Parametric methods

In situations where the image data is extremely noisy, and/or where a closed-form solution for f(q) is desired, a parametric method is used, in which a function g(f)that is the range-range function of a suitable response function f(q) is selected. The method amounts to a curve fitting problem in which the parameters of g are selected so that g best fits one or more joint histograms constructed from two or more differently exposed spatially registered images under analysis.

Various parametric models have been considered for this purpose [12]. For example, the one parameter model, $f(q)=exp(q^\Gamma)/e$is equivalent to applying gamma correction $\gamma = k^\Gamma$ to f in order to obtain $g = f(kq) = f^\gamma$, where $\Gamma$ is typically between 0 and 1/2. Note that when tonally registering images using this gamma correction, one makes an implicit assumption that f(q) does not go all the way to zero when q=0. Thus a better model is f=exp((1+q^b)^c) which also nicely captures the toe and shoulder of the response function.

However, one of the simplest models from a parameter estimation point of view is the classic response curve,

$$f(q)=\alpha + \beta q^\gamma,$$ (8)

used by photographers to characterize the response of a variety of photographic emulsions (and suitable also to characterize the response of electronic imaging devices which attempt to emulate the appearance of film), including so-called extended response film[15].

Proposition 3.2   The comparametric plot corresponding to   the standard photographic response function (8) is a straight line. The slope is $k^\gamma$, and the intercept is $\alpha (1-k^\gamma)$.
Proof: $g(f(kq))=f(kq)=\alpha + \beta (kq)^\gamma$ Re-arranging to eliminate q gives $g= k^\gamma(\alpha+\beta q^\gamma) + \alpha(1-k^\gamma)$ so that:

$$g=k^\gamma f + \alpha(1-k^\gamma)$$ (9)

This result suggests that f(q) can be determined from two or more differently exposed images by applying linear regression to the joint histogram, J₁₂, of the images, I₁ and I₂.

When two successive frames of a video sequence are related through a group-action that is near the identity of the group, one may think of the Lie algebra of the group as providing the structure locally, so that (7) may be solved as follows: From (8) we have that the (generalized) brightness change constraint equation [28] is: $% g(f(q(x,t))) = f(kq(x,t)) = f(q(x+\Delta x,t + \Delta t)) = k^\gamma f(q(x,t)) + \alpha - \alpha k^\gamma = k^\gamma I(x,t) + \alpha(1-k^\gamma)$, where I(x,t)=f(q(x,t)). Combining this equation with the Taylor series representation:

$$I(x+\Delta x, t + \Delta t) = I(x,t) + \Delta x I_x(x,t) +\Delta t I_t(x,t)$$ (10)

where I_x(x,t) = (df/dq)(dq(x)/dx), at time t, and I_t(x,t) is the frame difference of adjacent frames, we have:

$$k^\gamma I + \alpha (1-k^\gamma) = I + \Delta x I_x + \Delta t I_t$$ (11)

Thus, the brightness change constraint equation becomes:

$$I + uI_x + I_t - k^\gamma I - \alpha(1-k^\gamma) = \epsilon$$ (12)

where, normalizing, $\Delta t = 1$.