Exercise · ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-005 · ★☆☆ Basic
Which statement necessarily follows from \(\varphi\) being a smooth diffeomorphic deformation?
Choose one option.
Show answer
- A. The inverse \(\varphi^{-1}\) is also smooth
Explanation
By definition, the inverse of a smooth diffeomorphic deformation is smooth. It does not force distance preservation or fixed points, and its nonvanishing Jacobian rules out folds.
Review the concept: Smooth diffeomorphic deformation
A smooth diffeomorphic deformation \(\varphi\) is a bijective, smooth transformation whose inverse is also smooth.
\[\varphi^{-1}\text{ is smooth}\]
More practice problems: smooth diffeomorphic deformation
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-001 A map φ:M→ N is a candidate to be a diffeomorphism, but its inverse turns out to be only continuous…
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-002 During a registration, the Jacobian of φ vanishes in a region and folds appear. Which property is c…
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-003 If f and g are composable diffeomorphisms, why does g∘ f keep that structure?
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-004 Two deformations reach the same reference points, but one squashes a region until it forms a fold.…
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-006 What is called a smooth, bijective transformation with a smooth inverse?
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-007 What does it mean for the Jacobian of φ to be nonzero at every point?
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-008 What is the velocity field in LDDMM?
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-009 What does the inverse function theorem guarantee for a smooth map with invertible Jacobian?
← ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-004 ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-006 →