Graphics Reference
In-Depth Information
about units here because it's the form of the computation we care about, not the
actual values.
To determine the sensor pixel's response to the incoming light, we multiply
each bit of light S ( x , y ) by the responsivity M ( x , y ) , and sum up over the entire
pixel:
value =
1
2
1
2
S ( x , y ) M ( x , y ) dy dx .
(18.3)
1
2
1
2
If we extend the definition of M to the whole plane of the sensor by defining M to
be 0 outside the unit square corresponding to pixel ( 0, 0 ) , we can rewrite this as
value 0,0 =
−∞
S ( x , y ) M ( x , y ) dy dx ,
(18.4)
−∞
which may appear more complicated, but actually will result in simpler formulas
elsewhere.
In a well-designed camera, the sensor responsivity should be the same for each
pixel. What does this mean mathematically? It means, for instance, that for sensor
pixel ( 2, 3 ) , we'll want to multiply S ( x , y ) M ( x
2, y
3 ) and integrate, that is,
value 2,3 =
−∞
S ( x , y ) M ( x
2, y
3 ) dy dx ,
(18.5)
−∞
and in general, the formula for sensor pixel ( i , j ) will be
value i , j =
−∞
S ( x , y ) M ( x
i , y
j ) dy dx .
(18.6)
−∞
This expression has the form of a product of a function S with a shifted func-
tion M , integrated; the resultant value is a function of the shift amount, ( i , j ) . That
is the essence of a convolution, and indeed, Equation 18.6 is almost the definition
of the convolution S
M of the two functions. Two small adjustments are needed.
First, since S and M are both functions on all of R 2 , their convolution is defined to
be a function on all of R 2 . The values we've described above are the restriction of
that function to the integer grid. Second, it's very convenient to have a definition
of convolution that makes f
f . For this to work out properly, there needs
to be an extra negation; that is, we want Equation 18.6 to have the form
value i , j =
−∞
g = g
S ( x , y ) M ( i
x , j
y ) dy dx .
(18.7)
−∞
We can arrange this by defining M ( x , y )= M (
y ) . (For a typical sensor, the
response function is symmetric, so M and M are the same.) This final form is just
a 2D analog of the 1D convolution. Simplifying to one dimension, we can now
define the convolution of two functions f , g : R
x ,
R :
g )( t )=
−∞
( f
f ( x ) g ( t
x ) dx .
(18.8)
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