Image Registration
Rigid, similarity and affine registration: the transform matrix, RMS displacement over the image, control-point target registration error (Fitzpatrick) and the Cramér–Rao bound for intensity-based alignment.
RMS displacement over the image
934.0 nm
9.34 px; largest at a corner 1833.1 nm
Control-point TRE (RMS over the image)
38.7 nm
0.387 px; centre 31.6, corner 50.0 nm
Expected fiducial residual FRE
92.2 nm
The fit residual, not the target error: a small FRE does not mean a small TRE
Intensity registration bound (per axis)
0.039 nm
0.00039 px: Cramér–Rao bound for the translation
Transform x' = A·x + t (pixels, origin at the image centre)
Control-point TRE vs number of pairs (RMS over the image)
Key Formulas
Displacement: ⟨|d|²⟩ = |t|² + (L²/12)·‖A − I‖²_F over a centred L × L image
TRE: TRE² = (FLE²/N)·(1 + m·d²/(2f²)), m = 1, 2, 4 for rigid, similarity, affine
FRE: FRE² = FLE²·(1 − p/(2N)), p = 3, 4, 6 parameters
Cramér–Rao: σ_t ≥ ℓ/(SNR·√N_px) per axis, from J = Σ∇f∇fᵀ/σ²
The fiducials are spread uniformly over the image, so f² = L²/6 and the RMS target sits at d = f. The 2-D TRE and FRE follow Fitzpatrick's first-order least-squares analysis with isotropic, independent localisation errors, for fiducials on a regular, centred, isotropic layout; random placement raises the affine TRE by ≈ 20 % at N = 10 (rigid ≈ 6 %). The bound assumes a noise-free reference, white Gaussian noise on the moving image, and a random texture whose gradient variance is σ_s²/ℓ². The bound assumes a noise-free reference and no interpolation bias; with two equally noisy images it is √2 larger. Elastic (B-spline) registration has a displacement field instead of a matrix and is not modelled.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
First-order least squares: 2-D TRE² = (FLE²/N)(1 + m·d²/(2f²)) and FRE² = FLE²(1 − p/(2N)) for isotropic, independent fiducial errors and fiducials on a regular layout over the image (random placement raises the affine TRE ≈ 20 %). The translation bound assumes a noise-free reference and a Gaussian-correlated random texture. Elastic registration is not modelled.
- Fitzpatrick J. M., West J. B., Maurer C. R. (1998). Predicting error in rigid-body point-based registration. IEEE Trans. Med. Imaging 17(5), 694–702. TRE and FRE from linearised least squares (3-D; the page uses the 2-D analogue). doi:10.1109/42.736021
- Robinson D., Milanfar P. (2004). Fundamental performance limits in image registration. IEEE Trans. Image Process. 13(9), 1185–1199. Section II-A: Fisher information of a translation. doi:10.1109/TIP.2004.832923