Gaussian cross correlation

The function consists of a transmission decrease Sgg (t) at time zero due to ESA and a transmission increase Sgj(t) which rises in a step-like manner with a small delay and reflects the onset of the product emission. The total transient signal is convoluted with a Gaussian cross correlation CC(t). 

The cross-correlation function G is thus the transformed image intensity pattern displaced with respect to the origin by the average displacement coordinates. The peak position is found with sub-pixel resolution by means of a Gaussian interpolator as described by Lourenco and Krothapalli [8]. 

Fig. 5. Scans of the pump-probe time delay between the laser pump pulse and the x-ray probe of the transient X-ray absorption signal of [Run(bpy)3]2+ (the x-ray energy is set at the minimum of the signal in fig. 20.b) over several tens of nsec (a) and in the psec time domain (b). The latter represents the cross-correlation signal between the 200 fs laser pulse and the 70-80 ps long x-ray pulse (indicated by the Gaussian-shaped derivative of the fit curve to the data).

One can (at one s peril) freeze the elements of the cross correlation matrix at their initial value (which is the frozen Gaussian approximation) or evaluate their time dependence from the closed set of equations of motion... 

The set of measurements of the concentrations of the species in S,- is obtained as a function of the externally controlled concentrations of the species Ii and I2 at each of the selected time points. Figure 7.2 is a plot of the time series for each of the species in this system. One time point is taken every 10 s for 3,600 s. The effects of using a much smaller set of observations are discussed later. The first two plots are the time series for the two externally controlled inputs. The concentrations of Ii and I2 at each time point are chosen from a truncated Gaussian (normal) distribution centered at 30 concentration units with a standard deviation of 30 units. The distribution is truncated at zero concentration. The choice of Gaussian noise guarantees that in the long time limit the entire state-space of the two inputs is sampled and that there are no autocorrelations or cross-correlations between the input species. Thus all concentration correlations arise from the reaction mechanism. The bottom five times series are the responses of the species S3 to S7 to the concentration variations of the inputs. 

This approach to overlap can be extended to m-chain overlaps also, which show a nonlinear dependence on m at the transition point [30]. This suggests that eventhough the size exponent is = 1/2 Gaussian like, there is more intricate structure than the pure Gaussian chain. Overlaps of directed polymers on trees have been considered in Ref. [31]. A case of cross-correlation of randomness (each pol3rmer seeing a different noise) has been considered by Basu in Ref. [32]. 

From an analysis of cross-correlation PIV, Olsen and Adrian [7] found that for light-sheet PIV, the width of the correlation peak Aso, taken as the diameter of the Gaussian function measured at a height of l/e times the peak value, can be expressed as... 

The simple cross-correlation estimator is used extensively in the form of a matched filter implementation to detect a finite number of known signals (in other words, simultaneous acquisition of multiple chaimels of known signals). When these deterministic signals are embedded in white Gaussian noise, the matched filter (obtained from cross-correlation estimate at zero lag, k = 0, between the known signal sequence and the observed noisy signal sequence) gives the optimum detection performance (in the Bayes sense ). 

FIGURE 18.8 Cross-correlation-based method to detect the presence of spikes or action potentials in multiunit activity. The continuous waveform with the action potentials is shown in (b). The shapes of the action potentials to be detected are known a priori (deteiministic signal). Using a cross-correlation of the known signal [templates (c) and (d)] with the measured signal, the presence of action potentials is indicated by a vertical line in (a). This approach is optimal in the presence of white Gaussian noise. 

In contrast to the principal component (PCA)-based approach for detection and discrimination, the cross-correlation or matched-filter approach is based on a priori knowledge of the shape of the deterministic signal to be detected. However, the PCA-based method also requires some initial data (although the data could be noisy, and the detector does not need to know a priori the label or the class of the diHerent signals) to evaluate the sample covariance matrix K and its eigenvectors. In this sense, the PCA-based detector operates in an unsupervised mode. Further, the matched-filter approach is optimal only when the interfering noise is white Gaussian. When the noise is colored, the PCA-based approach will he preferred. 

In this paper, we propose the use of cross-correlation of the vessel profiles to estimate the change in the vessel diameter using synthetic images generated from the twin-Gaussian profile. We tested the efficacy of this method by evaluating the narrowing of the simulated vessel diameter. 

Figure 4 Typical cross-correlation function used to measure the radial velocity. This figure represents the mean of the spectral lines of the star 51 Peg. The position of the Gaussian function fitted (solid line) is a precise measurement of the Doppler shift to an accuracy of about 15 m s The width of the cross-correlation function reflects the star s rotational velocity. Reproduced with permission of Macmillan Magazines Ltd. from Mayor M and Queioz D (1995) Nature 378 355.

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