PSI - Issue 5

Francisco Barros et al. / Procedia Structural Integrity 5 (2017) 1260–1266 Francisco Barros et al./ Structural Integrity Procedia 00 (2017) 000 – 000

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assuming a discrete behaviour of the monitored surface. If stereoscopy is required, image acquisition is performed simultaneously from two or more distinct perspectives. In order to track the displacement of a subset, an iterative procedure is often adopted where a correlation criterion between a fixed subset in the reference image and a “moving” subset in the deformed image is evalua ted until a maximum is detected [1]. Different correlation criteria can be chosen as a similarity metric; the three most robust and widely used of which are: the zero-mean normalized cross-correlation (ZNCC), the zero-mean normalized sum of squared differences (ZNSSD) and the parametric sum of squared differences [2]. Other methods, however, instead of performing computations directly using light intensity values, rely on the evaluation of images in the frequency domain, by making use of 2D-Fourier Transforms and their mathematical properties, as proposed by Lu et al. [3]. The main advantage of these algorithms is the fact that they are capable of providing direct displacement results, thereby avoiding time-consuming iterative calculations. They are mainly based in somehow analogous Fourier Transform theorems, i.e. the Phase-Shift theorem, the cross-correlation theorem or the convolution theorem. The basic principles behind image correlation through frequency domain calculations are presented below. Let us consider the two-dimensional functions ( , ) and ( , ) , with dimensions × , that represent the grayscale image intensity in the pixel with coordinates and , for the reference and a deformed image, respectively. If a discrete approach is taken in consideration, as it is somehow done for finite element methods, and if only a translation transformation exists between the two images, then ( , ) = ( + , + ) , (1) where and represent translation or displacement values for both directions between the images. A statistical metric in the spatial domain, such as the ZNCC coefficient previously mentioned would be computed as ( , ) = ∑ ∑ [( ( , )− ̅)∙( ( , )− ̅)] =− 01 =−0 1 √∑ ∑ ( ( , )− ̅) 2 =− 01 =−0 1 ∙∑ ∑ ( ( , )− ̅) 2 =− 01 =−0 1 , (2) where ̅ and ̅ represent the images ’ overall mean. This operation is successively computed within an iterative procedure that can become time-consuming if and represent just two small subsets of a large number that constitute the entire number of images to analyse. However, if a spatial frequency spectrum approach is considered, the displacement computation can be reduced exclusively to non-iterative operations with the ability to determine displacement values of magnitudes up to half of the subset window dimensions. Let ℱ denotes the Fourier transform, and the operators " ⋆ " and " ∙ " the cross-correlation and point-wise multiplication, respectively. The cross-correlation theorem (3) states that: ℱ{ ⋆ } = ℱ̅̅̅{̅ ̅̅} ∙ ℱ{ } = ℱ{ } ∙ ̅ℱ̅̅{̅ ̅̅̅} , (3) where ℱ̅̅̅{̅ ̅̅} represents the complex conjugate of the Fourier transform of . Upon normalization, expression (3) returns the cross-power spectrum of and according to the Wiener-Khintchin theorem [4]. On the other hand, the discrete Fourier transform (DFT) of these signals can be described by ℱ{ ( , )} = ( , ) = 1 ∑ ∑ ( , ) − 2 ( + ) =− 01 =− 0 1 , (4) and ℱ{ ( , )} = ( , ) = 1 ∑ ∑ ( , ) − 2 ( + ) =− 01 =− 0 1 = ( , ) 2 ( + ) , (5)