Variation and Calibration Error in Electronic Imaging

In acquiring digital images, electronic systems not only detect optical signals but also convert them into a digital form for further image processing and exchange. Practical systems can introduce error during color calibration, and when acquiring image scene information. For large populations it is often assumed that the error can be modeled as a random variable having a zero mean. In the case of a single color instrument, camera or scanner, however, error due to deterioration of a physical standard, optical filter or detector will introduce a bias into the measurement or image data. This error is modified as the signals are transformed (processed) into their final form. An error-propagation method is shown to describe the influence of the dataprocessing path on the magnitude of bias error. This approach is related to the propagation of image noise, or variance. The analysis is applied in examples drawn from color-measurement and digital image processing. Introduction When making color measurements, we can think of instrument uncertainty as introducing an error into the (color) signal of interest. The same can be said of the acquisition of a digital image where pixel-to-pixel noise and color calibration errors can occur. We often assume that the error can be modeled as a random variable having a zero mean. For a single color instrument, however, a calibration error can introduce a consistent bias. Color-measurement signals and digital images are usually transformed between several common color spaces. Therefore, an analysis of the way signal transformations influence the magnitude of color-signal error is useful when comparing performance with system tolerances. These results can then be used to set limits for both instruments and physical standards. One tool for the evaluation of error or variation in signal processing is error-propagation analysis, where specific signal processing steps can be described by their corresponding transformations of the error statistics. The most common use of this method is the propagation of the second-order statistics, variance and RMS error. Livens addressed the combination of stochastic instrument errors and signal quantization. He showed that the combined variance is found by adding the effective variance of each noise source, as for independent sources. Gonzalez, et al., evaluated the practical limits to color accuracy in terms of the color-difference metric, * E94 ∆ . They compared results with and without color management, and pointed out requirements for ICC device profiles. In this paper, we address the propagation of first-order statistical error, bias. This can be applied to a consistent error component due to, e.g., instrument drift, deterioration of a physical standard, or signal quantization. Bias Error If an observed signal is subject to error, it can be expressed as the sum of true value and bias,