Restoration of Hardware Function of Spectral Colorimeters Using Regularization Methods

Abstract

The problem of determining the spectral characteristic of a controlled sample under conditions of limited a priori information using regularization methods is considered in the paper. A change in the state of the surface of optical elements significantly increases the light scattering, so it is necessary regularly to take into account the amount of scattered light in the light flux reflected from the surface and the measured and comparative samples. The conversion of the light flux into the electrical signal of the photodetector can also occur non-linearly. This requires the development of such measurement method that considers both the scattered light and various non-linearities of the measuring circuit. It is known that the mathematical model of measurement is described by the Fredholm integral equation of the first kind, its solution under the accepted assumptions is recommended to be sought in the form of a matrix equation using a recurring procedure. With regard to the fact that the estimation of the initial data errors in the equation is associated with certain difficulties, in the case under consideration, it is advisable to determine the regularization parameter based on the method of quasi-optimality. A characteristic disadvantage of the known analytical and experimental methods for determining the hardware function of a spectral device is that they do not take into account its change during operation. Since the actual hardware function of the device usually differs from the Gaussian curve, the use of hardware functions in the form of analytical dependencies does not always give the desired result, and for experimental methods, special equipment with a quasi-monochromatic radiation source is required. An algorithm for restoring the hardware function of a spectral device based on regular methods for solving ill-posed problems is proposed. The estimation of the matrix operator of the hardware function is proposed to be obtained on the basis of explicit least squares estimation algorithms. The expediency of choosing a value of the regularization parameter that minimizes the accepted characteristic of the accuracy of the solution is indicated.

Keywords

• colorimeters
• reflective samples
• transmissive samples
• photometric ball
• light scattering
• photometric scale
• calibration
• spectral characteristic
• hardware function
• fredholm equation
• incorrect tasks
• bayesian approach
• regularization parameter