specific classification problems in biol- ogy [LSZ10,LKSM14], where human annotation of the landmark points are either unavailable or are unreliable. Nevertheless, most immersion based representations are solved with numerically intensive computational schemes, except for a few special cases [SKJJ11, MMSY07]. Moreover, shapes are assumed to be locally smooth thus the distance function is sensitive to local noise. Alternatively to both landmark and immersion based representations, deformable-template mod- els emphasize the deformation space, which usually is a group, for curve represen- tations. Deformable-template models are generally good modelling approaches to represent a class-specific shape variation. The hypothesis is, as in active shape representation, that shape variations in a class-specific shape category are small with respect to a template shape. Consequently, the deformation space can be described with significantly fewer variables as opposed to the general deformation space– we will discuss PCA, which can be treated as deformable-template model, more specifically in Chapter5. Nevertheless, deformable-template models are not purely a shape representation framework, since the representation is conditioned on the shapes category and has to refer to a template shape for distance compu- tation. However, the general framework of quantifying distance as the minimum of the effort functional (2.27) is independent of the template notion; in Chapter3
we will show the relationship between the proposed metric and (2.27).
In this thesis, using a novel finite dimensional curved shape representation, we address the high computational cost and noise sensitivity associated with distance functions in immersed function-based representations. Furthermore, the dependence of finite-dimensional representations, landmark-based approaches, on manually annotated points is addressed.
Chapter 3
Deformation based curved shape
representation
3.1
Motivation
In this chapter, we present a novel curved shape representation framework that draws its motivation from the three distinct shape representation approaches dis- cussed in Chapter 2. The representation presented here aims to embody some of the advantages of the landmark based, immersion based, and deformable-template based representations in a single framework.
Similar to the landmark-based representation, we aim to develop a represen- tation with a computationally efficient distance function that is based on the approximation of a continuous curve by a finite set of points. Nevertheless, we do not depend on annotated points for the approximation as is done in landmark- based approaches. Instead, we use arc length based point sampling as an approxi- mation of a continuous curve with a known curve parametrization. Consequently, general questions which usually are framed in the immersed function based set- ting, e.g., deformation transportation and optimal parameter estimation, can be asked in the proposed curve representation, and approximate solutions can be provided without incurring the mathematical and computational complexity that comes along with infinite dimensional space representations. Finally, sim- ilar to deformable-template based approaches our representation is based on a deformation group. Hence, we exploit the advantages of the group structure in
our proposed framework. Nevertheless, we do not refer to a template shape. As a result, our framework can be used to address model-free problems like shape retrieval from a given dataset. However, distinctively to our representation, the notion of distance emphasizes the difference between intrinsic characteristics of curved shapes. In most of the earlier approaches discussed in Chapter2, distance is measured by the absolute deformation of each point– the difference between the two distance notions will be discussed in detail later. In that regard, what is pro- posed in this chapter is similar to the Frenet-Serret frame presented in [KMG98]. In summary, the main goals of the proposed curved shape representation are: 1) to develop a robust representation that leads to computationally efficient (closed form) geodesic curve and geodesic distance equations, 2) to develop a framework to handle optimal parameter estimation of curves and deformation analysis so that the representation can be used to later develop statistical models of a shape category.
The chapter is organized as follows: in Section 3.2, we present the proposed curved shape representation by starting from the data structure in Section 3.2.1
leading to the proposed representation in Section3.2.2. In Section3.3, we discuss distance metric and geodesic curves in the proposed curved shape representation space. In Section 3.4, the properties and interpretation of the proposed distance function is presented. Experimental results and comparisons are presented in Section3.5. The chapter concludes with final remarks in Section 3.6.