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```					What can be Known about the
Michael Grossberg and Shree Nayar

CAVE Lab, Columbia University
Partially funded by NSF ITR Award

ECCV Conference
May, 2002, Copenhagen, Denmark
Response         Scene Radiance: R        Image Plane      Response: u
function:
f(I)=u
0 255

Inverse                    I

response
function: g
u
g(u)=I                Response = Gray-level
Response Recovery from Images

What is measured? What is needed?                        What is recovered?
Images at different        Correspondence of gray-       Inverse Radiometric
exposures                  levels between images          Response, g

Exposure
I
Ratios
Gray-levels:
Image D
k3
Gray-levels:
k2                    Image C
u
Response
k1                  Gray-levels:
uB       Image B      Exposure Ratios
Gray-levels:                       k1   k2   k3
uA            Image A

Recovery Algorithms: S. Mann and R. Picard, 1995, P. E. Debevec, and
J. Malik, 1997, T. Mitsunaga S. K. Nayar 1999, S. Mann 2001,
Y. Tsin, V. Ramesh and T. Kanade 2001

Images at Different Exposures           Corresponding Gray-levels         Inverse Response g,
Exposure Ratio k

I

Geometric
Correspondences

Response
u

k1   k2   k3
Recovery
Algorithms

We eliminate the need for geometric                     We find:
correspondences:                                           • All ambiguities in recovery
Static Scenes           Dynamic Scenes                  • Assumptions that break them
Constraint Equations

IB
 Constraint on irradiance I:                         Brighter
image
 IB= kIA
 Constraint on g:
 g(uB)=kg(uA)                    Filter       IA
Darker
image

 Brightness Transfer Function T:            T
 uB=T(uA)
 Constraint on g in terms of T

g(T(uA))=kg(uA)
How Does the Constraint Apply?
I
kg(uA)    =     g(T(uA))
1
 Exposure ratio k
known
g(T(uA))
 Constraint makes                      kg(uA)

curve self-similar
1/k

g(uA)

u

0                 uA T-1(1)      T(uA)      1
Gray-levels
Self-Similar Ambiguity:
Can We Recover g?
I           Choose anything here
 Conclusions:        1
and
Constraint gives
copy
no information in

[T-1(1),1]
Regularity                                              1/k
assumptions break
ambiguity                                   1/k2
Known k: only
1/k3
Self-similar                                                            u
ambiguity
0                                T-1(1)         1
Gray-levels
Exponential Ambiguity:
Can We Recover g and k ?
Inverse Response Function gγ                       Exposure                      Brightness Transfer Function T
ratio
k=21/3

Gray-level Image B
I
k=21/2

γ=1/3                                      k=2
γ=1/2                                   k=22                                      T(M)=2M
γ=1                                  k=23
γ=2
γ=3

Response                    U                                       Gray-level Image A

T(u) = g -1(kg(u)) = g -γ(k- γg γ(u)) = T(u)

We cannot disambiguate (gγ, kγ) from (g, k) using T!
Obtaining the Brightness Transfer
Function (S. Mann, 2001)
Registered Static Images at Different        2D-Gray-level            Brightness Transfer
Exposures                        Histogram                     Function

Gray-level Image B
Regression

Gray-level Image A                                      Gray-level Image A

Scenes must be static.
Brightness Transfer Function without
Registration

Unregistered Images at   Brightness Histograms                  Brightness Transfer
Function
Different Exposures

Gray-level Image B
Gray-level Image B
Histogram
Specification

Gray-level Image A
Gray-level Image A

Scenes may have motion.
How does Histogram Specification
Work?
Cumulative Area

Histogram Equalization      Histogram Equalization

Histogram Specification

Gray-levels in Image A                   Gray-levels in Image B

Histogram Specification = Brightness Transfer Function
Results: Object Motion

1                                                              1                                                               1
0.9                                                            0.9          Recovered Response                                 0.9
0.8                                                            0.8                                                             0.8

0.7                                                            0.7                                                             0.7
0.6                                                            0.6                                                             0.6
0.5                                                            0.5       Macbeth Chart Data                                    0.5
0.4                                                            0.4                                                             0.4
0.3                                                            0.3                                                             0.3
0.2                                                            0.2                                                             0.2
0.1                                                            0.1                                                             0.1
0                                                              0                                                               0
0   0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1                      0    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1                      0   0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Red Response                                                    Green Response                                                  Blue Response
Recovered Inverse
1

Object and Camera                         0.9
0.8

0.7

Motion                                    0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Red Response
1
0.9       Recovered
0.8       Response

0.7
0.6      Macbeth Chart
0.5          Data
0.4
0.3
0.2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Green Response
1
0.9
0.8
0.7

0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Blue Response
Conclusions: What can be Known about
Inverse Response g from Images?
 A1:            Exposure ratio
k known        Self-similar Ambiguity
Need assumptions
Recovery                                                 on g and k to
of g from T                    Self-similar Ambiguity      recover g
+
Exposure ratio
Exponential Ambiguity
k unknown

 A2: In theory, we can recover exposure ratio directly
from Brightness Transfer Function T
 A3: Geometric correspondence step eliminated allowing
recovery in dynamic scenes:

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