0 Computational photography table 2

Computational Photography

 Summary of Relative Noise in White Balanced and Color Corrected Signals.

         QE Set        SB        TrSB    L B              SC           Tr SC   L C

                  397 000 000                    1038   250     082
         RGB      000 262 000     9.88   1.24    250     625    390    24.41   1.60
                  000 000 329                     082   390     778

                  397 000 000                    1822  1470     724
         RPB      000 100 000     8.26   0.84   1470    2168  1394     53.22   2.51
                  000 000 329                     724  1394    1332

                  202 000 000                    2505   643    471
         CMY      000 189 000     5.67   1.03    643    1045    828    51.34   1.90
                  000 000 175                    471    828    1584

  In the case where  2   P   G       G , where i  123, this simplifies to
                              C    E   i
                      g

                                                 
                                         G1  0   0
                                                 
                                K         0 G2   0   G  P                              (1.8)
                                  B                    E C
                                          0  0  G3

To focus on the relative sensitivity, the matrix S is defined by leaving out the factor of P  :
                                                B                                         C

           
                                          G1  0   0
                                                   
                                  SB      0  G2   0   GE                               (1.9)
                                          0   0  G3

The values on the diagonal of SB   show the relative noise levels in white balanced images
before color correction, accounting for the differences in photometric sensitivity. To finish
the comparison, the matrix S   is defined as MS  MT . The values on the diagonal of S     in-
                             C                  B                                      C
dicate the relative noise levels in color corrected images. The values L B  and L C indicate
the estimated relative standard deviation for a luminance channel based on Equation 1.4.

 As shown in Table 1.2, the TrSB  and L B      are smaller for CMY and for RPB than for
RGB, reflecting the sensitivity advantage of the broader spectral sensitivities.   However,
TrSC  and L C  are greater for RPB and CMY than for RGB, reflecting the noise ampli-
fication from the color correction matrix.  In summary, while optimal selection of spectral
sensitivity  is  important  for  limiting  noise,  a  well-selected  relatively  narrow  set  of  RGB
spectral sensitivies is close to optimum, as found in References [65] and [66]. Given these
results, it is tempting to consider narrower spectral bands for each color channel, reduc-
ing the need for color correction.  This would help to a limited extent, but eventually the
signal loss from narrower bands would take over.     Further, narrower spectral sensitivities
would produce substantially larger color errors, leading to lower overall image quality. The
fundamental problem is that providing acceptable color reproduction constrains the three
channel system, precluding substantial improvement in sensitivity.

Reference  [65] considers the possibility of reducing the color saturation of the image,
lowering the noise level at the expense of larger color errors.   However,  the concept of
lowering the color saturation can be applied with RGB quantum efficiencies as well.  Ref-
erence [66] shows that by allowing larger color errors at higher exposure index values, the

optimum set of quantum efficiencies changes with exposure index.  In particular, at a high 

exposure index, the optimum red quantum efficiency peaks at a longer wavelength and has 

less overlap with the green channel.    This is another way to accept larger color errors to 

reduce the noise in the color corrected image. 

Don't miss to read computational photography table 1!

0 Computational photography table

Summary  of  channel  sensitivity  and  color  correction  matrices. The  balance  gains  and  the
       sensitivity gain are respectively denoted by G  G  G   and GE .
                                             1  2  3

        QE Set   Channel Response     G  G  G          GE             M
                                         1  2  3

                                                               1558  0531  0027
        RGB      2616 3972 3159     1518 1000 1257    2.616  0078     1477  0399
                                                               0039  0508    1469

                                                               2000  1373    0373
        RPB      2616  10390 3159   3972 1000 3289    1.000  1062     3384 1322
                                                               0412  1248    1836

                                                             2554     2021   1533
        CMY      5134 5486 5929     1155 1081 1000    1.752    0941  1512    1571
                                                               1201   1783  1984

reflectance,  Qi is the quantum efficiency,  and IEI is the exposure index.  The additional
values are Planck’s constant h, the speed of light c, the spectral luminous efficiency function
V ,  and  normalization  constants  arising  from  the  definition  of  exposure  index. Using  a
relative spectral power distribution of D65 for the illuminant, a pixel size of l  22  m,
and a spectrally flat 100% diffuse reflector, the mean number of photo-electrons captured
in each pixel at an exposure index of ISO 1000 are shown under “Channel Response” in
Table 1.1.


The balance gains listed are factors to equalize the color channel responses.  The sensi-
tivity gain shown is calculated to equalize the white balanced pixel values for all sets of
quantum efficiencies. The color correction matrix shown for each set of quantum efficien-
cies was computed by calculating Equation 1.5 for 64 different color patch spectra, then
finding a color correction matrix that minimized errors between color corrected camera
data and scene colorimetry, as described in Reference [68].
 
The illustration compares the noise level in images captured at the same exposure index
and corrected to pixel value P  . For a neutral, the mean of the balanced pixel values is
                               C
the same as the color corrected pixel values.  Since the raw signals are related to the bal-
anced signal by the gains shown in Table 1.1, the original signal levels can be expressed as
follows:
                                      1    0     0        
                                                        P
                                    GE G1                 C
                                                     
                                            1
                                                           
                            P         0          0      P                            (1.6)
                                                     
                             O            GE G2           C
                                      0    0     1      P
                                                          C
                                                GE G3


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