GEOMETRIC OPERATIONS

Transformations and directions

•Affine (linear) transformations• Translation, rotation and scaling

•Non linear (Warping transformations)

•source to target (coordinate for each input image pixel)

I(x,y) O(x´,y´)

•target to source (coordinate for each output image pixel)

O(x,y) I(x´,y´) BETTER, WHY?

Translation

•Target to sourcex´ = x -Tx y´ = y -Ty

Translation

Rotation

•Target to sourcex´ = x cos(- y sin()

y´ = x sin(+ y cos()

Rotation

Scaling

•Target to sourcex´ = x / Sx

y´ = y / Sy

Scaling

General affine equations

Linear equations in general form

x´ = (x cos(+ y sin()) Sx +Tx

= (Sx cos(x + (Sx sin()) y +Tx

= a2x + a1y + a0

y´ = (-x sin(+ y cos()) Sy +Ty

= (-Sy sin(x + (Sy cos()) y + Ty

= b2x + b1y + b0

Warping• Increasing higher order terms increases accuracy

x2 y2

x3 y3

Warping and reference points

Warping transformed

Original with 4 reference points

Number of reference points

• 3 for first order warping• 6 for second order warping• 10 for third order warping

Number of reference points• Example: correction of a lense degradation by a third

order equation

x´= a9x3 + a8y3 + a7x2y + a6y2x + a5x2

+ a4y2 + a3x + a2y + a1xy + a0

y´= b9x3 + b8y3 + b7x2y + b6y2x + b5x2

+ b4y2 + b3x + b2y +b1xy +b0

Lens distortion filter

Curve bend filter

Sampling and resizing• In downsizing, first low pass filtering to avoid aliasing

(Nyquist theorem)

• Interpolation in resizing (up- and downsizing )

Low pass filtering and resizing

Group discussion• Discuss the application areas of geometric operations

Interpolation methods

•Nearest neighbor•Bilinear•Bicubic

Interpolation• Nearest neighbor

*

5 6

4

5

Address: 5.3, 4.4

Select pixel: column 5, row 4

Interpolation, Bilinear

*

5 6

4

5

a b

c d

Address: 5.3, 4.4

Weighted average of neighbour

pixels:

cd = (1-0.3)*c + 0.3 *d

ab = (1-0.3)*a + 0.3 *b

abcd= (1-0.4)ab + 0.4*cd

abcd is the final pixel value

Interpolation, Bicubic• Second order interpolation• In 16*16 neighbourhood, sometimes 64*64• Optimizing moving images

Nearest neighbour and bilinear

Nearest neighbor Bilinear

Anti-aliasing filter

Extra information abour geometric corrections in remote sensing

http://www.geo-informatie.nl/courses/grs20306/course/Schedule/Geometric-correction-RS-new.pdf