1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

function M= affine_least_square(x0,y0, x1,y1, x2,y2, xp0,yp0, xp1,yp1, xp2,yp2)

% This function finds the affine transform between three points

% an affine transformation with a 3x3 affine transformation matrix:

%

% [M11 M12 M13]

% [M21 M22 M23]

% [M31 M32 M33]

%Since the third row is always [0 0 1] we can skip that.

% [x0 y0 1  0  0  0 ]     [M11]   [xp0]

% [0  0  0  x0 y0 1 ]     [M12]   [yp0]

% [x1 y1 1  0  0  0 ]     [M13]   [xp1]

% [0  0  0  x1 y1 1 ]  *  [M21] = [yp1]

% [x2 y2 1  0  0  0 ]     [M22]   [xp2]

% [0  0  0  x2 y2 1 ]     [M23]   [yp2]

%         A            *    X   =  B

%A -> 6x6

%X -> 6x1 in fact

%X=A\B

A=[x0 y0 1  0  0  0 ; 0  0  0  x0 y0 1 ; x1 y1 1  0  0  0 ; 0  0  0  x1 y1 1 ; x2 y2 1  0  0  0;0  0  0  x2 y2 1];

B=[xp0; yp0; xp1;  yp1; xp2 ; yp2 ];

% X=A\B;

% this is what we need but accroding to https://www.mathworks.com/matlabcentral/newsreader/view_thread/170201

%The following is better:

X=pinv(A)*B;

M11 =X(1,1);

M12 =X(2,1);

M13 =X(3,1);

M21 =X(4,1);

M22 =X(5,1);

M23 =X(6,1);

M=[ M11 M12 M13; M21 M22 M23; 0 0 1];

end

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

clc;

clear all;

%This is a simple test that project 3 point with an affine trasnform and

%then tries to find the transformation:

% tform.T and M' should be equal

%Points are located at:

x0=0;

y0=0;

x1=1;

y1=0;

x2=0;

y2=1;

p1=[x0,y0; x1,y1; x2,y2];

%We rotate them (a roll-rotation around Z axes) theta degree, first

%building the rotation matrix

theta=24;

A11=cosd(theta);

A12=-sind(theta);

A21=sind(theta);

A22=cosd(theta);

%Translation part

Tx=1;

Ty=2;

tform = affine2d([A11 A12 0; A21 A22 0; Tx Ty 1]);

fprintf('3x3 transformation matrix:')

tform.T

[x,y]=transformPointsForward(tform,p1(:,1),p1(:,2));

fprintf('transformed points:')

[x,y]

xp0=x(1,1);

yp0=y(1,1);

xp1=x(2,1);

yp1=y(2,1);

xp2=x(3,1);

yp2=y(3,1);

%finding the tranformation matrix from set of points and their respective

%transformed points:

M=affine_least_square(x0,y0, x1,y1, x2,y2, xp0,yp0, xp1,yp1, xp2,yp2);

fprintf('3x3 affine transformation matrix from least_squares:')

M'

tform = affine2d(M');

[x,y]=transformPointsForward(tform,p1(:,1),p1(:,2));