Table 1.
Notation (frequently used acronyms and symbols).
| Notation | Description |
|---|---|
|
| |
| GN | Gauss–Newton |
| KKT system | Karush–Kuhn–Tucker system |
| PCG | preconditioned conjugate gradient method |
| PDE | partial differential equation |
| PDE solve | solution of hyperbolic PDEs of optimality systems (4.1) and (4.3) |
|
| |
| mR | reference (fixed) image (mR : Rd → R) |
| mT | template image (image to be registered; mT: Rd → R) |
| y | mapping (deformation; y : Rd → Rd) |
| v | velocity field (control variable; v : Rd × [0, 1] → Rd) |
| m | state variable (transported image; m : Rd × [0, 1] → R) |
| m1 | state variable at t = 1 (deformed template image; m1 : Rd → R) |
| λ | adjoint variable (transported mismatch; λ : Rd × [0, 1] → R) |
| f | body force (drives the registration; f : Rd × [0, 1] → Rd) |
| F1 | deformation gradient (tensor field) at t = 1 (F1 : Rd → Rd×d) |
| 𝒥 | objective functional |
| 𝒮 | regularization functional |
| 𝒜 | differential operator (first and second variation of 𝒮) |
| β | regularization parameter |
| γ | parameter that enables (γ = 1) or disables (γ = 0) the incompressibility constraint |
| nt | number of time points (discretization) |
| nc | number of coefficient fields (spectral Galerkin method in time) |
| nx | number of grid points (discretization; ) |
| g | reduced gradient (first variation of Lagrangian with respect to v) |
| ℋ | reduced Hessian (second variation of Lagrangian with respect to v) |