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Newton's approach starts from an initial value and refines this value using the assumption that is locally linear. A first order approximation of
yields:

(A2) 
with the Jacobian matrix and a small displacement. Under these assumptions minimizing
can be solved through linear leastsquares. A simple derivation yields

(A3) 
This equation is called the normal equation. The solution to the problem is found by starting from an initial solution and refining it based on successive iterations

(A4) 
with the solution of the normal equation A.3 evaluated at . One hopes that this algorithm will converge to the desired solution, but it could also end up in a local minimum or not converge at all. This depends a lot on the initial value .
Next: LevenbergMarquardt iteration
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Marc Pollefeys
20021122