Spatial Transformation Models

Roger P. Woods 1 Homogeneous Coordinates 465

UCLA School of Medicine 2 Rigid-Body Model 466

2.1 Two-Dimensional Case • 2.2 Three-Dimensional Case

3 Global Rescaling Transformation 480

4 Nine-Parameter Affine Model 482

5 Other Special Constrained Affine Transformations 484

6 General Affine Model 485

7 Perspective Transformations 487

8 Nonlinear Spatial Transformation Models 489

References 490

Spatial transformation models play a central role in any image registration procedure. These models impose mathematical constraints on the types of geometric distortions that can be imposed during the process of registration. It is fair to say that useful registration cannot be accomplished without some type of formal spatial transformation model. This chapter focuses on spatial transformation models that are linear. A variety of linear models can be used, ranging from rigid-body transformations that preserve all internal angles and distances to perspective models that distort all distances and angles while preserving colinearity. All linear spatial transformations can be expressed using matrix notation, and this chapter details the relationship between the formal parameters of the spatial transformation model and the elements of the corresponding matrix. Useful tools from matrix algebra are described, including singular value decomposition, eigenvector analysis, Schur decomposition, and computation of matrix logarithms and matrix exponentials. These tools are available in most matrix analysis software packages, and algorithms for their implementation are detailed in Golub and Van Loan [1]. These tools clarify the relationships between the transformation matrices and the underlying geometry and provide powerful strategies for diagnosing the geometric constraints imposed by any transformation matrix.

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