Dr. Engr. Sami ur Rahman Digital Image Processing Lecture 9: Rotation, Scaling, Shear, Affine Transformation

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

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

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

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

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

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

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

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

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

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

Interpolation University Of Malakand | Department of Computer Science | UoMIPS | Dr. Engr. Sami ur Rahman | 12 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

Shear University Of Malakand | Department of Computer Science | UoMIPS | Dr. Engr. Sami ur Rahman | 13 Shear: the deformation of a material substance in which parallel internal surfaces slide past one another Horizontal shearVertical shearNo shear

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

Affine Transformation University Of Malakand | Department of Computer Science | UoMIPS | Dr. Engr. Sami ur Rahman | 15 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).midpoint  Does not necessarily preserve angles or lengths.

Thanks for your attention