Comparing 3-D Interpolation Techniques

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Comparing 3-D Interpolation Methods - Nearest Neighbour, Cubic convolution, B-spline, Tri-linear

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Comparing 3-D Interpolation Techniques

  1. 1. Comparing 3-D Interpolation Methods Binu Enchakalody January, 12 2010
  2. 2. Interpolation Methods <ul><li>Nearest Neighbor </li></ul><ul><li>Tri-Linear </li></ul><ul><li>Cubic-Keys </li></ul><ul><li>Clamped Cubic Spline </li></ul><ul><li>Catmull-Rom Spline </li></ul><ul><li>Cubic B-Spline </li></ul>
  3. 3. Methods <ul><li>Vxyz  = V000 (1 - x) (1 - y) (1 - z) + </li></ul><ul><li>V100 x (1 - y) (1 - z) +  </li></ul><ul><li>V010 (1 - x) y (1 - z) +  </li></ul><ul><li>V001 (1 - x) (1 - y) z + </li></ul><ul><li>V101 x (1 - y) z +  V011 (1 - x) y z + </li></ul><ul><li>V110 x y (1 - z) + V111 x y z </li></ul>Tri-Linear Weights an interpolated intensity value, based on the distance from the nearest x,y and z pixels within a 2 x 2 x 2 neighborhood Nearest Neighbor Picks the intensity of the nearest x,y and z pixel within a 2 x 2 x 2 neighborhood
  4. 4. Piecewise-Cubic <ul><li>Basic Algorithm </li></ul><ul><li>[ t 0 t 1 t 2 t 3 ] x Basis Matrix x [a -1 a 0 a 1 a 2 ] ’, where 0 < t < 1 </li></ul><ul><li>Cubic-Keys </li></ul><ul><li>*Cubic Convolution Interpolation for Digital Image Processing - R.G. Keys </li></ul><ul><li>C2 Continuity (curvature) </li></ul><ul><li>4 Cubic Kernel </li></ul><ul><li>Dimension Separable </li></ul><ul><li>Catmull-Rom Spline </li></ul><ul><li>Different Basis Matrix </li></ul>
  5. 5. Piecewise Cubic-Spline <ul><li>Clamped Cubic-Spline </li></ul><ul><li>C2 Continuity (curvature) </li></ul><ul><li>4 Cubic Kernel </li></ul><ul><li>Dimension Separable </li></ul>General Equation
  6. 6. Piecewise Cubic-Spline <ul><li>B-Spline Interpolation </li></ul><ul><li>*A parallel B-Spline fitting algorithm – F.Cheng, A. Gosthashby </li></ul><ul><li>Uses Control Points defined by the neighboring intensity pixels </li></ul>
  7. 7. Calculating Control Points, C Parallel Algorithm A parallel B-Spline fitting algorithm – F.Cheng, A. Gosthashby
  8. 8. Original – Volume Center slice
  9. 9. Nearest Neighbour
  10. 10. Tri-Linear
  11. 11. Cubic-Keys
  12. 12. Catmull-Rom Spline
  13. 13. Clamped-Cubic Spline
  14. 14. B-spline
  15. 15. Analysis <ul><li>Rotated a 151 x 221 x 50 Test Matrix, 6 times in increments of 60 degrees </li></ul><ul><li>Along the Z-axis </li></ul><ul><li>Along the X-axis (permuted) </li></ul><ul><li>Recorded computation time for interpolating the data-set </li></ul><ul><li>Error Analysis on a 23 x 29 x 11 sub-matrix </li></ul><ul><li>Calculated MSE, Mean, Standard Deviation </li></ul>
  16. 16. Error analysis on test matrix, after 6 rotations of 60 ° - along Z-axis (top), along X-axis (bottom)
  17. 17. Approx. computation time@ rotation(MATLAB profiler)

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