Predicting expected gray level statistics of opened signals

A four-point polynomial interpolation (using Neville's algorithm [4]) is then performed at integer range values to give the binned predicted output cumulative distributions. The limits of the grey level ranges are taken from the ranges of the corresponding actual distributions (thus, the total probabilities of the two mass functions are not necessarily equivalent). Given the actual and predicted distributions (i.e., the probability and cumulative mass functions for each case), the errors between these distributions corresponding to each (b, , T , c t) combination are summarized by a mean square error and a maximum absolute error describing the discrepancies between the cumulative mass functions. The average and worst case root mean square errors encountered between the actual and predicted cumulative mass functions are 0.015 and 0.023, respectively. One should note that the mean square error does not indicate whether there is a bias or other structure in the error (e. g., it may be heaviest in the steepest tail), so it must be used with caution. Since we are interested in using the cumulative distributions in future algorithms, this statistic would be useful if the predicted cumulative probability at any grey level could be described by some zero-mean random variable given a particular set f b, , T , c t g of input variables. Considering our data, this assumption does not seem appropriate. Instead, we consider the maximum absolute errors that we make in predicting the cumulative grey level distributions of opened signals. The maximum absolute error between the predicted and actual grey level cumulative distribution functions indicates the largest prediction error that one is likely to make at any point in the distribution. As with the mean square error, the maximum absolute errors found in the characterization experiments do not exhibit much structure as functions of the input parameters. Therefore, we consider the collection of maximum absolute errors for each prediction in the characterization as a single sample. The largest of these errors encountered in the characterization is 0.066, their mean is 0.036, and their standard deviation is 0.011 (where the range of the cumulative probabilities is between zero and one). These statistics may be used to predict an approximate upper bound on the dierence between the actual and predicted grey level cumulative distributions of a pixel in an opened signal, regardless of the values of the input parameters (provided that they're within the range of the characterization). …