What can be Known about the Radiometric Response

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					What can be Known about the
Radiometric Response from Images?
Michael Grossberg and Shree Nayar




CAVE Lab, Columbia University
Partially funded by NSF ITR Award




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




Inverse                    I
              Irradiance




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




                                                            Irradiance
               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
    How is Radiometric Calibration Done?

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

                                                                                    I




                                                                           Irradiance
                         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)



                       Irradiance
  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



                      Irradiance
   [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
Irradiance




                     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
                         (Fake Irradiance)



          Histogram Equalization      Histogram Equalization


                     Histogram Specification



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


Histogram Specification = Brightness Transfer Function
             Results: Object Motion




             Recovered Inverse Radiometric Response Curves
             1                                                              1                                                               1
             0.9                                                            0.9          Recovered Response                                 0.9
             0.8                                                            0.8                                                             0.8
Irradiance




                                                               Irradiance




                                                                                                                               Irradiance
             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
Results:                   Radiometric Response Curves
                                          1

Object and Camera                         0.9
                                          0.8




                             Irradiance
                                          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




                             Irradiance
                                          0.7
                                          0.6      Macbeth Chart
                                          0.5          Data
                                          0.4
                                          0.3
                                          0.2
    Red Irradiance   Green Irradiance     0.1                          Blue Irradian
                                          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




                              Irradiance
                                          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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posted:9/7/2012
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