Problem 1: Develop a matrix equation for minimizing the cost in image registration (without summation indicates the sum over all points in a shape).
Cost = -T-
x - ax + by + t 12 y - bx + ay + t2
Cost = x - ax + by + t + y - bx + ay +
To find the optimal transformation that will align image 2 to image 1, take the partial derivatives of the above cost with respect to a, b, t1, and t2 and set these to 0. Express the four resulting equations in matrix form. ∂C/∂a = 0 ∂C/∂b = 0 ∂C/∂t1 = 0 ∂C/∂t2 = 0 For example, the first equation will appear as: ∂C/∂a = -2x(x - ax + by + t) - 2y(y - bx + ay + t) = 0 -2xx + 2ax + 2bx - 2yy + 2ay = 0 2x + 2y + 0b + 2t = 2xx + 2yy In matrix form, it can be written as: 2x + 2y 0 2x 2y xx + 2yy
The equation can be expressed as: At = b where t is the unknown transformation vector consisting of a, b, t1, and t2. It can be easily solved as 9-V.