Functional data analysis in shape analysis

Abstract

Mid-level processes on images often return outputs in functional form. In this context the use of functional data analysis (FDA) in image analysis is considered. In particular, attention is focussed on shape analysis, where the use of FDA in the functional approach (contour functions) shows its superiority over other approaches, such as the landmark based approach or the set theory approach, on two different problems (principal component analysis and discriminant analysis) in a well-known database of bone outlines. Furthermore, a problem that has hardly ever been considered in the literature is dealt with: multivariate functional discrimination. A discriminant function based on independent component analysis for indicating where the differences between groups are and what their level of discrimination is, is proposed. The classification results obtained with the methodology are very promising. Finally, an analysis of hippocampal differences in Alzheimer's disease is carried out.

Hippocampal shape analysis in Alzheimer's disease using functional data analysis

“…Many of them involve a type of preprocessing (sometimes implicit) of the functional data (see for a comparison of different methods for univariate functions and for multivariate functions with one argument). One possible regularization approach is to concentrate on the first few principal components (PCs) as in or some other finite‐dimensional representation of the data, as ICA, which has given better results than PCA and other alternatives in previous literature . So, once the hippocampi are represented in a basis (SPHARM), we can carry out the FDA, beginning with exploring the hippocampal variability by PCA and ICA, and using them for the discriminant analysis.…”

“…Functional linear discriminant can be used if the objective is also to discriminate between different groups and to understand the way in which these groups differ. The coefficients () for ICA components will constitute the feature vector used for the classification step, as made in for univariate functions and for multivariate functions with one argument. The scores for functional PCs can also be used, although in , the results were not so good as for ICA.…”

“…The coefficients () for ICA components will constitute the feature vector used for the classification step, as made in for univariate functions and for multivariate functions with one argument. The scores for functional PCs can also be used, although in , the results were not so good as for ICA. The use of PCA is quite common before the classifier is applied, such as in or (although they did not apply PCA on the coefficients but on the landmarks: points estimated on the surfaces).…”

Realtime Classification of Fish in Underwater Sonar Videos

“…need to be described with respect to their shape. The approach that is proposed in this paper is comparable with Epifanio and Ventura‐Campos () and references given there; see also Stoyan and Stoyan (). They made use of functional data analysis to describe shapes; see also Ramsay and Silverman (, ) or Kindratenko ().…”

Resampling Techniques for Estimating the Distribution of Descriptive Statistics of Functional Data

“…Recent computer technology facilitates the presence of functional data, whose graphical representations are in the form of curve (Hyndman and Shang, 2010), image (Locantore et al, 1999), or shape (Epifanio and Ventura-Campos, 2011). The monographs by Silverman (2002, 2005) have greatly popularized the functional data analysis (FDA), offering a number of appealing case studies and presenting many parametric statistical methods.…”