The document discusses metamerism in color and spatial vision. Metamerism occurs when different illuminants produce the same cone response triples, leading to the perception of identical colors. For spatial vision, the document proposes using derivatives-of-Gaussians and jet similarity measures based on inner products to address metamerism between image features classified by non-linear circuits. An inner product structure is needed to ensure jet similarity conforms to the linearity of the measurement process. Comparing images by comparing their jets using a scale-space inner product provides an approximation that is 1% accurate for flat spectra and 0.1% for natural images.

1. Metamerism? What metamerism?
Lewis D Griffin
Computer Science,
University College London

2. Metamerism in Colour Vision
cone sensitivity
functions
artificial illuminant
natural illuminant
9.1
The cone response triple
8.8
is the same for both illuminants.
7.2
http://www.onlandscape.co.uk/2012/02/the-myth-of-universal-colour/#/

3. Metamerism in local Spatial Vision
2.1
-1.2
.
=
2.1
-0.6
-3.2 -2.8
-3.4
Derivatives-of-Gaussians are
a good model of V1 simple cells
0.1
0.7
1.4
0.1
-0.2
a jet
2.2
0.8
4.5

4. Why is Metamerism a problem?
2.1
-1.2
=
.
2.1
-3.2
-3.4
-0.6
-2.8
0.1
-0.2
Metamery Class
Non-linear feature
classifier circuitry
Q: How should this
work, given this?
symmetry groups
J2,3
edge
bar
J7
……
……
0.7
T-junction
J12,2,7
1.4
0.1
2.2
0.8
4.5

5. Need to decide when jets
are similar.
Jet similarity should
conform to the linearity of
the measurement process.
Therefore what is needed is
an Inner Product structure.
The Beezer 1962
2.4
2.1


0.7
1.3
Inner Product
6.3

6. There is an infinity of possible Inner Products on jets…
The dot product
j0
j1
:
k0
k1
j2
 
j,k
k2
j1 k1
j2 k 2
1

0 k
0
j0 k 0
1
T
j
0
0
0
0
1
Gram Matrix based
 
j,k
9
T
j
0
0
6
30
3
3
6
3

k
0
0
12
5
The scale-space Inner Product
 
j,k
2
T
j
0
0
0
0
2
2
0
0
4

k
…but this one is best

7. A Jet Space IP induces an Image IP
A way to measure how similar jets are, is equivalent to a rule to measure how
similar images are
Dot product
1
0
0
T 0
j
0
1
0
0
1
0 
k
0
0
T
0
0
0
1
Gram Matrix
Scale-Space

8. Image IPs can also be expressed in the Frequency Domain
Dot product
T
spatial
domain
T
frequency
domain
Gram Matrix
Scale-Space

9. measure images
with filters to make jets
then
compare jets using the
‘scale space’ inner product
 
j,k
j0 k 0
then
filter images
2
4
j1 k1
2
=
6
j2 k 2
6
j3 k 3
then
window images
compare images using this
fourier inner product
compare them
using a standard
inner product
= compare images using this
fourier inner product

10. How good is the approximation?
≈
• 1.0% error for images with flat spectra
• 0.1% error for ‘natural’ images with 1/f spectra.

11. Approximately equal ‘views from the inside’
Simple cell assembly
fuzzy
window
frosted
glass

12. 2.1
-1.2
=
.
2.1
-3.2
-3.4
-0.6
-2.8
0.1
0.7
-0.2
1.4
0.1
2.2
0.8
Non-linear feature
classifier circuitry
symmetry groups
J2,3
edge
bar
J7
……
……
Filtering has
no effect on
symmetries!
T-junction
J12,2,7
4.5

13. +2
+1
+1
+2
+1
+1

14. Metamerism? What metamerism?
Approximately equal ‘views from the inside’
Simple cell assembly
fuzzy
window
Thank you for your attention
frosted
glass