2. Spatial Transformations
Affine transformations of the support of the image f (x, y )
Scaling: x = cx x and y = cy y → f (x , y ) = f (x, y ).
Rotation: x = x cos θ − y sin θ and
y = x cos θ + y sin θ → f (x , y ) = f (x, y ).
Translation: x = x + tx and
y = y + ty → f (x , y ) = f (x, y ).
Shear: x = x + sy y and y = y → f (x , y ) = f (x, y ).
All these collected together can be represented in a matrix
form using Homogeneous coordinates, as follows:
x cos θ/cx /1 sin θ/sy /0 0/tx x
y = − sin θ/sx /0 cos θ/cy /1 0/ty y (1)
1 0 0 1 1
IT472: Lecture 5 2/18
3. Spatial Transformations
Affine transformations of the support of the image f (x, y )
Scaling: x = cx x and y = cy y → f (x , y ) = f (x, y ).
Rotation: x = x cos θ − y sin θ and
y = x cos θ + y sin θ → f (x , y ) = f (x, y ).
Translation: x = x + tx and
y = y + ty → f (x , y ) = f (x, y ).
Shear: x = x + sy y and y = y → f (x , y ) = f (x, y ).
All these collected together can be represented in a matrix
form using Homogeneous coordinates, as follows:
x cos θ/cx /1 sin θ/sy /0 0/tx x
y = − sin θ/sx /0 cos θ/cy /1 0/ty y (1)
1 0 0 1 1
IT472: Lecture 5 2/18
4. Spatial Transformations
Affine transformations of the support of the image f (x, y )
Scaling: x = cx x and y = cy y → f (x , y ) = f (x, y ).
Rotation: x = x cos θ − y sin θ and
y = x cos θ + y sin θ → f (x , y ) = f (x, y ).
Translation: x = x + tx and
y = y + ty → f (x , y ) = f (x, y ).
Shear: x = x + sy y and y = y → f (x , y ) = f (x, y ).
All these collected together can be represented in a matrix
form using Homogeneous coordinates, as follows:
x cos θ/cx /1 sin θ/sy /0 0/tx x
y = − sin θ/sx /0 cos θ/cy /1 0/ty y (1)
1 0 0 1 1
IT472: Lecture 5 2/18
5. Spatial Transformations
Affine transformations of the support of the image f (x, y )
Scaling: x = cx x and y = cy y → f (x , y ) = f (x, y ).
Rotation: x = x cos θ − y sin θ and
y = x cos θ + y sin θ → f (x , y ) = f (x, y ).
Translation: x = x + tx and
y = y + ty → f (x , y ) = f (x, y ).
Shear: x = x + sy y and y = y → f (x , y ) = f (x, y ).
All these collected together can be represented in a matrix
form using Homogeneous coordinates, as follows:
x cos θ/cx /1 sin θ/sy /0 0/tx x
y = − sin θ/sx /0 cos θ/cy /1 0/ty y (1)
1 0 0 1 1
IT472: Lecture 5 2/18
6. Spatial Transformations
Affine transformations of the support of the image f (x, y )
Scaling: x = cx x and y = cy y → f (x , y ) = f (x, y ).
Rotation: x = x cos θ − y sin θ and
y = x cos θ + y sin θ → f (x , y ) = f (x, y ).
Translation: x = x + tx and
y = y + ty → f (x , y ) = f (x, y ).
Shear: x = x + sy y and y = y → f (x , y ) = f (x, y ).
All these collected together can be represented in a matrix
form using Homogeneous coordinates, as follows:
x cos θ/cx /1 sin θ/sy /0 0/tx x
y = − sin θ/sx /0 cos θ/cy /1 0/ty y (1)
1 0 0 1 1
IT472: Lecture 5 2/18
7. Implementation issues
For a given affine transformation matrix A, A · (x, y )t is not
always an integer.
Is it possible that due to rounding off
A · (x1 , y1 )t = A · (x2 , y2 )t
Solution:
Instead of using a forward mapping, let’s work with the inverse
mapping.
A−1 : (x , y )t → (x, y )t
We can scan the output image coordinates and see where they
come from, and accordingly assign them grey values.
IT472: Lecture 5 3/18
8. Implementation issues
For a given affine transformation matrix A, A · (x, y )t is not
always an integer.
Is it possible that due to rounding off
A · (x1 , y1 )t = A · (x2 , y2 )t
Solution:
Instead of using a forward mapping, let’s work with the inverse
mapping.
A−1 : (x , y )t → (x, y )t
We can scan the output image coordinates and see where they
come from, and accordingly assign them grey values.
IT472: Lecture 5 3/18
9. Implementation issues
For a given affine transformation matrix A, A · (x, y )t is not
always an integer.
Is it possible that due to rounding off
A · (x1 , y1 )t = A · (x2 , y2 )t
Solution:
Instead of using a forward mapping, let’s work with the inverse
mapping.
A−1 : (x , y )t → (x, y )t
We can scan the output image coordinates and see where they
come from, and accordingly assign them grey values.
IT472: Lecture 5 3/18
10. Implementation issues
For a given affine transformation matrix A, A · (x, y )t is not
always an integer.
Is it possible that due to rounding off
A · (x1 , y1 )t = A · (x2 , y2 )t
Solution:
Instead of using a forward mapping, let’s work with the inverse
mapping.
A−1 : (x , y )t → (x, y )t
We can scan the output image coordinates and see where they
come from, and accordingly assign them grey values.
IT472: Lecture 5 3/18
11. Implementation issues
For a given affine transformation matrix A, A · (x, y )t is not
always an integer.
Is it possible that due to rounding off
A · (x1 , y1 )t = A · (x2 , y2 )t
Solution:
Instead of using a forward mapping, let’s work with the inverse
mapping.
A−1 : (x , y )t → (x, y )t
We can scan the output image coordinates and see where they
come from, and accordingly assign them grey values.
IT472: Lecture 5 3/18
12. Implementation issues
For a given affine transformation matrix A, A · (x, y )t is not
always an integer.
Is it possible that due to rounding off
A · (x1 , y1 )t = A · (x2 , y2 )t
Solution:
Instead of using a forward mapping, let’s work with the inverse
mapping.
A−1 : (x , y )t → (x, y )t
We can scan the output image coordinates and see where they
come from, and accordingly assign them grey values.
IT472: Lecture 5 3/18
13. Examples
Figure: (top-left) Image of Lena (top-right) Image rotated by 23◦
(bottom-left) Shear sx = 1.2, (bottom-right) Shear sx = −1.2
IT472: Lecture 5 4/18
14. End of Chapter 2
Chapter 3: Intensity transformations and Spatial filtering
(Image enhancement in the spatial domain)
IT472: Lecture 5 5/18
15. End of Chapter 2
Chapter 3: Intensity transformations and Spatial filtering
(Image enhancement in the spatial domain)
IT472: Lecture 5 5/18
16. Image enhancement
Image enhancement is a pre-processing step that makes the
input image better suited for further processing. For example,
for segmentation, recognition, or simply better for somebody
to view the image.
Intensity transformations: Depends only on the intensity value
at a point: g (x, y ) = T [f (x, y )], can be also written as
s = T [r ], where r and s are the input and output grey values
respectively.
IT472: Lecture 5 6/18
17. Image enhancement
Image enhancement is a pre-processing step that makes the
input image better suited for further processing. For example,
for segmentation, recognition, or simply better for somebody
to view the image.
Intensity transformations: Depends only on the intensity value
at a point: g (x, y ) = T [f (x, y )], can be also written as
s = T [r ], where r and s are the input and output grey values
respectively.
IT472: Lecture 5 6/18
18. Intensity transformations
Image negative: s = L − 1 − r
Powers, nth roots, Log transformations etc..
IT472: Lecture 5 7/18
19. Intensity transformations
Image negative: s = L − 1 − r
Powers, nth roots, Log transformations etc..
IT472: Lecture 5 7/18
20. Intensity transformations
Image negative: s = L − 1 − r
Powers, nth roots, Log transformations etc..
IT472: Lecture 5 7/18
21. Applications of Intensity transformations
Intensity transformations are used frequently for Contrast
enhancement.
Contrast measures how much the object color/grey value
differs from its surroundings.
Objective definitions:
Weber contrast: I −Ib
Ib
Michelson contrast: IImax −Imin
max +I
min
M N
RMS contrast: i=1 j=1 (Iij − ¯)2
I
Appropriate transformation must be chosen depending on
what grey values you want to enhance and what is the
content of the input image.
IT472: Lecture 5 8/18
22. Applications of Intensity transformations
Intensity transformations are used frequently for Contrast
enhancement.
Contrast measures how much the object color/grey value
differs from its surroundings.
Objective definitions:
Weber contrast: I −Ib
Ib
Michelson contrast: IImax −Imin
max +I
min
M N
RMS contrast: i=1 j=1 (Iij − ¯)2
I
Appropriate transformation must be chosen depending on
what grey values you want to enhance and what is the
content of the input image.
IT472: Lecture 5 8/18
23. Applications of Intensity transformations
Intensity transformations are used frequently for Contrast
enhancement.
Contrast measures how much the object color/grey value
differs from its surroundings.
Objective definitions:
Weber contrast: I −Ib
Ib
Michelson contrast: IImax −Imin
max +I
min
M N
RMS contrast: i=1 j=1 (Iij − ¯)2
I
Appropriate transformation must be chosen depending on
what grey values you want to enhance and what is the
content of the input image.
IT472: Lecture 5 8/18
24. Applications of Intensity transformations
Intensity transformations are used frequently for Contrast
enhancement.
Contrast measures how much the object color/grey value
differs from its surroundings.
Objective definitions:
Weber contrast: I −Ib
Ib
Michelson contrast: IImax −Imin
max +I
min
M N
RMS contrast: i=1 j=1 (Iij − ¯)2
I
Appropriate transformation must be chosen depending on
what grey values you want to enhance and what is the
content of the input image.
IT472: Lecture 5 8/18
25. Applications of Intensity transformations
Intensity transformations are used frequently for Contrast
enhancement.
Contrast measures how much the object color/grey value
differs from its surroundings.
Objective definitions:
Weber contrast: I −Ib
Ib
Michelson contrast: IImax −Imin
max +I
min
M N
RMS contrast: i=1 j=1 (Iij − ¯)2
I
Appropriate transformation must be chosen depending on
what grey values you want to enhance and what is the
content of the input image.
IT472: Lecture 5 8/18
26. Applications of Intensity transformations
Intensity transformations are used frequently for Contrast
enhancement.
Contrast measures how much the object color/grey value
differs from its surroundings.
Objective definitions:
Weber contrast: I −Ib
Ib
Michelson contrast: IImax −Imin
max +I
min
M N
RMS contrast: i=1 j=1 (Iij − ¯)2
I
Appropriate transformation must be chosen depending on
what grey values you want to enhance and what is the
content of the input image.
IT472: Lecture 5 8/18
27. Applications of Intensity transformations
Intensity transformations are used frequently for Contrast
enhancement.
Contrast measures how much the object color/grey value
differs from its surroundings.
Objective definitions:
Weber contrast: I −Ib
Ib
Michelson contrast: IImax −Imin
max +I
min
M N
RMS contrast: i=1 j=1 (Iij − ¯)2
I
Appropriate transformation must be chosen depending on
what grey values you want to enhance and what is the
content of the input image.
IT472: Lecture 5 8/18
28. Contrast enhancement
We will frequently use the log-transformation to view the
Fourier spectrum of an image.
IT472: Lecture 5 9/18
29. Contrast enhancement
We will frequently use the log-transformation to view the
Fourier spectrum of an image.
IT472: Lecture 5 9/18
36. Bit-plane slicing
Every pixel needs 8 bits (assuming gray values from 0 - 255).
Every bit plane can be thought of as a binary image.
IT472: Lecture 5 16/18
37. Bit-plane slicing
Every pixel needs 8 bits (assuming gray values from 0 - 255).
Every bit plane can be thought of as a binary image.
IT472: Lecture 5 16/18
38. Bit-plane slicing
Every pixel needs 8 bits (assuming gray values from 0 - 255).
Every bit plane can be thought of as a binary image.
IT472: Lecture 5 16/18
40. Bit-plane slicing
Can be used for image compression.
Figure: (top row) Reconstructions from bit planes (bottom) Original
image
IT472: Lecture 5 18/18
