ABSTRACT
Nomenclature ............................................................................................................................. 537 References.................................................................................................................................... 538
Infrared thermography is widely used to measure the thermal diffusivity of materials. A thermographic experiment usually consists in illuminating the front face of a sample by a light beam (point like, line, or other motifs) and detecting the thermal response of the sample on either its front or its rear face. The light beam intensity can be Dirac-like, modulated, pseudorandom, etc. The thermal behavior of the sample is observed using an infrared camera with a focal plane array of infrared detectors. The development of infrared video cameras with fast data acquisition (thousands of images=s) and high lateral resolution (tens of micrometers) provides powerful tools for fast materials testing. The drawback of these cameras is the huge amount of noisy data to be treated. To overcome such a difficulty, experimental conditions are usually chosen so that simple analytical solutions of the heat conduction problem exist. For instance, the diffusivity of homogeneous materials can be easily studied by the so-called slope method when illuminating a thin but large enough sample placed in a vacuum chamber by a point-like or a line-like intensity modulated laser beam (see, e.g., Mendioroz et al. 2009 and references within). Otherwise (i.e. heterogeneous materials, other lighting motifs), the estimation process (heat conduction model inversion) becomes a tricky task: computing time and memory resources required when using standard approaches, i.e., diffusivity estimation by minimization of a chosen residuals norm defined over the whole spatial domain, are generally huge while results are very sensitive to noise when using point-by-point least squares estimation approaches (see Chapter 7). In such a sense, mathematical tools allowing significant reduction of the data set dimension, as well as noise control and parsimonious estimations, could be an interesting alternative for extending characterization of materials based on Infrared thermography to situations with unknown analytical solutions. Proper orthogonal decomposition techniques are widely used for multivariate data reduc-
tion inmany areas of application. The reduction starts by choosing an appropriate orthogonal basis allowing identification of some few dominant components (referred to as dominant directions, eigenfunctions, or modes). A low-dimensional approximate description of the whole set of data is thus obtained by projecting the initial high-dimensional set on the dominant eigenfunctions. The choice of the basis makes the main difference among methods. When dealing with regular signals, those arising out of spectral decomposition of the
energy matrix (or covariance matrix) of the multivariate data give the best results. This means that it provides the lowest dimension for a given approximation precision or, alternatively, the best precision for a given dimension. Such a method has been developed about 100 years ago by Pearson (1901) as a tool for graphical data analysis and redeveloped several times since then in different areas of application (Hotteling 1933, Karhunen 1946, Loève 1955), that it has assumedmany names such as principal components analysis (PCA), Karhunen-Loève decomposition (KLD), singular value decomposition (SVD), etc. PCA=KLD=SVD is very commonly used today in image processing and signal processing problems for compression and noise reduction (Deprettere 1988). It is also widely used for signals classification, data clustering, and information retrieval problems (Berry et al. 1995, Everitt and Dunn 2001, Everitt et al. 2001). Powerful model reduction techniques based on PCA=KLD=SVD have been also proposed for low-dimensional description of problems described by partial differential equations, mostly in the field of turbulent flows (Berkoz et al. 1993, Holmes et al. 1996). In thermal analysis, SVD-based methods have been developed for efficient reduction of linear and nonlinear heat transfer problems (Ait-Yahia and Palomo del Barrio 1999, 2000; Palomo del Barrio 2000; Dauvergne and Palomo del Barrio
sources (Park and Jung 1999, 2001, Palomo del Barrio 2003a,b). For a fairly comprehensive introduction to PCA=KLD=SVD, we recommend the books by Jolliffe (1986) and Deprettere (1988). For more details on the mathematics and computation, good references are Golub and Van Loan (1996), Strang (1998), Berry (1992), and Jessup and Sorensen (1994). This chapter focuses on the use of KLD techniques in association with infrared thermog-
raphy for reliable and parsimonious thermal characterization of materials. In Section 14.2, the KLD of infinite-dimensional and finite-dimensional problems is defined. Functions and signal considered are, respectively, space-time-dependent functions and multivariate time series. In the framework of thermal analysis, functions represent the thermal field while multivariate time series are data taken from thermal field sampling. The property of KLD to provide the closest r-dimensional approximation for an infinite-dimensional problem or the closest rank-r approximation for a rank-n (n> r) matrix is used. A numerical example illustrates the application of KLD for this problem. In Section 14.4, measurement noise propagation through KLD is analyzed and two KLD-based filters are described. The application of KLD for data filtering is highlighted using a numerical example. A KLDbased method for reliable estimation of the diffusivity of homogeneous materials is described and tested in Section 14.5, while Section 14.6 focuses on thermal characterization of heterogeneous materials. This chapter includes unpublished experiments and results.
