1. Digital Image Processing & Pattern Analysis (CSCE 563) Geometric Transformations
Prof. Amr Goneid
Department of Computer Science & Engineering
The American University in Cairo

2. Geometric Transformations
Image Cropping
Interpolation
Image Resizing
Image Rotation
Affine Transform
Image Registration
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3. Image Cropping
Cropping refers to the removal of the outer parts of an image to select a rectangular ROI.
In MATLAB, The bounding rectangle is
rect = [col,row,w,h]
B = imcrop or
[B , rect] = imcrop;
uses mouse to define area
B = imcrop(A,rect);
Returns the cropped image defined
by the bounding rectangle A B h w
col,row
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4. Example: Top Left Quadrant
load treesc;
[n,m] = size(X);
rect = [1,1,m/2,n/2];
B = imcrop(X,rect);
subplot(2,1,1);
imshow(X,map);
subplot(2,1,2);
imshow(B,map);
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5. Interpolation 2-D Interpolation Used in resizing and rotation
Nearest Neighbor Interpolation (nearest)
Bilinear Interpolation (bilinear)
Bicubic Interpolation (bicubic)
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6. Interpolations
Nearest Neighbor
Bilinear
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7. Example:Bilinear interpolation (From Wikipedia)
bilinear interpolation is an extension of linear interpolation for interpolating functions of two variables on a regular grid.
The key idea is to perform linear interpolation first in one direction, and then again in the other direction.
to find the value of the unknown function f at the point P = (x, y). It is assumed that we know the
value of f at the four points
Q11 = (x1, y1), Q12 = (x1, y2),
Q21 = (x2, y1), and Q22 = (x2, y2).
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8. Example:Bilinear interpolation
Linear interpolation in the x-direction
interpolating in the y-direction
This yields
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9. Bilinear interpolation
In a coordinate system in which the four points are at (0, 0), (0, 1), (1, 0), and (1, 1), then in matrix form
The result of bilinear interpolation is independent of the order of interpolation. If we had first performed the linear interpolation in the y-direction and then in the x-direction, the resulting approximation would be the same.
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10. Resizing using Interpolation
No Interpolation
Resize by 183%
Original
With Interpolation
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11. Resizing B = imresize(A,m,’method’) m = ratio Example: load clownc;
B = imresize(X,3,’nearest’);
C = imresize(X,0.3,’nearest’);
subplot(3,1,1); imshow(X,map);
subplot(3,1,2); imshow(B,map);
subplot(3,1,3); imshow(C,map);
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12. Rotation B = imrotate(A,angle); or B = imrotate(A,angle,’crop’);
angle is anti-clockwise (degrees)
Example:
load amber256;
C = imrotate(X,45,'crop');
subplot(2,1,1);
imshow(X,map);
subplot(2,1,2);
imshow(C,map);
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13. Affine Transform
Let f be an image defined over (w,z) coordinate system, g is a geometric transformation of f with:
(x , y) = T{(w , z)}
Example:
(x , y) = T{(w , z)} = (w/2 , z/2)
This is shrinking f by ½ in both spatial dimensions.
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14. Affine Transform
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15. Affine Transform
Commonly used transform is the Affine Transform, in matrix form:
[x y 1] = [w z 1] T = [w z 1]
Can produce scaling, translation, rotation and shear.
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16. Affine Transform
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17. Affine Transform MATLAB Example (scaling): T = [2 0 0; 0 3 0; 0 0 1];
tform = maketform(‘affine’ , T);
wz = [1 1; 3 2];
xy = tformfwd(wz , tform);
Result is xy = [2 3; 6 6];
Also:
wz2 = tforminv(xy , tform); % inverse transform
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18. Affine Transform Examples: Scale Horizontally by 3 and vertically by 2
Shear
T2 = [1 0 0; ; 0 0 1];
Tscale = [ ; 0 2 0; 0 0 1];
Trot = [cos(pi/4) sin(pi/4) 0; -sin(pi/4) cos(pi/4) 0; 0 0 1];
Tshear = [1 0 0; ; 0 0 1];
T3 = Tscale * Trot * Tshear;
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19. Affine Transform
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20. Affine Transform
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21. Image Registration
Image registration is the process of overlaying two or more images of the same scene taken at different times, from different viewpoints, and/or by different sensors.
Applications: Change detection, Image fusion, Target recognition, Target localization, Depth perception, Image mosaicing, Motion estimation
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22. Image Registration CP CP
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23. Image Registration CP CP CP CP
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24. Image Registration (References)
Image registration methods: a survey
Barbara Zitova´*, Jan Flusser
Image Registration
Image Registration in MATLAB
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