Smoothness and monotonicity constraints

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

Thus the estimate of $F$ can be constrained in both smoothness and monotonicity. A smoothness constraint may be formulated by appending $N-L_s$ extra rows to ${\bf A}$ and the same number of extra zeros to the end of $\bf K$, where $L_s$ is the length of the appropriate smoothness filter, $\bf s$, and then solving (12). The extra rows appended to $\bf A$ are constructed as follows: Let ${\bf A_s}\in \R^{N-L_s\times N}$ denote the portion appended to $\bf A$, and create $\bf A_s$ as a toeplitz matrix in which the first $L_s$ 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 $\lambda [1, -2, 1]$ for which the effect of appending the corresponding $\bf A_s$ to $\bf A$ is to impose a penalty for nonzero second derivatives (inflection) of the curve $F$. The amplitude, $\lambda$, 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, $F$, from two differently exposed images, are shown in Fig 2(a).

Figure 2: (a) Response functions estimated from the data for no smoothing (solid line), smoothing with $\lambda =100$ (dotted line), and smoothing with $\lambda =1000$ (dotted line). (b) Derivatives of the response functions make the effects of smoothing more evident. The derivative of the response function is called the certainty function, because it shows the sensitivity of pixel integer output as a function of changes in light [7]. Slight ripple in the response function becomes magnified and visible as periodicity components in these certainty functions. Note how the harmonic contributions have a period of $K$ (normalized to stepsize $K=1$ in this plot).

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