Image Processing

1
Image Processing Computer Vision

• ?????????????????????????
• Two Dimensional Geometric Transformation

2
Image Transformation

• ???????????????? (Image Transformation)
  ????????????????????????????????????
  ?????????????? ????? Template matching, Shape
  Matching, Image Search, Generalized Hough
  Transform
• ?????????? Image Transformation ????
  ??????????????? ????????????, ?????????????????,
  ????????????? ???????????????????????????????

3
??????????????????????????

• ???????????? (Translation) ???????????????????????
  ?????????????????????
• ?????????? (Rotation) ????????????????????????????
  ??? ???? clockwise(???????) ????
  counterclockwise(???????) ????????????????????????
  ??????????
• ????????????????? (Scaling) ???????????????? ????
  ??????????????????????????????????????????????????
  ?????????????
• ????????? (Shearing) ?????????????????????????????
  ???????????????????????? ?????????????????????????
  ????????????? x ??? y

4
???????????? (Translation)

• ?????????????????????????????????????????????????
  x(Tx) ???????????? y(Ty)
• ????????????????????????? (x, y) ???????????????
  (x, y) ?????????????????????????????????? ??????

Tx ?????????????? x ?????????
x x Tx y y Ty
Ty ?????????????? y ??????????
????????????????????????????????? Matrix
???????????????????? P P T ?????
???
5
???????????? (Translation)
???????????????? (x, y) ???????? (x, y) ????
6
?????????? (Rotation)
??????????????????????????? xy ???????????????????
????? (Pivot Point) ?????????? Origin (0,0)
7
?????????? (Rotation)
???????????????????????????????????????
???????????

1. ????????????????????
2. ????????????????????
3. ??????????????????????? (origin) (0,0)
4. ?????????????????????????? (??? x,y ?? ?)

8
?????????? (Rotation)

• ???????? (x,y) ?????????????? ????????????????????
  ?????????????????????????????
• ???????? (x,y) ???????????? ??????????????????????
  ???????????

9
?????????? (Rotation)
?????????????????? (x,y) ????????????????? ? ?
10
?????????? (Rotation)
x r cos (f) y r sin (f) x r cos (f ?) y
r sin (f ?) Trig Identity x r cos(f)
cos(?) r sin(f) sin(?) y r sin(f) sin(?) r
cos(f) cos(?) Substitute x x cos(?) - y
sin(?) y y sin(?) x cos(?)
11
?????????? (Rotation)
x x cos(?) - y sin(?) y y sin(?) x cos(?)
??????????????? ????????????????? Matrix
????????? P R.P
P
P
R
12
?????????? (Rotation)

• ?????????? ? ????? ? ?????????????????????????????
  ????????? 3 ?????????????
• 1. ??????????? (xp,yp) ????? (0,0)
  ?????????????????????????????
• ???????????????

13
?????????? (Rotation)

• 2. ??????????? (xp,yp) ?????????????? (0,0)
  ????????????????? ???
• (x,y) ?????????????? ???????????????????????
  (x,y) ????
• 3. ?????????????????????(origin)
• 4. ??????????????????????????? xp ??? yp
• 5. ??????????????? Pivot ?? ? ????????????
  origin ???

x x - xp y y - yp
x (x xp)cos? - (y - yp)sin? xp y (y
yp)sin? (x-xp)cos? yp
14
???????????????? (Scaling)

• ????????????????????????????????? Scaling factor
  ?????? Sx ??? Sy ???????????????????????????? x
  ??? y ???????????

0 lt Sx,Sy lt 1 ???????????????????? Sx,Sy gt
1 ????????????????????? Sx Sy ????????????????
?????????????????????? Sx ? Sy ??????????????????
???????????????????????
???????????????? ?????????????????????????????????
???? (fix point) ?????????
15
???????????????? (Scaling)
???????????????????????? Sx ??? Sy ??????????
16
???????????????? (Scaling)

• ??????????? Scaling ?????????? ??????

P
x x . Sx y y . Sy
P
????????????????????????????? Matrix
???????????????? ??? P S.P
??????????????????????????????????????????????????
????????
17
???????????????? (Scaling)

• ???????????????????????? Fixed ????????????????
  Origin
• 1. ?????????????????????? origin
• 2. ?????????????????????? origin
• 3. ???????????? Fixed point ??????????
  ?????????????? ??????

x (x - xf).Sx xf y (y - yf).Sy yf
18
???????????????? (Scaling)

• 4. ??????????????????? ???
• 5. ??????????????????????? ??????????? Matrix
  ??????

x xSx xf(1 - Sx) y ySy yf(1 - Sy)
19
????????? (Shearing)

• ????????????????????????????????????????????????
  ???
• 1. ??????????????? x
• 2. ??????????????? y

- ??????????????? y ??????????????????? (x,y) ??
(x,y) ??????
x x y Shy.x y
20
????????? (Shearing)
- ??????????????? x ??????????????????? (x,y) ??
(x,y) ??????
x Shx.y x
y y
21
Homogeneous Coordinate

• ????????????????????????????????????????????? x
  ??? y ???????????? Cartesian coordinate
  ?????????? cartesian ???????????????????????? ?
  ????????????????????????????????? Matrix
  ???????????????????????????????????????????
  Matrix ???????????????????????????????????????????
  ???????????????? Matrix ???????
  ?????????????????????? ????????????? Homogeneous
  coordinate ???? ???????????? ??????

22
Homogeneous Coordinate

• ??????? Matrix ????????????????? ? ??????????
  ??????

??????????
????????????
?????????????
?????????
23
Combining Transformation

• ??????????????????????? ??????????????? 2 ?????
• 1. ?????????? (Tx1,Ty1)
• 2. ?????????? (Tx2,Ty2)?????????????????????????
  ????? ??????

24
Combining Transformation

• ????????????????????? ???????????????????????
• 1. ????????????? R(?1)
• 2. ????????????? R(?2)
• ??????????????????????????????????

25
Combining Transformation

• ????????????????????? ?????????????????? 2 ?????
• 1. ???????????? S(Sx1,Sy1)
• 2. ???????????? S(Sx2,Sy2)
• ?????????????????????????????????????

26
Example of Composition

• ??????????????? Rotation ??????? Translation
• ??????????????????????????? 45 ???? ????????? a
  ???????????

a
a
27
Example of Composition

• ?????????????????????????????? ???????????????????
  ?????????????????????????
• ????????????????????????? ????????????????????????
  ??

a
a
a
Wrong
Correct T(-3) R(45) T(3)
R(45)
28
Example of Composition

• ????????????????????? ? ????????? origin
  ???????????
• 1. ?????????????? a ???????? origin T(-3)
• 2. Rotate ??????? 45 ???? R(45)
• 3. ?????????????????????????? T(3)

a
a
a
a
29
Example of Composition

• ???????????????????? Matrix ???? ? ?????????