= Image registration
{wiki}
<Image registration> estimates a coordinate transformation aligning corresponding image content. If $I$ is the moving image and $J$ is the reference, one simple model chooses a transformation $T$ to minimize
$$
\int_\Omega|I(T(x))-J(x)|^2\,dx+\lambda R(T).
$$
The first term compares image values; $R$ penalizes unwanted deformation. A translation model $T(x)=x-a$ has only a displacement vector to estimate, while a spatially varying deformation needs many more parameters and stronger <regularization>.
The transformation direction is part of the convention: this expression samples the moving image at $T(x)$ for each reference location. The sampling requires <interpolation> when $T(x)$ falls between pixel centers. Boundary handling must also specify what happens outside the image domain.
Intensity agreement does not prove correct correspondence. Repeated patterns can produce several alignment minima, and a change of imaging modality can alter intensities even at matching physical locations. The transformation family, similarity objective, interpolation rule and deformation penalty are distinct modeling decisions in this <inverse problem>.
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