Need help with Image Registration and Cubic Spline Interpolation

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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

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