Spatial transformations

tiepoints

z

w

y

xT

f(w,z) g(x,y)

original distorted

Affine transform

1

0

0

111

3231

2221

1211

tt

tt

tt

zwzwyx T

MATLAB code:

T=[2 0 0; 0 3 0; 0 0 1];tform=maketform('affine', T);tform.tdata.Ttform.tdata.TinvWZ=[1 1; 3 2];XY=tformfwd(WZ, tform);WZ2=tforminv(XY, tform);

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 ijhold onplot(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; .2 1 0; 0 0 1];T3 = [cos(pi/4) sin(pi/4) 0; -sin(pi/4) cos(pi/4) 0; 0 0 1];

Apply spatial transformations to images

Inverse mapping then interpolation

original distorted

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 imagesto show the differences.

original Tiepoints afterdistortion

Nearest neighbor

Bilinear interp.

distorted

distorted

restored

restored

original distorted

Diff. restored

(UsingPreviousSlide)

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 theoriginal checkerboard.

tform=cp2tform(input_points, base_points, 'projective');gp=imtransform(g, tform);

Project#3: iris transformation

Polar coordinate to Cartesian coordinate Check the reference paper

Image Topology

Motivation How many rice?

Outline

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

Neighborhood and adjacency

4-neighbors

8-neighbors

P QP and Q are 4-adjacent

P

QP and Q are 8-adjacent

Path, connected and components

P

Q

P and Q are 4-connectedThe above set of pixels are 4-connected.=> 4-component

P

Q

P and Q are 8-connectedThe above set of pixels are 8-connected.=> 8-component

Component labeling

Two 4-components. How to label them?

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 labelNeighbor not labeled => new label

1

2 1 3

4 3

5 4

Component labeling algorithm

1

2 1 3

4 3

5 4

{1,2} and {3,4,5} are equivalent classes of labels, which aredefined by adjacent relation.

{1,2} => 1{3,4,5} => 2

1

1 1 2

2 2

2 2

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)

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 )

.X

1

2

4

8

16

32

64

128

256

= 3

Lookup tables for neighborhood (cont.)

Performing any possible 3x3 neighborhood operation is equivalent to table lookupInput Output

0

1

1

2

511 2

… …

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);

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;