PCA has been widely used in a variety of fields and situations. It has been applied to reduce the dimensions of data and to extract their most meaningful components and structures. Dynamic speckle sequences are collections of frames taken at a given temporal rate. The light source is coherent and speckle patterns change dynamically due to variations in the observed specimen. However, as far as the object is globally static along the measurement, frames show correlation among them. This correlation is analyzed and sliced by PCA. From a geometrical point of view, PCA can be seen as a rotation transformation from an -dimensional coordinate system (the original frames) to a new -dimensional coordinate system, where the new frames show no correlation. A dynamic speckle sequence has frames, , each one having pixels organized as a rectangular image. Each rectangular frame is rearranged as a vector containing as many elements as pixels in one frame, . This reshaping of data is fully reversible and rectangular frames can be retrieved. To properly evaluate the desired parameters, we have transformed each vector containing each frame to zero-mean vectors. This is easily done by subtracting the mean of each frame to each one, , where denotes averaging. The vectors in the sequence are organized as an matrix, and its covariance matrix, , is calculated as an matrix. The elements of the variance matrix of the original data set, , are calculated as the scalar product of the zero-mean vectors representing frames and . Due to the character of the acquired dynamic speckle images, this matrix is not diagonal and shows correlations among original frames. PCA works by diagonalizing this covariance matrix, .4,5 The results obtained from PCA are eigenvalues (appearing at the diagonal of the new diagonalized matrix), ; principal components, (uncorrleated frames); and eigenvectors, (describing the transformation from original to-and-from principal component frames), where runs from 1 to (we have chosen to denote the index for the principal components and for the original frames). Eigenvalues describe the variance associated with each principal component. Most of the time it is more useful to evaluate the relative weight of each eigenvalue, . The algorithm involved in the calculation of the principal components organizes eigenvalues in decreasing order (). Then the relative importance of a given principal component decreases when increasing its rank, . Principal components, , can be interpreted as speckle pseudoimages showing no correlation among them. Finally, eigenvectors can be organized as an unitary matrix describing the rotation between the original frames, , and the principal components, . In summary, PCA allows to move from a correlated set of frames (original speckle sequences) to an uncorrelated sequence (principal components).4,5 This transformation is described as Display Formula
(1)where are the components of eigenvector .