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144 3 Image processing

f g f g
h h

f g f g
h h

x x x x

Figure 3.44 Image warping involves modifying the domain of an image function rather than its range.

y projective
similarity
translation

Euclidean affine
x

Figure 3.45 Basic set of 2D geometric image transformations.

Transformation Matrix # DoF Preserves Icon

translation I t 2 orientation
2×3

rigid (Euclidean) R t 3 lengths
2×3

similarity sR t 4 angles
2×3

affine A 6 parallelism

2×3
˜
projective H 8 straight lines
3×3
Table 3.5 Hierarchy of 2D coordinate transformations. Each transformation also preserves the properties listed
in the rows below it, i.e., similarity preserves not only angles but also parallelism and straight lines. The 2 × 3
T
matrices are extended with a third [0 1] row to form a full 3 × 3 matrix for homogeneous coordinate transforma-
tions.