^{1,a)}and Ioannis Bertsatos

^{1}

### Abstract

A maximum likelihood method for estimating remote surface orientation from multi-static acoustic, optical, radar, or laser images is presented. It is assumed that the images are corrupted by signal-dependent noise, known as speckle, arising from complex Gaussian field fluctuations, and that the surface properties are effectively Lambertian. Surface orientation estimates for a single sample are shown to have biases and errors that vary dramatically depending on illumination direction. This is due to the signal-dependent nature of specklenoise and the nonlinear relationship between surface orientation, illumination direction, and fluctuating radiance. The minimum number of independent samples necessary for maximum likelihood estimates to become asymptotically unbiased and to attain the lower bound on resolution of classical estimation theory are derived, as are practical design thresholds.

I. INTRODUCTION

II. RADIOMETRY

III. THE LIKELIHOOD FUNCTION AND MEASUREMENT STATISTICS

IV. CLASSICAL ESTIMATION THEORY AND A HIGHER ORDER ASYMPTOTIC APPROACH TO INFERENCE

V. INFERRING LAMBERTIAN SURFACE ORIENTATION

A. MLEs of surface orientation and albedo

B. The angle of incidence

C. 3D surface orientation and albedo

VI. CONCLUSIONS

### Key Topics

- Illumination
- 21.0
- Albedo
- 18.0
- Surface measurements
- 10.0
- Medical imaging
- 9.0
- Speckle
- 9.0

## Figures

Resolved surface patch.

Resolved surface patch.

Probability densities given real, imaginary, or Gaussian constraint on parameter estimate.

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.

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 (

*θ*= π/4,

_{n}*φ*=

_{n}*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 s

_{3}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 [

**i**

_{a}^{-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 s

_{3}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).

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 (

*θ*= π/4,

_{n}*φ*=

_{n}*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 s

_{3}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 [

**i**

_{a}^{-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 s

_{3}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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