Elongations near intensity maxima: a cue for shading?
Michael Langer
Daria Gipsman
School of Computer Science,
McGill University,
Montreal, Canada
1. Linear vs. quadratic shading
2. Shading elongations
The standard Lambertian shading model says that the intensity
of reflected light varies with the cosine of the angle between
the light source and the surface normal.
I ( x , y )  L  N ( x, y )
 
The light vector and surface normal can be written respectively
as:

L  ( Lx , Ly , Lz )

N ( x, y ) 
(
Z Z
,
,1)
x y
2
where Z(x,y) is the surface depth map.
2
 Z   Z 
      1
 x   y 
Intensity can be approximated as the sum of a linear and a
quadratic component (Pentland 1989)
 1   z 2  z 2  
z
z
I ( x , y )  Lx  Ly
 Lz  1         
 2   x   y   
x
y



linear
quadratic
Linear shading produces intensity elongations in the direction
perpendicular to (Lx, Ly). These elongations seem to play a
role in perception of illumination direction. (Pentland 1982)
Quadratic shading also produces intensity elongations. These
elongations occur at maxima of the quadratic component of the
intensity, namely at points where the surface normal is (0,0,-1).
To see why, without loss of generality, suppose (x,y) = (0,0) is a
point whose normal is (0,0,-1). In a small neighborhood of this
point, we can approximate the depth map as:
Z ( x, y )  a x 2  b y 2
The quadratic component of the image intensity near this point
obeys:
I ( x, y)  1  2 (a 2 x 2  b 2 y 2 )
3. Sparse codes and Gabor filtered shading
It is believed that oriented structures in images (such as from
occluding edges) produce “sparse codes”, in that the response
histograms of Gabor filtered images have longer tails than a
Gaussian. Do elongated intensities that arise from linear and
quadratic shading produce such “sparse codes” ?
We checked whether the Gabor response histograms (i.e.
probabilities) are better fit with a Gaussian distribution (not
sparse) or a two sided Laplacian distribution (sparse), namely,
whether the log histograms of responses are better fit by a
parabola or by a pair of lines.
Gaussian

log e
Laplacian
response2
2 2
 response
2
2
2
log e
|
response

|
  | response
|

Linear shading does not yield a sparse code
If | a | and | b | are not equal, then the aspect ratio of the
intensities is exaggerated, relative to the aspect ratio of the
depth.
The result: Intensities are elongated near intensity maxima.
Quadratic shading does yield a sparse code
depth map
Z(x,y)
I(x,y)
linear only
quadratic only
Conjecture: the quadratic component of shading
may play a role in distinguishing shading vs. nonshading.