Weighting the solution by the certainty function

It is expected that pixel values near the middle of their range in the image will change more rapidly for a given change in exposure than pixel values close to the extremes. For example, when a pixel is saturated (e.g. 255 in the case of an eight bit camera) no additional light will change the pixel value. Therefore, we expect that the midtones in the image will give greater certainty in estimating $f$ than will the shadows and highlights. This intuition is made formal by the so-called certainty functionsmannwyckoff[3], which are the derivatives of the response functions.

The certainty functions may therefore be used to weight the columns of ${\bf A}$, and the corresponding entries of the vector $\bf K$, when solving for $\bf F$: A_w=A (w1)
A_w^TA_wF - A_w^TK_w = 0 where $\circ$ denotes Hadamard multiplication⁵, $\bf w$ is a column vector in which each entry is made up of $c(n)$ where $c$ is the certainty function, and $n$ is the column index in which the quantity $-1$ appears for that row.

In practice, we do not know the response function (this is the very entity we are trying to estimate) so we also do not know a-priori, its derivative, the certainty function, which we need to use in the weighting.

Figure 7: (a) One of 19 differently exposed pictures of test pattern. (b) Each of the 19 exposures produced 11 ordered pairs in a plot of $f(Q)$ as a function of $Q$. (c) Shifting these 19 plots left or right by the appropriate $K_i$ allowed them all to line up to produce the ground truth known-resonse function $f(Q)$.

However, we can use a certainty function that has the general shape of the derivative of a typical response function. It has been found that the solution $F$ is not particularly sensitive to the shape of the certainty function. A Gaussian weighting is generally used. If desired, once $F$ is found, $F^{-1}$ can be differentiated and used as the certainty function to weight the columns of $\bf A$ to re-estimate $F$. This procedure can be repeated again (e.g. giving rise to an iterative estimation of $F$).