Signal and Image Alignment during the Application of Functional Data Analysis. Practical Examples of Chemometrics and Biometrics

The Functional Data Analysis (FDA) has been applied in the field of digital image processing in order to obtain low dimensional representations with a high discriminative power and a relatively low computational cost. The functional representation by means of basis functions expansion is used to standardize and simplify the mathematical and computational processing of images and signals, improving in many cases the accuracy in the classification process. Usually, as a result of instrumental errors in the process of capturing the information or the physical nature of the phenomena itself, shifts or deviations may occur. These problems arisen during the acquisition of images and signals might hinder their analysis and comparison. In order to solve these problems, an essential step in image or signal processing by means of Functional Data Analysis (FDA) is the alignment: usually a critical step to obtain good results in the whole process. The objective pursued in the alignment process is the correction of signal shifts in time, the position of the origin and axes of the coordinate system and scaling in space, so that a comparison can be established. The alignment process commonly requires a pre-processing for smoothing and noise removal, selection of the best patterns for alignment, and the implementation of appropriate methods for each data type. The methods and steps for alignment are different for each data type and are related to their own geometry. In order to assess the impact of the introduction of an alignment step in the classification accuracy, different methods are described and applied to several types of signals and images. The corresponding experimental results are shown and discussed for each case study.