Graphics Reference
In-Depth Information
univariate polynomial in the pixel value,
N
n
=
0
c
n
M
n
I
=
f
(
M
)=
.
Only the coefficients need to be determined in order to model the response func-
tion. There is no theoretical reason to believe that camera response functions
are polynomials; however, the model is justifiable because
any
function can be
uniformly approximated by a polynomial.
The authors describe their method to compute the polynomial coefficients as
radiometric self-calibration
The process works on a sequence of
q
Q
images, each taken from the same vantage point under the same lighting condi-
tions. It is assumed that the only variation is in the exposure time and the aperture
size. If the pixels are indexed by
p
,
M
p
,
q
and
I
p
,
q
denote the pixel value and ideal
exposure, respectively, of pixel
p
in image
q
. Suppose the ratio
R
q
=
=
1
,
2
,...,
e
q
/
e
q
+
1
between two images
q
and
q
+
1 in the image is known. Then, for each pixel
p
,
I
p
,
q
I
p
,
q
+
1
=
f
(
M
p
,
q
)
R
2
=
M
p
,
q
+
1
)
,
f
(
and therefore
N
n
=
0
c
n
M
p
,
q
=
R
q
N
n
=
0
c
n
M
p
,
q
+
1
.
(6.6)
Equation (6.6) is a system of linear equations in
c
n
and
R
q
over all the pixels
p
.
With a known value of
R
q
, this system is an overdetermined linear system; a so-
lution that minimizes the error in the least squares sense is used. This amounts to
minimizing a system of quadratic equations, which can be done exactly by setting
the derivatives with respect to the variables
c
n
to zero and solving the resulting
system of linear equations. An extra constraint is applied so that the response
curve of the maximum pixel value is normalized.
In practice, the ratios
R
q
are not known, but they can be estimated by taking the
ratio of pixel values in the middle range. Mitsunaga and Nayar employ an iterative
method. It starts with an approximation to each
R
q
, computes the polynomial
coefficients, then constructs a new value of
R
q
using the approximation to
f
.The
value of
R
q
is computed from the values of
I
using the polynomial approximation
to
f
. The process is repeated until the computed error term is sufficiently small,
or stops changing. More precisely, each iteration starts with an approximation
R
(
k
−
1
)
, computes the polynomial coefficients
c
(
k
n
, then updates
R
q
from
q
n
=
0
c
(
k
n
M
p
,
q
N
∑
R
(
k
)
=
∑
p
0
c
(
k
n
M
p
,
q
+
1
.
q
n
∑
=
