Nonparametric 1-D temperature restoration in lossy media using Tikhonov regularization on sparse radiometry data

Microwave thermometry has the potential to characterize thermal gradients in lossy materials down to a few centimeters depth. The problem of retrieving temperature profiles from sets of brightness temperatures is studied using Galerkin expansion of one-dimensional (1-D) temperature profiles combined with Tikhonov regularization and predefined boundary conditions. From a priori knowledge of the temperature field shape, smooth Chebyshev polynomials are used as basis functions in the series expansion. The proposed estimator does not require iterative calculations that are normally performed using conventional numerical methods for signal parameter estimation and is, thus, very fast. Noise effects versus bandwidth limitations (smoothness of solutions) are studied in terms of four performance indexes defined in the text. In general, statistical spread of the temperature estimator increases with increasing number of Chebyshev polynomials. Systematic deviation from true values (bias) decreases as the number of Chebyshev polynomials increases. Results show that smooth temperature profiles can be reproduced using 6-7 Chebyshev polynomials. With additional constraints such as boundary conditions and maxima localization, a three-frequency-band radiometric scan is sufficient to produce acceptable results in regions with low thermal gradients. As the spatial variability of the 1-D temperature profile increases, more radiometric bands (5-6) are required to provide nonbiased estimates.