Estimating camera response function

From (4), we may find $F$ by minimizing the sum of squared errors: =_x _y ( F(f_1(x)) - F(f_2( x )) + K ) ^2

Since $F$ can only be found up to a single unknown scalar constant, $F(N)$ is fixed at zero³, where $N$ is the number of greyvalues (typically $N=256$).

Equation 4 gives a set of equalities of the form: ${\bf A} {\bf F} = - {\bf K}$, where ${\bf A}\in \R^{L+1\times N}$, and $L$ is the number of pixels in one of the images $f_i$. The first $L$ rows of ${\bf A}$ are constructed by inserting $1$ in the column index corresponding to the pixel value of $f_1$ and inserting $-1$ into the column index corresponding to the pixel value of $f_2$: A(x+wy,f_1(x,y))&=&1
A(x+wy,f_2(x,y))&=&-1 where $w$ is the width of one of the images, and the last row of $\bf A$ is all zeros except its last entry which is 1: A(L+1,N)=1 All unspecified entries of matrix $\bf A$ are zero. Vector $\bf K$ is constructed by placing the value $K$ in the first $L$ entries and $0$ in the last entry. This is an overdetermined system of equations.

The solution that minimizes the error $\varepsilon = \Vert AF + K\Vert^2$ in (9) is the maximum likelihood solution according to the noise model of (5), assuming $n_{q_i} >> n_{f_i}$ [7], and is given by: ddF = 2A^TAF + 2A^TK = 0 giving F=(A^TA)^-1A^T (-K), assuming additive white Gaussian noise. This solution gives us a way of estimating the camera response function from two or more differently exposed pictures of overlapping subject matter.

Although this system is massively overdetermined, the constraints follow a comparametric form that admits solutions having sinusoidal components (e.g. solutions tend to be ``wavy'') [7]. (Indeed, comparametric forms are very similar to difference equations as can be seen from the form of (4) which constrains the function over an interval step size of $K_i$ allowing ripples in the solution.)