keystone-rogass17052013.pdf

Coarse Bisection

Analogy:

• Imagine you are trying to find the exact weight of an object using a balance scale, but you don't have any reference weights. To estimate the weight, you start by placing the object on one side of the scale and adding an arbitrary weight on the other side. Let's call this initial guess "Weight A." Now, you observe that the scale tips to one side, indicating that Weight A is not the correct weight.
• To improve your estimate, you try another weight that is slightly heavier than Weight A, let's call it "Weight B." You observe how the scale responds to Weight B. If the scale tips less than it did with Weight A, you can infer that the object's weight is likely closer to Weight B than Weight A. On the other hand, if the scale tips more with Weight B, you can infer that the object's weight is likely closer to Weight A than Weight B.
• Now, you have narrowed down the possible range of weights to two intervals: one with Weight A and the other with Weight B. Since you are looking for a more accurate estimate, you decide to further refine the intervals. You choose two new weights, one closer to Weight A and the other closer to Weight B, and you repeat the process. Let's call these new weights "Weight A1" and "Weight B1."
• Again, you observe how the scale responds to these new weights. If the scale tips less with Weight A1 compared to Weight B1, you can infer that the object's weight is likely closer to Weight A1 than Weight B1. Conversely, if the scale tips more with Weight A1, you can infer that the object's weight is likely closer to Weight B1.
• By iteratively repeating this process and choosing weights with smaller intervals, you eventually narrow down the range to a smaller and smaller region until the scale stops tipping significantly. At this point, you have a good estimate of the object's weight, and you consider it your final solution.

Method:

• In the context of keystone estimation, coarse bisection is a numerical optimization technique used to iteratively refine the estimation of the differential keystone distortion coefficients. It aims to efficiently narrow down the possible range of coefficients for a more accurate estimation by dividing the coefficient space into intervals and iteratively adjusting the intervals based on the observed phase shifts.
• The term "bisection" refers to the process of dividing an interval into two parts, and "coarse" indicates that the intervals between the guesses are relatively large at the beginning of the process.
• To generate two additional guesses, you can use the initial guess and modify its coefficients slightly to create variations in the overall slope of the polynomial. This process is typically achieved by scaling the initial coefficients up or down while maintaining the same relative relationships between the coefficients.