Adaptive regularized filtering for BRDF model inversion and land surface albedo retrieval based on spectrum cutoff technique

Remote sensing of surface reflective properties: Role of regularization and a priori knowledge

“…(7), only the first p terms play a role in the minimization of the residual norm. When the number of observations is insufficient or the angular distribution is poor, noise components inr are divided by small singular values, their contribution to the retrieved BRDF parameters is amplified [16]. Hence, the task of the retrieval algorithm is to filter out the noise-dominated components of the solution and thus to retrieve only that part of the BRDF parameters about which information is present in the measurements.…”

“…Here, ı i are the singular values of H. Advantage of (14) over (7) is that the regularized solution makes use of a filter [16] …”

“…In Tikhonov regularization method, can be interpreted as the information index, which is defined as the ratio of variance of the measurement errors to that of the parameters to be retrieved [16]. Conceptually, it depends on the noise level and the information content in the measurements.…”

“…First, the initial value of is determined using the signal-to-noise ratio (SNR) of optical sensors [16] and the reflectance noises [25]. Then, with this initial value, Newton's iterative method is employed to solve the discrepancy equation.…”

“…However, crude selection of regularization parameters would limit the algorithm's efficiency and its applications. Cui et al [16] improved the method for choosing regularization parameters and modified the algebra spectrum of the BRDF kernel matrix (i.e., replace tiny singular values with positive values after performing singular value decomposition (SVD) of the BRDF kernel matrix) to stable the BRDF model parameters estimates using the spectrum cut-off technique. Actually, the difference between these various approaches relies on how rigorously additional information is mathematically processed, and in particular, the uncertainties associated to this additional information.…”