3D Volume Rotation Using Shear Transformations

13 years 4 months ago

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We presenta group of methods of decomposingan arbitrary 3D volume rotation into a sequenceof simple shear(i.e., regular shift) operations. We explore different types of shear operations: 2D beam shear, a shear in one coordinate based on the other two coordinates; 2D slice shear, a shear of a volume slice (in two coordinates) according to the third coordinate; and2D slicebeam shear, the combination of a beam shear and a slice shear. We show that an arbitrary 3D rotation can be decomposed into four 2D beam shears. We use this decomposition as a basis to obtain the decomposition sequence of 3D rotation into four 2D slice shears or three 2D slice-beam shears. Moreover, we observe that two consecutive slice shears can be achieved by shifting beams in 3D space, a transformation we call a 3D beam shear. Therefore, an arbitrary 3D rotation can be decomposed into only two 3D beam shears. Because of the regularity and simplicity of the shear operation, these decompositions are suitable for impl...

Baoquan Chen, Arie E. Kaufman

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