Optimally Resolving Lambertian Surface Orientation

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Optimally Resolving Lambertian Surface Orientation

• Ioannis Bertsatos
• Research Advisor Nicholas C. Makris
• Massachusetts Institute of Technology

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Motivation

• Resolve surface slope and shape from intensity
  images corrupted by speckle noise
• Sonar images of the ocean environment
• Relative motion between source, surface
• and receiver results in field fluctuations
• Optical and radar images of planetary bodies
• Natural light sources (sun, stars, light bulbs!)
• Machine vision

Examples
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Overview

• Scattering from Rough Surfaces
• Radiometry, Lambertian Surfaces
• Measurement Statistics
• Asymptotic Optimality Conditions
• 1D Surface Slope Estimation
• 2D Albedo and Slope Estimation
• 3D General Surface Orientation Estimation

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Surface Types

• Smooth surfaces Specular reflection
• Rough surface Scattering function
• (Scale of roughness gt ?incident)
• Lambertian surfaces perfect diffuse scatterers
• (All incident energy reflected uniformly)

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Radiometry
subtended solid angle d?i cos?i dA / r2 ? cos?i
dO
intercepted flux dFi I cos?i dO
irradiance dFi d?i cos?i dA ? Fi Ei cos?i dA

dFi (Li cos?i dO) dA ? Li dEi / (cos?i dO)
Lr dEr / (cos?r dO)
dFr Lr cos?r dO dA
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dFr Lr d?r dA Lr cos?r dO dA
Total flux reflected
For a Lambertian surface, Lr constant
For non-Lambertian surfaces, ? ?(?i,fi,?r,fr)
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Approach

• Sonar images of remote surfaces resolved by a
  pencil beam (2D sonar) array
• Attain specified design criteria from 2D sonar or
  optical measurements
• Design specifications are met when both
  conditions are satisfied
• Estimate becomes unbiased, and attains the
  Cramer-Rao Lower Bound (CRLB)
• CRLB is within the design specifications

Goals
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Surface Orientation from Reflected Intensity

• Lamberts Law
• Measure
• Non-linear estimation problem

surface radiance
random
known constant
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Measurement PDF

• a a1 aj orientation parameters
• R R1 R2 Rk RN
• Rk are Independent Identically Distributed (IID)
  measurements corrupted by signal-dependent noise
  (speckle) due to Circular Complex Gaussian Random
  (CCGR) field fluctuations
• ltRkgt ? sk(a)
• µk is a measure of the number of independent
  intensity fluctuations in the measurement Rk
• Can be interpreted as the signal-to-noise ratio
• SNR ? ltRkgt2/var(Rk) µk

N. C. Makris, 1997
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Necessary Conditions for Asymptotic Optimality of
MLE

• Maximum Likelihood Estimate (MLE)
• For large sample sizes or SNR, the MLE becomes
  unbiased, and attains the Cramer-Rao Lower Bound
  (CRLB)
• Using asymptotic expansions of the type
• We can derive necessary conditions on SNR

for unbiased estimate for minimum variance
estimate
Naftali and Makris, 2001
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Slope Estimation

• parameter a ?i
• lt R gt s(?i) ? cos?i

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• To attain design specifications on resolution of
  ?i
• Determine necessary SNR to achieve unbiased,
  minimum variance estimate, then
• Specify SNR so that the CRLB is within the design
  specs

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Illustrative Example

• For the 1D case, the necessary conditions on SNR
  introduced earlier become
• If ?i 10o, then in order to achieve an
  unbiased, minimum variance estimate, we require
  that
• SNRv is the limiting value. For SNR ? SNRv the
  estimate asymptotically attains minimum variance,
  so that
• If the design criteria specify a maximum variance
  of 1o, then we require that
• If, instead, ?i 80o, then to satisfy the same
  design criteria we now only require that

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2D Parameter estimate Albedo and Slope

• Incident directions must be distinct so that 2
  independent equations exist for ? and ?i
• 2 unknowns incident angle (?) and albedo (?)
• Assume s1, s2 as shown
• It is impossible to invert for both parameters in
  this arrangement
• s1 cannot be collinear to s2

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3D Surface Orientation Estimation

• Define S source matrix
• ST s1 s2 sk sN
• lt Rk gt ? sk(a) ? skT n
• lt R gt ? S n
• 3 orientation
• parameters
• a ?, ?, f, or
• a ?, p, q, or
• a nx, ny, nz

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Incident Directions must be Distinct

• 3 unknowns 3 illumination vectors
• If all si are coplanar, then we cannot estimate
• the orientation
• parameters
• Need 3 distinct
• illumination vectors
• (i.e. 3 equations) to estimate all the unknowns
• s1 (s2 ? s3) ? 0 ? si must span some
  volume

all si on this plane
n
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Illustrative Examples

• Estimate of a ?, p, q, given
• Specified
• Fixed
• Varying

• ? 1/p

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Illustrative Examples

• Estimate of a ?, ?, f, given
• ? 1/p
• In order to achieve an unbiased, minimum variance
  estimation, we require that
• SNRv is the limiting value. For SNR ? SNRv the
  estimate asymptotically attains minimum variance,
  so that
• If design criteria require that the maximum
  allowable variance is 1o, then

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Shallow s1, s2 Variable s3
Steep s1, s2 Variable s3
SNR for minimum variance estimation
SNR for minimum variance estimation
(dB)
(dB)
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Conclusions

• Optimality conditions for the estimation of
  Lambertian surface orientation can be derived
  from asymptotic expansions of the MLE.
• For slope estimation (with known albedo), it is
  shown that it is more efficient to illuminate the
  surface from shallow grazing angles.
• For 3D estimation it is shown that the incident
  vectors must also span some volume.
• Given fixed source/illumination vectors spanning
  non-zero volume, it is possible to come up with
  specific SNR conditions, so that any resolution
  criteria for surface orientation are met.

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Background

• Stochastic behavior of optical, radar, and
  acoustic fields received from fluctuating sources
  and/or scatterers can be described by Circular
  Complex Gaussian Random (CCGR) statistics
• Intensity images derived from CCGR fields exhibit
  signal-dependent noise, or speckle
• Lamberts Law is often appropriate to model
  surface radiant intensity
• Maximum likelihood estimators (MLE) for
  Lambertian surface orientation are used
• N. C. Makris, 1997
• Optimality criteria for MLE are applied
• Naftali and Makris, 2001

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Physical Model

• Assume a planar surface with wave-scale roughness
  in free space. A surface patch of scale gtgt ? is
  resolved. It is characterized by a specific
  orientation wrt a fixed coordinate system
• Sonar images are obtained a source illuminates
  the target surface and reflected intensity is
  measured at a receiver

Lamberts Law

• Surface Radiance, L, is defined as radiant power
  in a given direction, per unit solid angle, per
  unit projected area of the source, as viewed from
  the given direction
• Lambertian surfaces are defined as those surface
  for which, L, is constant throughout space
• The measured reflected intensity does not depend
  on the observation direction, but contains
  information about the angle of incidence of the
  illumination direction