Optimally Resolving Lambertian Surface Orientation

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Optimally Resolving Lambertian Surface Orientation

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

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Motivation

• Resolve surface slope and shape from intensity
  images corrupted by speckle noise
• 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
source
receiver
surface normal
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Overview

• Orientation from Reflected Intensity
• Measurement Statistics
• Asymptotic Optimality Conditions
• 1D Surface Slope Estimation
• 2D Albedo and Slope Estimation
• 3D General Surface Orientation Estimation
• This research is a combination and extension of
  earlier work
• by N. C. Makris, 1997 and Naftali and Makris,
  2001

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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

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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
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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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Review

• 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.

Conclusion