Image Morphing & Spatial Transformations

Image Morphing & Spatial Transformations

Gonzalez & Woods • DIP

According to Digital Image Processing by Gonzalez & Woods, image morphing belongs to the class of Geometric Spatial Transformations (Image Warping) combined with intensity interpolation. A spatial transformation modifies the spatial relationship between pixels in an image.

1. Two-Step Mapping Mechanism

A complete morphing process relies on two fundamental operations applied to image coordinates $(x, y)$:

• Spatial Coordinate Transformation (Warping): Mapping spatial coordinates $(x, y)$ to new coordinates $(x', y')$ using transformation equations.
• Intensity Interpolation (Gray-Level Mapping): Assigning pixel values to the newly mapped coordinates using methods like Nearest-Neighbor, Bilinear, or Bicubic Interpolation.

2. Affine & Matrix Transformations

Forward spatial mapping transforms coordinates via linear combination matrices:

// General Affine Transformation Matrix (Gonzalez & Woods)
[x' y' 1] = [x y 1] * T

T = | t₁₁ t₁₂ 0 | (Rotates, scales, shears, and translates)
| t₂₁ t₂₂ 0 |
| t₃₁ t₃₂ 1 |

3. Tie-Points & Mesh-Based Warping

When the transformation cannot be modeled globally by a single matrix, Tie-Points (Control Points) are established across quadrangle or triangular meshes (Delaunay Triangulation) over both images.

For a triangular region with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$, the mapped coordinates are uniquely determined using affine coefficient solvers:

x' = c₁x + c₂y + c₃
y' = c₄x + c₅y + c₆

// Solved via 3 non-collinear tie-points per triangle pair

4. Inverse Mapping vs. Forward Mapping

Mapping Type	Mechanism	Key Advantage / Disadvantage
Forward Mapping	Maps directly from source $(x, y)$ to destination $(x', y')$	Causes hole artifacts or overlap when multiple pixels map to one destination.
Inverse Mapping	Iterates target coordinates $(x', y')$ backward to source $(x, y)$	Guarantees every output pixel is filled via bilinear/bicubic interpolation.

5. Intensity Cross-Dissolving

Once both images are spatially warped to an intermediate control-point geometry at morph stage $t \in [0, 1]$, pixel intensities are combined:

f_morph(x, y, t) = (1 - t) · f_A(x_A', y_A') + t · f_B(x_B', y_B')

Notes Created By Bijan Krishna Paul Digital Image Processing (DIP) Syllabus