Issue 35

E. Dall’Asta et alii, Frattura ed Integrità Strutturale, 35 (2016) 161-171; DOI: 10.3221/IGF-ESIS.35.19 163 the problem of finding the correspondences deals with the minimization of a global cost function extended usually to all image pixels). Least Squares Matching Least Square Matching (LSM) method consists in the minimization of the squared differences of the grey values between patch and template. LSM uses a functional model, which is able to provide matches with sub pixel accuracy while accounting for some deformation of the patch block with respect to the template window. In this approach, the matching process consists in the estimation of the function parameters that maximize the similarity between the template and the (deformed) patch, using a least squares solution [16]. Given two image points, the least squares matching considers the two conjugate image regions as discrete two-dimensional functions: the template f(x 1 ,y 1 ) and the patch g(x 2 ,y 2 ). The patch is transformed applying both radiometric and geometric adjustments to obtain a new patch g’(x 2 ,y 2 ). The matching process establishes a correspondence minimizing the L2-norm of the residual vector e(x 1 ,y 1 ):       1 1 2 2 1 1 f x , y g x , y e x , y   (1) Radiometric changes (due to contrast and brightness variations of intensity values in the slave image) are modelled in the patch function as:       2 2 0 1 2 2 g x , y r 1 r g x , y      (2) where r 0 and r 1 are two parameters accounting for brightness and contrast changes in the slave image, respectively. Geometric corrections are considered by means of a geometrical parametric transformation:         2 2 2 1 1 2 1 1 g x , y g x x , y , y x , y  (3) The following affine transformation model is commonly used in DIC applications: 2 1 1 2 1 3 2 1 1 2 1 3 x a x a y a y b x b y b          (4) where (a 1 , a 2 , b 1 , b 2 ) model shape differences between patch and template, while (a 3 , b 3 ) are the shift parameters. Such transformation is considered as the optimal choice for taking into account the large anisotropic deformation of the slave image and for limiting, at the same time, the numerical complexity (and consequently instability) of the geometric model. While more simplified shape function models becomes rapidly not affordable to identify the exact localization of the template centre, even with isotropic scale deformations, higher order transformations require more observation to be appropriately estimated (and consequently a wider template area which can present strong localized deformation especially with composite high deformable materials). The very same behaviour is shown quantitatively in [17], even if in that case the matching algorithm is applied to planetary surface reconstruction. Radiometric and geometric correction parameters are then estimated solving, for   e x1, y1 = min, the following least squares system, obtained by substituting the transformed functions in Eq. 1:             1 1 1 1 0 1 2 1 1 2 1 1 0 1 1 2 3 1 2 3 f x , y e x , y r r g x x , y , y x , y g r , r a , a a , b , b , b      (5) The function   0 1 1 2 3 1 2 3 g r , r , a , a , a , b , b , b is linearized and the system is commonly solved by Gauss-Markov least squares estimation model. Semi-Global Matching The Semi-Global Matching method [15] performs a pixel-wise matching, considering both the image similarity and the displacement continuity, by the concurrent application of regularization constraints (in terms of adiacent pixels displacement). It realizes the minimization of a global cost function, combining matching costs along indipendent one- dimensional paths from all directions through the image. The costs extracted by each path, refered to a particular displacement value, are summed for each pixel and possible displacement (also referred as disparity) value. Finally, the