sLecture

↳ Topic 2: Tomographic Reconstruction

↳ Intro

↳ CT

↳ PET

↳ Co-ordinate Rotation

Excerpt from Prof. Bouman's Lecture

Accompanying Lecture Notes

• Define the counter-clockwise rotation matrix

the matrix rotates vector $ v_0 $ in a 2-D real space by angle $ \theta $ in a fixed coordinate system.

$ \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} $

• Define the new coordinate system $ (r,z) $

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

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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Back to the "Bouman Lectures on Image Processing" by Maliha Hossain