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### Smoothness and monotonicity constraints

We can make an inference that the true is probably smooth, and, as mentioned previously, must be semimonotonic.

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 solving (12). The extra rows appended to are constructed as follows: Let denote the portion appended to , and create 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: A_s = [

] 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).

The rippling is particularly evident when we consider the derivatives of the response functions as shown in Fig 2(b).

Next: Computational efficiency Up: Estimating camera response function Previous: Estimating camera response function
Steve Mann 2002-05-25