A deconvolution method for the reconstruction of underlying profiles measured using large sampling volumes

Xiong, Y.-S.; Withers, P.J.

doi:10.1107/S0021889806012210

Figure 2
The right-hand panels, (b), (d) and (f), show the underlying smooth function, $[\sin(\pi x)/(\pi x)]$ (dashed line), along with the actual microstructurally fluctuating ( $[\sigma_{f}]$ = 0.1) values (open symbols). The convolution of the smooth profile with the sensor is shown in the left-hand panels, (a), (c) and (e) (dashed line), along with the noisy ( $[\sigma_y]$ = 0.1) measurements (full symbols). In (a) a conventional $[\chi^{2}]$ minimization has been applied (solid line) to give an estimate of the underlying profile (solid line) in (b). In (c) a $[0.01\|{\bf w}\|^{2}]$ weight decay has been added to the $[\chi^{2}]$ cost function, while in (e) a Bayesian algorithm has been used to estimate the underlying function (dashed line) in (f). The number of points at which the underlying function f is specified, N (= 200), is not important so long as the integrations in equation (2)

are accurate enough numerically; for clarity, only a fraction of the 200 points are shown (by open symbols) in the right-hand figures. The insets in (c), (d), (e) and (f) compare the predicted values (triangles) and the data points (circles) against the noiseless true profile. All the predictions (triangles) of the Bayesian algorithm have associated error-bars which have been omitted for clarity, but a typical error-bar is shown at the bottom-right of the inset (f).

JOURNAL OF
APPLIED
CRYSTALLOGRAPHY

ISSN: 1600-5767

Volume 39| Part 3| May 2006| Pages 410-424

doi:10.1107/S0021889806012210

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