Linear Depth Estimation from an Uncalibrated, Monocular Polarisation Image
William A. P. Smith1
Department of Computer Science
1
Ravi Ramamoorthi2
2
Silvia Tozza3
Center for Visual Computing
Overview
3
Solving for depth
Laplacian smoothness prior
Illumination estimation from a polarisation image
1
Least squares solution over all pixels:
For the correct light source vector s:
Non-convex but alternating assignment/optimisation usually obtains global optimum
Intensity
0.6
Transformation between two
possible polarisation normals
1.2
2.5
1
2
0.8
1.5
0.6
1
GBR transformation so
solution ambiguous
0.4
0.2
3
1.4
One of the possible
polarisation normals
0.8
0
0
/4
/2
3/4
5pi/4
3/2
7/4
2
Polariser angle
(b) Per-pixel weight
Stack all per-pixel equations in large linear system
Decomposing polarimetric measurements to a polarisation image
wsmooth = 0.1
(a) Azimuth from boundary
Background
Convexity prior
Finite difference approximation of surface gradient
wsmooth = 0
Shape-from-polarisation
Dipartimento di Matematica
0.35
3
0.3
2.5
0.25
1.5
0.15
"%
$ 0.5
0.05
0
1
0.1
0.5
0.2
Experimental Results
2
0.2
0.4
%
$ Diffuse shading cue
-
-
-&)(*'
-&)(*'
!$
!$
0.6
0.4
-
-&*(,'
"%
$ 0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Zenith angle
#
#
First linear equation in surface gradient (diffuse)
Specular pixels
First linear equations in surface gradient (specular)
"
!
"
!
Degree of polarisation
Specular
Diffuse
0.8
[1] Miyazaki,D.,Tan,R.T.,Hara,K.,Ikeuchi,K.: Polarization-based inverse rendering from a single view.
In: Proc. ICCV. (2003) 982–987
[2] Atkinson, G.A., Hancock, E.R.: Recovery of surface orientation from diffuse polarization. IEEE
Trans. Image Process. 15(6) (2006) 1653–1664
[3] Mahmoud, A.H., El-Melegy, M.T., Farag, A.A.: Direct method for shape recovery from polarization
and shading. In: Proc. ICIP. (2012) 1769–1772
1
"
!
"
!
#
#
Specular dominant pixel labelling
Phase angle cue
We don’t disambiguate locally
We write a linear equation that is
satisfied by either interpretation
Disambiguated globally when we
solve for depth
Summary
Second linear equation in surface gradient
Specular pixels are phase shifted by π
Azimuth aligns with Imin
%
$ -
-&)(+'
Positives:
  Monocular
  Passive
  Uncalibrated illumination (can be spherical harmonic)
  Can be made single shot
  Depth directly by solving large, sparse linear system
"%
$
Degree of linear polarisation cue
Degree of polarisation and shading cue
%
$ "%
$ %
$
-
-&*(,'
Assumptions:
  Dielectric material (non-metallic)
  Orthographic projection
  Reflectance can be characterised as specular or
diffuse dominant
  Known index of refraction (assumed = 1.5)
  Uniform albedo