## 5 Spatially-Resolved Spectroscopic Technique for Optical Property Measurement
### Algorithms and Instrumentation  
#### 5.2.1 Inverse Algorithms and Instrumental Design Parameters  
Accurate measurement of optical properties depends on the implementation of inverse algorithms and the instrumental design. A proper inverse algorithm needs to be developed and optimized for parameter estimation for the diffusion model. The inverse algorithm here refers to estimating the absorption and scattering coefficients from the measured diffuse reflectance by means of the curve fitting of diffusion model, using a non-linear least squares optimization procedure. Non-linear least squares algorithms begin with initial guesses of model parameters, proceed iteratively by using searching methods for solution updating, and finally converge to the best estimates that minimize the sum-of-squares (SS) of the difference between the true and predicted values. Gauss-Newton and Levenberg-Marquardt algorithms are often used for solving non-linear least squares problems [114], which can be readily implemented in commercial software like Matlab (The Mathworks, Inc., Natick, MA, USA) using its built-in functions. In implementing the inverse algorithm for estimating \(\mu_{\text{a}}\) and \(\mu^{\prime}_{\text{s}}\), [PERSON] and [PERSON] [55] proposed a three-step curve fitting procedure. First, both \(\mu_{\text{a}}\) and \(\mu^{\prime}_{\text{s}}\) were treated as unknown parameters, and their estimates were obtained by fitting the diffusion model. Second, the \(\mu^{\prime}_{\text{s}}\) values were updated by fitting the \(\mu^{\prime}_{\text{s}}\) values obtained in the first step with the function \(\mu^{\prime}_{\text{s}}=a\lambda^{-b}\), where \(a\) and \(b\) are parameters and \(\lambda\) is the wavelength [115]. Last, the \(\mu^{\prime}_{\text{s}}\) values from the second step were inserted into the diffusion model, and the curve fitting procedure was repeated to obtain the estimates of \(\mu_{\text{a}}\) values with \(\mu^{\prime}_{\text{s}}\) being known. The three-step procedure, compared to one-step curve fitting, can reduce the fitting noise for the two parameters and thus improve the fitting accuracy and stability. [PERSON] et al. [104] further explored the effects of the data transformation and weighting methods on parameter estimation. Several methods, including logarithm and integral transformation and relative weighting, were used to pre-process the diffuse reflectance profiles generated by Monte Carlo (MC) simulations based on a total of 29 combinations of \(\mu_{\text{a}}\) and \(\mu^{\prime}_{\text{s}}\) that are typical for food and horticultural products. It was found that these pre-processing methods greatly reduced the estimation errors of \(\mu_{\text{a}}\) and \(\mu^{\prime}_{\text{s}}\), compared to those obtained from the original diffuse reflectance, and that the  
Figure 19: Principle of the spatially-resolved technique for optical property measurement. logarithm transformation achieved the best results with the average estimation errors of 10.4% and 6.6% for \(\mu_{\mathrm{a}}\) and \(\mu_{\mathrm{s}}^{\prime}\), respectively. The authors also reported that \(\mu_{\mathrm{s}}^{\prime}\) can be estimated more accurately than \(\mu_{\mathrm{a}}\) based on sensitivity analysis for the two parameters. The diffusion models are derived based on the use of an infinitely small pencil beam incident normally on the semi-infinite turbid medium and are only valid when the source-detector distance exceeds a certain limit [97]. Hence, optimization of the instrumental system should consider the effects of light beam including size and shape, and source-detector distance including minimum and maximum source-detector distances, so as to achieve accurate measurement of optical properties. [PERSON] and [PERSON] [104] examined these factors in the development of the hyperspectral imaging-based spatially-resolved technique for optical property measurement. Three types of light beams, i.e., infinitely small, Gaussian and circular flat, were investigated based on MC simulations.