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Computational Colour Vision

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http://www.colourtech.org

Computational Colour Vision

Stephen Westland

Centre for Colour Design Technology

University of Leeds

June 2005

Oxford Brookes University

Computational Colour Vision

Introduce some basic concepts - the physical basis of colour

Phenomenology of colour perception (the problem)

Computational approaches to how colour vision works

Computational and psychophysical studies of transparency perception

E(l)P(l)

E(l)

C(l) = E(l)P(l)

The colour signal C(l) is the product at each wavelength of the power in the light source and the reflectance of the object

P(l)

The Physical Basis of Colour

Cone spectral sensitivity

S

∫

L = E(l)P(l)FL(l)dl

∫

M = E(l)P(l)FM(l)dl

M

∫

S = E(l)P(l)FS(l)dl

L

∫

L = E(l)P(l)FL(l)dl

Cone Responses

∫

M = E(l)P(l)FM(l)dl

∫

S = E(l)P(l)FS(l)dl

Each cone produces a univariant response

L

S

M

Colour perception stems from the comparative responses of the three cone responses

Colour is a perception – ‘the rays are not coloured’

P

Indoors (100 cd/m2)

Outdoors (10,000 cd/m2)

0.01

1

100

0.99

99

9900

Colour Constancy

Objects tend to retain their approximate daylight appearance when viewed under a wide range of different light sources

The visual system is able to discount changes in the intensity or spectral composition of the illumination

WHY? / HOW?

noon

sunset

X

X

L1 / L1W = L2 / L2W

M1 / M1W = M2 / M2W

S1 / S1W = S2 / S2W

e1 = De2

L1W/L2W

M1W/M2W

S1W/S2W

0

0

0

0

0

0

L1

M1

S1

L2

M2

S2

D =

e1 =

e2 =

Computational Explanation

∫

∫

L1 = E1(l)P(l)FL(l)dl

L2 = E2(l)P(l)FL(l)dl

∫

∫

M1 = E1(l)P(l)FM(l)dl

M2 = E2(l)P(l)FM(l)dl

∫

∫

S1 = E1(l)P(l)FS(l)dl

S2 = E2(l)P(l)FS(l)dl

brightest pixel is white

grey-world hypothesis

Practical Use – Colour Correction

Camera RGB values vary for a scene depending upon the light source

colour correction

In order to correct the images we need an estimate of the light source under which the original image was taken

∫

L = E(l)P(l)FL(l)dl

∫

M = E(l)P(l)FM(l)dl

∫

S = E(l)P(l)FS(l)dl

Colour Constancy

Adaptation is too slow to explain colour constancy

“Any visual system that achieves colour constancy is making use of the constraints in the statistics of surfaces and lights” – Maloney (1986)

Is it possible for the visual system to recover the spectral reflectance factors of the surfaces in scenes from the cone responses?

Basis Functions

P(l)S wiBi(l)

Using a process such as SVD or PCA we can compute

a set of basis functions Bi(l) such that each reflectance

spectrum may be represented by a linear sum of

basis functions - a linear model of low dimensionality.

If we use n basis functions then each spectrum can

be represented by just n scalars or weights.

Original

1 BF

P(l) = w1B1(l)

Original

1 BF

2 BF

P(l) = w1B1(l) + w2B2(l)

Original

1 BF

2 BF

3 BF

P(l) = w1B1(l) + w2B2(l) + w3B3(l)

How many Basis Functions are Required?

About 99% of the variance

can be accounted for by

a 3-D model (Maloney &

Wandell, 1986)

But what proportion of

the variance do we need to

account for?

6-9 basis functions are required

Simultaneous Contrast

original

original covered by filter

original with small filter

ei,1/ei,2 = e'i,1/e'i,2

(Foster)

Ratio under second light source

Ratio under first light source

i = {L, M, S}

Colour Constancy - spatial comparisons

“For the qualities of lights and colours are perceived by the

eye only by comparing them with one another” (Alhazen, 1025)

“… object colour depends upon the ratios of light reflected

from the various parts of the visual field rather than on

the absolute amount of light reflected” (Marr)

L1

=k

L2

L’1

=k

L’2

Spatial Comparison of Cone Excitations

Retinex – Land and McCann (1971)

Foster and Nascimento (1994)

e’1

e’2

e1

e2

Transparency Perception

e1/e2 = e’1/e’2

(Ripamonti and Westland, 2001)

What is transparency?

An object is (physically) transparent if some proportion of

the incident radiation that falls upon the object is able to

pass through the object.

What is perceptual transparency?

Perceptual transparency is the process ‘of seeing one object

through another’ (Helmholz, 1867)

Physical transparency is neither a necessary or sufficient

condition for perceptual transparency (Metelli, 1974)

Even in the complete absence of any physical transparency

it is possible to experience perceptual transparency

Perceptual transparency

Research Questions

What mechanisms could drive perceptual transparency?

What are the chromatic conditions that cause transparency?

Could transparency and colour constancy be linked?

Perceptual transparency

T(l)

e'i,1

e'i,2

Transparency and Spatial Ratios

ei,1

ei,2

ei,1/ei,2 = e'i,1/e'i,2

Experimental

Computational analysis to investigate whether for physical

transparency the cone ratios are preserved

Psychophysical study to investigate whether the invariance

of spatial ratios can predict chromatic conditions for

perceptual transparency

Psychophysical study to compare the performance of the

ratio-invariance model when the number of surfaces

is varied

P'(l) = P(l)[T(l)(1-b)2]2

(Wyszecki & Stiles, 1982)

Physical Model of Transparency

(1-b)b2P3T6

(1-b)PT2

(1-b)bP2T4

b

T

opaque surface P

1. A pair of surfaces P1(l) and P2(l) were randomly selected

2. A filter was randomly selected (defined by a gaussian

distribution)

3. The cone excitations were computed for the surfaces

viewed directly (under D65) and through the filter

P1(l)

P2(l)

s

ei,1/ei,2 e'i,1/e'i,2

lm

Monte Carlo Simulation

4. Steps 1-3 repeated 1000 times

e

e

i,1

i,2

e

e

'

'

i,1

i,2

Monte Carlo Results

e'i,1/e'i,2

ei,1/ei,2

e

e

i,1

i,2

e

e

'

'

i,1

i,2

Monte Carlo Results

The ratios are approximately invariant

Invariance is slightly better for the S cones

Invariance decreases as the spectral transmittance

decreases

xB

xA

xQ

xP

g

Convergence

xA

xB

xQ

xP

xP= a xA+ (1-a) g

xQ= a xB+(1- a) g

Da Pos, 1989, D’Zmura et al., 1997

Psychophysical Stimuli I

(a)

convergent

(deviation 0)

(b)

invariant

(deviation = 0)

deviationi = 1 - [ei,1/ei,2]/[ e'i,1/e'i,2]

5

3

d'

1

-1

-3

0

0.1

0.2

0.3

0.4

0.5

0.6

LMS deviations

L

M

S

Log. (L)

Log. (S)

Log. (M)

Psychophysical Results I

d'<0 indicates subjects' preference for convergent filter; d'=0 no preference; d'>0 indicates subjects' preference for invariant filter

Psychophysical Stimuli II

Psychophysical Results II

Conclusions

Computational and pyschophysical studies show

that the invariance of cone-excitation ratios

may be a useful cue driving transparency

perception

Colour constancy and transparency perception

may be related. Could they result from

similar mechanisms, perhaps even similar

groups of neurones?

There are still a mass of unsolved problems in computational colour vision including the relationship between cone excitations and the actual sensation of colour

xB

xA

xQ

xP

g

xA

xB

xQ

xP

xP= a xA+ (1-a) g

xQ= a xB+(1- a) g

xP= a xA

xQ= a xB

Cone excitations are transformed by a

a diagonal matrix whose diagonal elements

are all equal

xA

xP= b xA

xQ= b xB

xB

xQ

xP

Cone excitations are transformed by a

a diagonal matrix whose diagonal elements

are not necessarily all equal

The two models can be made to be the same

if the convergence model has no additive

component and if the invariance model has

equal cone scaling