Since can only be found up to a single unknown scalar constant, is fixed at zero3, where is the number of greyvalues (typically ).
Equation 4 gives a set of equalities of the form:
and is the number of pixels in one of the images .
The first rows of are constructed by inserting in
the column index corresponding to the pixel value of
into the column index corresponding to the pixel value of :
A(x+wy,f_2(x,y))&=&-1 where is the width of one of the images, and the last row of is all zeros except its last entry which is 1: A(L+1,N)=1 All unspecified entries of matrix are zero. Vector is constructed by placing the value in the first entries and in the last entry. This is an overdetermined system of equations.
The solution that minimizes the error in (9) is the maximum likelihood solution according to the noise model of (5), assuming , 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'') . (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 allowing ripples in the solution.)