11.0 WARPING, MORPHING, AND LAGRANGIAN APPROACHES⧉
Straightening a façade, aligning a burst and moving a face landmark all change where image samples contribute. A warp specifies that geometry; resampling determines the output values. Keeping those two tasks separate lets us reuse one sampling implementation for many kinds of edit.
Chapter 11.1, Warping defines forward and backward maps, boundaries and interpolation. Chapter 11.2, Antialiasing for complex transforms handles footprints that stretch or rotate across the image. A destination pixel may cover many source samples, so a sharper interpolator alone cannot prevent aliasing.
Chapter 11.3, Morphing combines correspondence with appearance blending. Chapter 11.4, Morphable models restricts possible shapes using a learned population model, while Chapter 11.5, Shape-preserving warping penalizes local deformation. These constraints can improve an edit without guaranteeing physical validity or preventing every fold.
Chapter 11.6, Perspective distortion and its correction separates planar rectification, virtual camera rotation and local perceptual adjustments. None can recover unseen surfaces from a single photograph without additional information or inference.
The distinction between changing values and moving content is related to two descriptions of fluid motion. An Eulerian description observes a fixed location, like a weather station measuring the air that passes it. A Lagrangian description follows material, like a balloon drifting with that air. A tone curve changes values on a fixed image grid. A geometric warp instead describes where scene content moves. The implementation may still visit a fixed output grid through backward mapping; the Lagrangian analogy concerns the motion being represented. Video compression shows why the distinction matters in practice. Coding each frame as its per-pixel difference from the previous one is an Eulerian scheme, and it wastes bits whenever something moves, since a moving edge produces large differences at every pixel it crosses. Motion-compensated coding, as in the MPEG and H.264 families, predicts each block from where its content was in the previous frame and codes only the correction, which is cheap when the motion is well described (Chapter 30.2, Video compression and motion compensation).
That separation also explains the organization of these chapters. One question is how to specify the map: a few parameters, sparse landmarks, a deformation mesh or a dense field. Another is how to apply it to sampled data. Translation, lens correction, stabilization and a facial morph can share a resampling engine even though their maps come from different models. Matching pixels across space and time addresses estimating those correspondences; this part develops their use to transport image content.
Contents of this part
- 11.1 Warping
- 11.1.1 Warping is a domain transform: move where, not what
- 11.1.2 Forward vs inverse warping: why output-driven wins
- 11.1.3 Resampling: a quick reminder
- 11.1.4 Specifying the warp I: parametric models and the degrees-of-freedom (DOF) ladder
- 11.1.5 Specifying the warp II: free-form warps from sparse correspondences
- 11.1.6 Specifying the warp III: feature lines (Beier–Neely)
- 11.1.7 Liquify: brush-driven warping
- 11.2 Antialiasing for complex transforms
- 11.2.1 When the footprint stops being a square: the warp Jacobian
- 11.2.2 EWA: the elliptical weighted average
- 11.2.3 Feline and anisotropic MIP probing: EWA on a GPU budget
- 11.2.4 Other footprint integrators: summed-area tables and ripmaps
- 11.2.5 Footprint estimation, clamping, and when to reach for it
- 11.2.6 Jacobians of common warps
- 11.3 Morphing
- 11.3.1 Why a cross-dissolve isn't enough: the ghosting motivation
- 11.3.2 Two interpolations: domain (shape) and range (color)
- 11.3.3 The morphing recipe (combine both)
- 11.3.4 Field morphing: the Beier–Neely line-pair warp
- 11.3.5 Mesh / triangulation morphing (the alternative warp)
- 11.3.6 View morphing: the geometrically-correct in-between of two views
- 11.3.7 Recap and significance
- 11.4 Morphable models
- 11.4.1 Step 1: dense correspondence, the precondition for averaging shapes
- 11.4.2 Step 2: Procrustes alignment, factor out pose so PCA sees shape
- 11.4.3 Step 3: PCA, a mean and a basis of eigen-deformations
- 11.4.4 Step 4: shape and appearance are separate bases
- 11.4.5 Step 5: fitting a new photo, analysis-by-synthesis
- 11.4.6 Step 6: once fitted, edit by moving in the space
- 11.4.7 Step 7: from linear PCA to neural priors
- 11.4.8 Recap and significance
- 11.5 Shape-preserving warping
- 11.5.1 Why plain interpolation shears: the rigidity gap
- 11.5.2 As-rigid-as-possible (ARAP) manipulation
- 11.5.3 Linear blend skinning and why the weights decide everything
- 11.5.4 Bounded biharmonic weights (BBW)
- 11.5.5 ARAP interpolation: rigid-as-possible morphing between two poses
- 11.5.6 Moving least squares: the simpler cousin
- 11.5.7 Recap and significance
- 11.6 Perspective distortion and its correction
- 11.6.1 Keystoning is projection, not a lens flaw
- 11.6.2 The fix is a homography: re-render the façade fronto-parallel
- 11.6.3 The optical alternative at capture: tilt-shift / Scheimpflug
- 11.6.4 Cropping wide-angle photographs: recenter with a homography, don't just crop
- 11.6.5 A different perspective distortion: wide-angle portraits, and a content-aware fix
- 11.6.6 Beyond faces: correcting arbitrary objects (MaDCoW)
- 11.6.7 A montage has no single viewpoint: a family in a box
- 11.6.8 Recomposing perspective after capture: computational zoom
- 11.6.9 The catch: resampling cost, and "only a plane rectifies exactly"
- 11.6.10 Where this sits: one map, then transport
- 11.6.11 Robustness of perspective to the viewer's viewpoint