sLecture

↳ Topic 2: Tomographic Reconstruction

↳ Intro

↳ CT

↳ PET

↳ Co-ordinate Rotation

Excerpt from Prof. Bouman's Lecture

Accompanying Lecture Notes

Motivation: Before introducing FST some background

$ A_{\theta} $ is the counterclockwise rotation matrix given by
$ A_{\theta}=\begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} $

The matrix rotates vector $ v_0 $ in a 2-dimensional real space by angle $ \theta $ in a fixed coordinate system. Notice that this is equivalent to keeping the vector fixed and rotating the coordinate system clockwise by $ \theta $. This equivalence is illustrated in figure 1.

Fig 1: bottom left: ccw rotation of vector; top right: cw rotation of coordinate axes

Vector

Let us define a new coordinate system $ (r,z) $ where
$ \begin{bmatrix} x \\ y \end{bmatrix} = A_{\theta}\begin{bmatrix} r \\ z \end{bmatrix} $

i.e. vector $ [r,z]' $ is rotated counterclockwise angle $ \theta $ to produce vector $ [x,y]' $

Figure 1 shows the geometric interpretation of the rotation.

Fig 1: Geometric Interpretation

• Inverse Transformation

$ \begin{bmatrix} r \\ z \end{bmatrix} = A_{-\theta}\begin{bmatrix} x \\ y \end{bmatrix} $

References

• C. A. Bouman. ECE 637. Class Lecture. Digital Image Processing I. Faculty of Electrical Engineering, Purdue University. Spring 2013.

• E. W. Weisstein, "Rotation Matrix." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RotationMatrix.html. May 8th, 2013 [May 21st, 2013]

Questions and comments

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