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Digital Image Processing Lecture 9: Rotation, Scaling, Shear, Affine Transformation

Digital Image Processing Lecture 9: Rotation, Scaling, Shear, Affine Transformation. Courtesy. Gonzalez and Woods. Transformation. Transformations: Move and rotate objects, scaling, stretching Euclidean Transformations

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Digital Image Processing Lecture 9: Rotation, Scaling, Shear, Affine Transformation

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  1. Digital Image ProcessingLecture 9: Rotation, Scaling, Shear, Affine Transformation

  2. University Of Malakand | Department of Computer Science | UoMIPS| Dr. Engr. Sami ur Rahman | 2 Courtesy Gonzalez and Woods

  3. Transformation Transformations: Move and rotate objects, scaling, stretching Euclidean Transformations The Euclidean transformations are the most commonly used transformations. An Euclidean transformation is either a translation, a rotation, or a reflection. The angles and lengths remain constant. University Of Malakand | Department of Computer Science | UoMIPS| Dr. Engr. Sami ur Rahman | 3

  4. Translation University Of Malakand | Department of Computer Science | UoMIPS| Dr. Engr. Sami ur Rahman | 4

  5. Translation University Of Malakand | Department of Computer Science | UoMIPS| Dr. Engr. Sami ur Rahman | 5

  6. Translation University Of Malakand | Department of Computer Science | UoMIPS| Dr. Engr. Sami ur Rahman | 6

  7. Rotation x = r cos (ᶲ) y = r sin (ᶲ) x’ = r cos (ᶲ + ) y’ = r sin (ᶲ + ) Trig Identity… x’ = r cos(ᶲ) cos() – r sin(ᶲ) sin() y’ = r cos(ᶲ) sin() + r sin(ᶲ) cos() Substitute… x’ = x cos() - y sin() y’ = x sin() + y cos() Sin (ᶲ + ) = sin ᶲcos  + cosᶲsin  Sin (ᶲ- ) = sin ᶲcos  -cos sinᶲ cos(ᶲ+ ) = cosᶲcos  - sin sinᶲ cos(ᶲ- ) = cosᶲcos  +sin sinᶲ (x’, y’) (x, y) r y’ = r sin (ᶲ + ) r  y = r sin (ᶲ) ᶲ x’ = r cos (ᶲ + ) x = r cos (ᶲ) University Of Malakand | Department of Computer Science | UoMIPS| Dr. Engr. Sami ur Rahman | 7

  8. Rotation University Of Malakand | Department of Computer Science | UoMIPS| Dr. Engr. Sami ur Rahman | 8

  9. Scaling Scaling: Resizing an image University Of Malakand | Department of Computer Science | UoMIPS| Dr. Engr. Sami ur Rahman | 9

  10. Scaling University Of Malakand | Department of Computer Science | UoMIPS| Dr. Engr. Sami ur Rahman | 10

  11. Scaling Rescaling and interpolation University Of Malakand | Department of Computer Science | UoMIPS| Dr. Engr. Sami ur Rahman | 11

  12. Interpolation • Interpolation: Constructing new data points from existing data points • Types of interpolation • Nearest neighbor interpolation • Linear interpolation • Bilinear interpolation • Polynomial interpolation • Piecewise constant interpolation • Spline interpolation University Of Malakand | Department of Computer Science | UoMIPS| Dr. Engr. Sami ur Rahman | 12

  13. Shear Shear: the deformation of a material substance in which parallel internal surfaces slide past one another No shear Horizontal shear Vertical shear University Of Malakand | Department of Computer Science | UoMIPS| Dr. Engr. Sami ur Rahman | 13

  14. Shear Horizontal shear Vertical shear University Of Malakand | Department of Computer Science | UoMIPS| Dr. Engr. Sami ur Rahman | 14

  15. Affine Transformation • Affine transformation or affine map or an affinity: • A transformation which preserves straight lines (i.e., all points lying on a line initially still lie on a line after transformation) • Preserves ratios of distances between points lying on a straight line (e.g., the midpoint of a line segment remains the midpoint after transformation). • Does not necessarily preserve angles or lengths. University Of Malakand | Department of Computer Science | UoMIPS| Dr. Engr. Sami ur Rahman | 15

  16. Thanks for your attention

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