Need help with Image Registration and Cubic Spline Interpolation

Hi guys! I was asked to implement a problem which had the details represented in the image. I am new to image processing and am having difficulty understanding them. I don't understand as to how do they get g'(x) = x + Ө* as Ө* is undefined.

What is the significance of cubic B-Spline here? How do we get the value of pixel x, do we just assume randomly?

Please help me get started, I am interested in learning them , but I am confused with this.

Thank you