Advanced Mathematical and Computational Tools in Metrology by P. Ciarlini, M. G. Cox, F. Pavese, G. B. Rossi

By P. Ciarlini, M. G. Cox, F. Pavese, G. B. Rossi

This quantity collects refereed contributions in response to the shows made on the 6th Workshop on complex Mathematical and Computational instruments in Metrology, held on the Istituto di Metrologia "G. Colonnetti" (IMGC), Torino, Italy, in September 2003. It presents a discussion board for metrologists, mathematicians and software program engineers that might motivate a more beneficial synthesis of talents, functions and assets, and promotes collaboration within the context of ecu programmes, EUROMET and EA tasks, and MRA requisites. It includes articles by means of a major, all over the world workforce of metrologists and mathematicians all in favour of dimension technological know-how and, including the 5 past volumes during this sequence, constitutes an authoritative resource for the mathematical, statistical and software program instruments essential to glossy metrology.

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The criteria to recognize the absence/presence of features (edges) are discussed in the next section. 28 The 2D data fitting procedure of the second step constructs each piece of s as a tensor product of splines of the same order, namely a bilinear spline in homogeneous sub-domains, a bicubic spline, otherwise. In each one of the two cases, the optimal knot displacement is identified by means of an heuristic procedure described in [2] to solve the non linear mimimization problem. 3. Calibration of the tests for feature recognition Let us emphasize that the exact recognition of the type of a sub-domain Dj enables the method to use the appropriate spline space to get an invisible restoration, since the special feature in Djis preserved.

This mathematical procedure emphasizes the horizontal contours to get the thickness measurements in the Grey-level representation. However, neglecting every vertical detail might cause misleading information in predicting a reasonable position of the border when it crosses the vertical anomaly. We now consider the anomaly that occurs in the left side of the image at ks = 33 that has constant width w = 5. To get the measurements using the digitized image ( 4096 Grey-level scale), we need to accurately compute the horizontal contours: we apply the three-step method, before the multi-level wavelet analysis with compression, developed in the PTB Laboratory.

And T = V(X) = T(X)TT(X),and V = aT ax,. The matrix T satisfies TTT + TTT = V and can be determined by differentiating the algorithm2 to compute T : Suppose now that V = I Set P ( i , j ) = V ( i , j ) ,for all i 2 j. (k, k ) = P ( k , k ) / ( 2 T ( k 7k ) ) and P ( 1 : k-l,k)=(T(l:k-1,k)-T(k,k)T(l:k-l,k))/T(k,k). j For j = k - 1 : -1 : 1, set T(l :j , j ) = T(l : j , j ) - T(1 : j , k ) T ( j ,k ) - F(1: j , k ) T ( j ,k ) . This algorithm can be easily vectorized to compute all the partial derivatives of T ( X )simultaneously.

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