COMPARAMETRIC EQUATIONS, QUANTIGRAPHIC IMAGE
DIFFERENTLY EXPOSED IMAGES OF THE SAME SUBJECT MATTER As previously mentioned, comparison of two or more differently exposed
4.3.9 Power of Root over Root Plus Constant Correction of Images Although affine correction was an improvement over zeta correction, which itself
was an improvement over gamma correction, affine correction still has the two shortcomings listed above. Therefore another form of image exposure correction is proposed, and it will be calledpower of root over root plus constant correction.
This new exposure correction is unicomparametric (bounded in normalized units between 0 and 1) and also has a parameter to control the softness of the transition into the toeand shoulderregions of the response function, rather than the hard clipping introduced by (4.54).
As with affine correction,power of root over root plus constant correctionwill be introduced first by its solution, from which the comparametric equation will be derived. The solution is
f (q)=
ebqa ebqa+1
c
, (4.55)
35 30 25 20 15 10 5 0
50 100 150 200 250
Q (Quantity of light) (a )
f (response byte)
Estimated camera response function with ground truth data
14 13 12 11 10 9 8 7 6
0 50 100 150 200 250
Q (Quantity of light) (b )
f (response byte)
Estimated camera response function with ground truth data
Figure 4.15 The standard power law photographic response function (4.51) can only fit the response of the imaging apparatus over a narrow region of exposure latitude. (a) Best fit over the full 37/3 F-stops is poor. (b) Best fit over an interval of ten thirds of a stop is satisfactory.
Although this region of exposures is typical of conventional photography, a feature of cybernetic photography is the use of deliberate massive overexposure and underexposure. Indeed, the human eye has a much wider exposure latitude than is suggested by the narrow region over which the power law model is valid. Therefore a new model that captures the essence of the imaging system’s response function in regions of extreme exposure is required.
which has only three parameters. Thus no extra unnecessary degrees of freedom (which might otherwise capture or model noise) have been added over and above the number of degrees of freedom in the previous model (4.51).
An intuitive understanding of (4.55) can be better had by rewriting it:
f =
1
(1+e−(alog(q)+b))c ∀q=0,
0 forq=0.
(4.56)
written in this form, the soft transition into the toe (region of underexposure) and shoulder (region of overexposure) regions is evident by the shape this curve has if plotted on a logarithmic exposure scale,
f = 1
(1+e−(aQ+b))c, (4.57) where Q=log(q); see Figure 4.16.
The (4.57) model may, at first, only seem like a slight improvement over (4.51), given our common intuition that most exposure information is ordinarily captured in the central portion that is linear on the logarithmic exposure plot.
Q D
Certainty
Q D
(a )
(b ) (c )
Q
Figure 4.16 Example of response functions 1/(1+e−(alog(q)+b))c, which have soft transition into the toe region of underexposure and shoulder region of overexposure. Traditionally these responses would be on film, such as the Wyckoff film, having density ‘D’ as a function of log exposure (i.e.,Q=log(q)). (a) Response functions corresponding to three different exposures.
In the Wyckoff film these would correspond to a coarse-grained (fast) layer, denoted by a dashed line, that responds to a small quantity of light, a medium-grained layer, denoted by a dotted line, that responds moderately, and a fine-grained layer, denoted by a solid line, that responds to a large quantity of light. Ideally, when the more sensitive layer saturates, the next most sensitive layer begins responding, so each layer covers one of a set of slightly overlapping amplitude bins. (b) Tonally aligning (i.e., tonally ‘‘registering’’ by comparadjustment), the images creates a situation where each image provides a portion of the overall response curve. (c) The amplitude bin over which each contributes is given by differentiating each of these response functions, to obtain the relative ‘‘certainty function.’’ Regions of highest certainty are regions where the sensitivity (change in observable output with respect to a given change in input) is maximum.
However, it is important that we unlearn what we have been taught in traditional photography, where incorrectly exposed images are ordinarily thrown away rather than used to enhance the other images! It must be emphasized that comparametric image processing differs from traditional image processing in the sense that in comparametric image processing (using the Wyckoff principle, as illustrated in Fig. 4.4) the images typically include some that are deliberately underexposed and overexposed. This overexposure of some images and underexposure of other images is often deliberately taken to extremes. Therefore the additional sophistication of the model (4.55) is of great value in capturing the essence of a set of images where some extend to great extremes in thetoeorshoulderregions of the response function.
Proposition 4.3.4 The comparametric equation of which the proposed photo-graphic response function (4.55) is a solution, is given by
g(f )= f kac (√c
f (ka −1)+1)c, (4.58)
whereK =log(k).
Again, note that g(f ) does not depend on b, which is consistent with our knowledge that the comparametric equation captures the information off (q)up to a single unknown scalar proportionality constant.
Therefore we may rewrite (4.55) in a simplified form f (q)=
qa qa+1
c
, (4.59)
where b has been normalized to zero, and where it is understood that q >0, since it is a quantity of light (thereforef is always real). Thus we have, for q,
q= a
√c
f (q) 1−√c
f (q). (4.60)
From this simple form, we see that there are two degrees of freedom, given by the free parametersaandc. It is useful and intuitive to consider the slope of the corresponding comparametric equation (4.58),
dg
df = kac
(√c
f (ka−1)+1)c+1 (4.61) evaluated at the origin, which iskac.
If the exposure,k, is known we can resolve the two degrees of freedom of this function into a slope-at-the-origin termkac, and a sharpness term proportional to a/c. Thus we can vary the product ac to match the slope at the origin. Then,
once the slope at the origin is fixed, we can replace a with a* sharpness and replace c with c* sharpness where sharpness typically varies from 1 to 200, depending on the nature of the camera or imaging system (e.g., 10 might be a typical value for sharpness for a typical camera system).
Once the values of a and c are determined for a particular camera, the response function of that camera is known by way of (4.60). Equation (4.60) provides a recipe for converting from imagespace to lightspace. It is thus implemented, for example, in the comparametric toolkit as function pnm2plm (from http://wearcam.org/cement), which converts images to portable lightspace maps.
It should also be emphasized that (4.60) never saturates. Only whenqincreases without bound, does f approach one (the maximum value). In an actual camera, such as one having 8 bits per pixel per channel, the value 255 would never quite be reached.
In practice, however, we know that cameras do saturate (i.e., there is usually a finite value of qfor which the camera will give a maximum output). Thus the actual behavior of a camera is somewhere between the classic model (4.51) and that of (4.60). In particular, a saturated model turns out to be the best.
4.3.10 Saturated Power of Root over Root Plus