Resolved surface patch.
Probability densities given real, imaginary, or Gaussian constraint on parameter estimate.
Comparison of the necessary sample sizes to achieve (a) a correlation greater than 0.99 between and , (b) an unbiased estimate, and (c) an estimate that attains the minimum possible variance.
A visualization for a corresponding to the surface gradient parameterization for a 3D measurement vector R with σ N negligible and Lambertian surface defined by the polar coordinate parameterization (θn = π/4, φn = π/4, ρ = 0.6). Incident vectors s 1 and s 2 are fixed at [θ 1 = 0.3662π (≈ 65.9°), φ 1 =1.8524π (≈333°)] and [θ 2 = 0.3662π, φ 2 = 0.3524π (≈ 63.4°)], respectively, but s3 is allowed to vary as in (a) where the positive z-axis is central and points out of the page, the direction of the positive x-axis is the right-most point, and that of the positive y-axis is the top-most point, each marked by x and y, respectively. (b) The bound [ia -1]11 on the x-gradient, a 1 = p, including full 3D coupling. Only values where s 3 · n is positive and the Lambertian surface is in view from the positive z-axis are shown. The incident vectors s 1 and s 2, and the surface normal n are marked by their respective symbols. The black cross marks the center (θ = 0, φ = 0). Optimal resolution (b), minimum necessary sample size for unbiased (c), and minimum variance (d) estimation, occurs when s3 is tangent to the Lambertian surface closest to the negative x-axis (extreme left), for the p-bound, so as to maximize the volume of incident vectors. Poorest resolution and largest sample sizes occur when the volume of incident vectors approaches zero, as realized along the dark arc in (b), (c), and (d). (c) The sample size necessary for the MLE to be effectively unbiased, from Eq. (18). (d) The sample size necessary, from Eq. (19), for to effectively attain the bound given in (b).
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