Thus the estimate of can be constrained in both smoothness and
monotonicity. A smoothness constraint may be formulated
by appending extra rows to and the same
number of extra zeros to the end of ,
where is the length of the appropriate
smoothness filter, , and then
The extra rows appended to are constructed as follows:
denote the portion appended to ,
as a toeplitz matrix in which the first elements
of the first row are the filter coefficients, and the remaining
rows are appropriately shifted versions of the coefficients:
] In order to achieve smoothness, the filter needs to be a highpass filter. (The intuition for this comes from the fact that by ``looking'' at the function through a highpass filter, this makes it ``expensive'' for the curve to have high frequency (non-smoothness) content since the right hand side vector for this portion of the matrix equations is zero.)
The simplest filter is a three-tap filter for which the effect of appending the corresponding to is to impose a penalty for nonzero second derivatives (inflection) of the curve . The amplitude, , of the filter, determines how heavily the smoothness constraint is weighted. Additionally, a monotonicity constraint may be imposed using Quadratic Programming (QP).
Examples of determining the response function, , from two differently exposed images, are shown in Fig 2(a).