ECE 468: Digital Image Processing
Lecture 2
Prof. Sinisa Todorovic
1
Outline
•Image interpolation
•MATLAB tutorial
•Review of image elements
•Affine transforms of images
•Spatial-domain filtering
2
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Digital Image Processing Lecture 2: Affine Transforms and Spatial-Domain Filtering - Prof., Study notes of Electrical and Electronics Engineering
The second lecture notes for the digital image processing course (ece 468) at the university of x. The notes cover topics such as image interpolation, matlab tutorial, review of image elements, affine transforms of images, and spatial-domain filtering. Explanations, formulas, and examples using homogeneous coordinates.
Typology: Study notes
Pre 2010
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ECE 468: Digital Image Processing
Lecture 2
Prof. Sinisa Todorovic
1
Outline
Image interpolation
MATLAB tutorial
Review of image elements
Affine transforms of images
Spatial-domain filtering
Image Interpolation
Bilinear N = 1
Bicubic N = 3
f (x, y) =
N
i=
N
j=
a
ij
x
i
y
j
3
MATLAB Image Processing Toolbox
Image Structure
7
Pixels, 4-adjacency, 8-adjacency, m-adjacency
Path -- directed, undirected, loop
Region = Connected set of pixels
Region boundary, inner and outer contour
Foreground - background
Edge = Connected pixels with high derivative values
Interest points: T-junction, Y-junction
Highlights or specularities
Lambertian surface = isotropic reflectance
Specular surface = zero reflectance except at an angle
Image Elements
9
2D Translation
displacement
[
x
y
]
=
source: S. Savarese
t =
[
t
x
t
y
]
13
2D Translation
displacement
[
x
y
]
=
source: S. Savarese
t =
[
t
x
t
y
]
P
′
= P + t =
[
x + t
x
y + t
y
]
[
1 0 t
x
0 1 t
y
]
x
y
2D Translation
displacement
homogeneous
coordinates
[
x
y
]
=
source: S. Savarese
t =
[
t
x
t
y
]
P
′
= P + t =
[
x + t
x
y + t
y
]
[
1 0 t
x
0 1 t
y
]
x
y
13
2D Translation
displacement
homogeneous
coordinates
P
′
x + t
x
y + t
y
1 0 t
x
0 1 t
y
x
y
[
x
y
]
=
source: S. Savarese
t =
[
t
x
t
y
]
P
′
= P + t =
[
x + t
x
y + t
y
]
[
1 0 t
x
0 1 t
y
]
x
y
2D Scaling
=
[
x
y
]
=
[
s
x
x
s y
y
]
s
x
x
s
y
y
s
x
0 s
y
x
y
source: S. Savarese
14
2D Scaling
=
[
x
y
]
=
[
s
x
x
s y
y
]
s
x
x
s
y
y
s
x
0 s
y
x
y
source: S. Savarese
scaling matrix
2D Scaling
=
[
x
y
]
=
[
s
x
x
s y
y
]
s
x
x
s
y
y
s
x
0 s
y
x
y
source: S. Savarese
scaling matrix
S
14
2D Scaling + Translation
P
′
= S · P
P
′′
= T · P
′
P
′′
= (T · S) · P
Is the ordering important?
source: S. Savarese
2D Rotation
counter-clockwise
by angle θ
source: S. Savarese
θ
P =
[
x
y
]
x x
′
y
′
y
P
′
[
cos θ x − sin θ y
sin θ x + cos θ y
]
x
′
y
′
cos θ − sin θ 0
sin θ cos θ 0
x
y
16
2D Rotation
counter-clockwise
by angle θ
source: S. Savarese
θ
P =
[
x
y
]
x x
′
y
′
y
P
′
[
cos θ x − sin θ y
sin θ x + cos θ y
]
x
′
y
′
cos θ − sin θ 0
sin θ cos θ 0
x
y
rotation matrix
2D Rotation
counter-clockwise
by angle θ
source: S. Savarese
θ
P =
[
x
y
]
x x
′
y
′
y
P
′
[
cos θ x − sin θ y
sin θ x + cos θ y
]
x
′
y
′
cos θ − sin θ 0
sin θ cos θ 0
x
y
rotation matrix
R
16
2D Rotation + Scaling + Translation
x
′
y
′
1
=
cos θ − sin θ 0
sin θ cos θ 0
0 0 1
1 0 t
x
0 1 t
y
0 0 1
s
x
0 0
0 s
y
0
0 0 1
x
y
1
scaling
matrix
S
translation
matrix
rotation
matrix
t R
x
′
y
′
[
R S t
]
x
y
Re-writing the Equation of Transformation
x
′
y
′
1
=
t
11
t
12
t
13
t
21
t
22
t
23
0 0 1
x
y
1
20
Re-writing the Equation of Transformation
x
′
y
′
1
=
t
11
t
12
t
13
t
21
t
22
t
23
0 0 1
x
y
1
x i
·t
11
- y
i
·t
12
- 1 ·t
13
- 0 ·t
21
- 0 ·t
22
- 0 ·t
23
= x
′
i
0 ·t
11
- 0 ·t
12
- 0 ·t
13
- x
i
·t
21
- y
i
·t
22
- 1 ·t
23
= y
′
i
Re-writing the Equation of Transformation
x
′
y
′
1
=
t
11
t
12
t
13
t
21
t
22
t
23
0 0 1
x
y
1
x i
·t
11
- y
i
·t
12
- 1 ·t
13
- 0 ·t
21
- 0 ·t
22
- 0 ·t
23
= x
′
i
0 ·t
11
- 0 ·t
12
- 0 ·t
13
- x
i
·t
21
- y
i
·t
22
- 1 ·t
23
= y
′
i
x
i
y
i
1 0 0 0
0 0 0 x
i
y
i
1
t
11
t 12
t
13
t
21
t
22
t
23
=
x
′
i
y
′
i
20
Summary of Affine Transforms
Addition
Multiplication
Basic Operations on Images
24
Example: Averaging Noisy Measurements
g ¯(x, y) =
K
K
i=
g
i
(x, y)
g(x, y) = f (x, y) + η(x, y)
Example: Shading Correction
g(x, y) = f (x, y)h(x, y)
26
Example: Masking
g(x, y) = f (x, y)h(x, y)