Digital Picture Processing by Azriel Rosenfeld, Avinash C. Kak

By Azriel Rosenfeld, Avinash C. Kak

The swift price at which the sector of electronic photograph processing has grown some time past 5 years had necessitated wide revisions and the creation of themes now not present in the unique version.

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If two scales are needed, the projection is said to be dimetric and if three are needed, the projection is said to be trimetric. 8 Euler Angles and Rotation in Space One way of applying rotations in 3D is to perform successive rotations around one of the basis axes, referred to as rotation by Euler angles. The matrix needed in each case is akin to performing a rotation in 2D [Eq. 2)]—with one of the coordinates remaining invariant, which results in the following matrices:   1 0 0 Rx =  0 cos φx − sin φx  , 0 sin φx cos φx   cos φy 0 sin φy , 0 1 0 Ry =  − sin φy 0 cos φy   cos φz − sin φz 0 cos φz 0  Rz =  sin φz 0 0 1 Unlike rotations in the plane, rotations in 3D are not commutative.

3 2 1 Orthogonal View Transformation Consider a virtual eye situated at E(ex , ey , ez ). The eye models an observer or a virtual camera and we wish to render, or draw, a representation of threedimensional objects on a two-dimensional surface that would be a faithful rendition of what would be captured by the virtual eye. This problem has long been considered by artists, but here we do not study the general problem, which involves perspective (see Chapter 11), but only orthographic projection. Our objective is to define a plane, called the view plane or the image plane, in 3D and to project points on it.

1) where x1 A = x2 x3 y1 y2 y3 1 1 , 1 x21 + y12 C = x22 + y22 x23 + y32 x1 x2 x3 q p2 1 1 , 1 x21 + y12 B = x22 + y22 x23 + y32 y1 y2 y3 1 1 , 1 x21 + y12 D = x22 + y22 x23 + y32 x1 x2 x3 y1 y2 . y3 q p1 q p2 p1 p2 p3 on circle boundary p3 outside circle 22 G EOMETRIC P REDICATES It is clear that the determinant in Eq. 1) vanishes if the point P (x, y) coincides with any of the three points P1 (x1 , y1 ), P2 (x2 , y2 ), or P3 (x3 , y3 ). 2 that A = 0 if and only if the three given points are not colinear.

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