During a registration, the Jacobian of φ vanishes in a region and folds appear. Which property is compromised?

Smooth invertibility and, therefore, the one-to-one correspondence

The existence of a formula for φ

The possibility of measuring any pixel

Exercise · ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-002 · ★★☆ Intermediate

During a registration, the Jacobian of \(\varphi\) vanishes in a region and folds appear. Which property is compromised?

Choose one option.

Show answer

If the Jacobian vanishes, the differential is no longer invertible, the map loses its smooth inverse and folds appear.

A smooth diffeomorphic deformation \(\varphi\) is a bijective, smooth transformation whose inverse is also smooth.

\[\varphi^{-1}\text{ is smooth}\]