1. Spatial transformations
tiepoints
f(w,z)
g(x,y)
distorted
original

2. Affine transform MATLAB code: T=[2 0 0; 0 3 0; 0 0 1];
tform=maketform('affine', T);
tform.tdata.T
tform.tdata.Tinv
WZ=[1 1; 3 2];
XY=tformfwd(WZ, tform);
WZ2=tforminv(XY, tform);

3. Exercise#1 % generate a grid image
[w, z]=meshgrid(linspace(0,100,10), linspace(0,100,10));
wz = [w(:) z(:)];
plot(w, z, 'b'), axis equal, axis ij
hold on
plot(w', z', 'b');
set(gca, 'XAxisLocation', 'top');
EX. Apply the following T to the grids, and plot all results
T1 = [3 0 0; 0 2 0; 0 0 1];
T2 = [1 0 0; ; 0 0 1];
T3 = [cos(pi/4) sin(pi/4) 0; -sin(pi/4) cos(pi/4) 0; 0 0 1];

4. Apply spatial transformations to images
Inverse mapping then interpolation
distorted
original

5. Apply spatial transformations to images (cont.)
MATLAB function
Ex#2
g=imtransform(f, tform, interp);
f=checkerboard(50);
Apply T3 to f with ‘nearest’, ‘bilinear’, and ‘bicubic’
interpolation method. Zoom the resultant images
to show the differences.

6. Tiepoints after original distortion restored distorted
Nearest neighbor
restored
distorted
Bilinear interp.
restored
distorted

7. original
distorted
(Using
Previous
Slide)
restored
Diff.

8. Image transformation using control points
Ex#3, for original f and transformed g in Ex#2, using cpselect tool to select control points:
cpselect(g, f);
With input_points and base_points generated
using cpselect, we do the following to reconstruct the
original checkerboard.
tform=cp2tform(input_points, base_points, 'projective');
gp=imtransform(g, tform);

9. Project#3: iris transformation
Polar coordinate to Cartesian coordinate
Check the reference paper

10. Image Topology

11. Motivation
How many rice?

12. Outline Neighbors and adjacency Path, connected, and components
Component labeling
Lookup tables for neighborhood

13. Neighborhood and adjacency
4-neighbors
8-neighbors P Q
P and Q are 4-adjacent Q
P and Q are 8-adjacent P

14. Path, connected and componentsQ
P and Q are 8-connected
The above set of pixels are 8-connected.
=> 8-component Q P
P and Q are 4-connected
The above set of pixels are 4-connected.
=> 4-component

15. Component labeling
Two 4-components. How to label them?

16. Component labeling algorithm
Scan left to right, top to down.
Start from top left foreground pixel.
2. Check its upper and left neighbors.
Neighbor labeled => take the same label
Neighbor not labeled => new label 1 2 1 3 4 3 5 4

17. Component labeling algorithm
{1,2} and {3,4,5} are equivalent
classes of labels, which are
defined by adjacent relation.
{1,2} => 1
{3,4,5} => 2 1 2 1 2 3 4 5

18. Ex#4: component labeling
MATLAB code
Threshold the rice.tiff image, then count the number of rice. Using 4- and 8-neighbors. Show the label image and the number of rice in the title.
i=zeros(8,8);
i(2:4,3:6)=1;
i(5:7,2)=1;
i(6:7,5:8)=1;
i(8,4:5)=1;
bwlabel(i,4)
bwlabel(i,8)

19. Lookup tables for neighborhood
For a 3x3 neighborhood, there are 29=512 possible
For each possible neighborhood, give it a unique number(類似2進位編碼) 1 8 64
= 3 2 16
128 .X 4 32
256

20. Lookup tables for neighborhood (cont.)
Performing any possible 3x3 neighborhood operation is equivalent to table lookup
Input
Output 1 2 1 … … 2
511

21. Ex#5: table lookup
Find the boundary that are 4-connected in a binary image
Ex#5: Find the 8-boundary of the rice image using lookup table
f=inline('x(5) & ~(x(2)*x(4)*x(6)*x(8))');
lut=makelut(f,3);
Iw=applylut(II, lut);

22. Ex#6: table lookup Build lookup tables for the following cases
Show your result with the following matrix
a=zeros(8,8);
a(2,2)=1;
a(4:8,4:8)=1;
a(6,6)=0;