Updating the maximum

The maximum of $q$ numbers can be written as an update equation:

$$\bigvee_{k=0}^{q-1} x_k = x_{q-1} \vee \bigvee_{k=0}^{q-2} x_k$$ (10)

i.e, the maximum of the $k^{th}$ value and the maximum of the first $k-1$ values. The important thing to realize from the above equation is that we don't need to know all the values in advance, i.e. we can update the maximum as and when we get a new value. Consequently, we can perform the maximum operation on the images of Eq. 9 one at a time. Also, we don't need to store all $k-1$ values to compute the $k^{th}$ value. This reduces computer memory requirements and in practice, improves the speed of execution.

In Section 2.8, we use this update method to compute the maximum for both Eq. 1 and Eq. 9. Recall that in the original method, the maximum is of the results of matching the filter at a particular pixel value while in the modified technique, the maximum is of the results of matching all the pixels. Consequently, the memory usage of the algorithm at the time of the maximum operation is higher in the modified technique than in Wolfson et al. (1999). Hence, the effect of updating the maximum as each filtered image is available instead of attempting to store 18 filtered images is significant in the modified technique.