Derivative of canopy reflectance with respect to leaf density: A probabilistic rendering framework

Estimating the derivative of canopy reflectance on leaf area density ( μ ) is crucial in quantifying how vegetation structure changes, for example caused by crop growth or leaf senescence, influence remote sensing observations. We propose a probabilistic approach that estimates it using a radiative transfer model based on the rendering expression: each leaf survives with probability P (Bernoulli thinning) so that an initial density μ 0 is reduced to μ = μ 0 P . Then, a change of P implies a change in μ , as their derivatives are linked. The new rendering expression for a given leaf thinning configuration, while taking expectation over leaf presence, leads to a closed form of the expected radiance and its derivative with respect to P . Based on Monte Carlo path tracing, we develop an unbiased estimator for both quantities. Its accuracy is similar to that of state-of-the-art techniques (i.e., finite difference and importance sampling) with less noisy derivatives, which highlights its efficiency.