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Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Mathematics in Image Processing
Georges KOEPFLER
MAP5, UMR CNRS 8145
UFR de Mathematiques et Informatique
Universite Rene Descartes - Paris 5
45 rue des Saints-Peres
75270 PARIS cedex 06, France
Mathematics in Image Processing – p. 1/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Outline
• Natural and Numerical Images
• Mathematical Representation
• Image Processing Tasks
• Denoising
• Segmentation
• Compression
Mathematics in Image Processing – p. 2/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Natural vs. Numerical Images
50 100 150 200 250
50
100
150
200
250
Mathematics in Image Processing – p. 3/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Natural vs. Numerical Images
50 100 150 200 250
50
100
150
200
250
• The human eye depends on a
finite number of cells;
Mathematics in Image Processing – p. 3/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Natural vs. Numerical Images
50 100 150 200 250
50
100
150
200
250
• The human eye depends on a
finite number of cells;
• Classical camera photos have a natural grain
i.e. resolution, approached by current
numerical cameras;
Mathematics in Image Processing – p. 3/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Natural vs. Numerical Images
50 100 150 200 250
50
100
150
200
250
• The human eye depends on a
finite number of cells;
• Classical camera photos have a natural grain
i.e. resolution, approached by current
numerical cameras;
⇓
natural or digital image ?we process only digital images
(correctly sampled)
Mathematics in Image Processing – p. 3/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Numerical Imagesdiscrete grid
70 80 90 100 110 120 130 140 150
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Mathematics in Image Processing – p. 4/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Numerical Imagesdiscrete grid
70 80 90 100 110 120 130 140 150
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123
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115
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110
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121
127
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139
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96
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103
132
117
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128
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Mathematics in Image Processing – p. 4/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Numerical Imagesdiscrete grid
70 80 90 100 110 120 130 140 150
75
80
85
90
95
100
105
110
115
120
125
80 82 84 86 88 90 92 94 96 98 100
95
100
105
110
115
126
122
117
105
85
61
52
38
39
37
26
22
35
44
32
28
25
26
29
24
132
124
107
96
100
94
89
73
61
55
30
21
44
43
43
37
29
37
43
45
118
113
102
93
70
38
32
36
56
46
27
19
44
47
54
61
56
63
65
87
114
93
88
58
17
4
6
10
13
7
12
22
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87
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105
123
135
143
104
89
58
13
8
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8
14
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22
28
35
45
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83
97
111
115
128
105
77
46
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11
8
15
13
31
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56
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92
97
107
89
66
33
18
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102
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107
115
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64
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101
127
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109
121
117
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42
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27
42
104
128
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68
83
86
76
103
123
125
126
64
56
47
26
13
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66
88
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59
64
73
76
77
71
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108
124
121
119
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68
61
33
18
41
139
204
144
77
78
76
70
72
73
95
122
124
118
116
65
75
78
46
24
30
42
91
104
75
84
85
93
97
95
110
125
115
108
113
65
72
89
64
34
40
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132
147
146
120
102
128
143
116
123
127
110
104
113
65
70
76
65
44
49
59
56
77
150
172
168
171
142
119
126
121
114
107
113
73
71
91
88
61
60
96
127
143
191
232
215
203
148
128
128
125
109
112
123
79
76
99
109
88
69
101
140
167
170
194
223
195
137
121
120
110
107
115
123
88
88
90
127
123
88
64
99
128
152
194
237
221
145
120
125
115
110
121
141
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92
115
144
129
90
75
100
133
152
144
121
109
121
127
120
131
139
138
85
96
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84
103
132
117
86
90
114
128
118
120
127
139
146
154
156
153
167
76
84
96
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84
99
122
119
95
92
116
136
146
152
157
164
169
167
161
163
8085
9095
100105
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110
115
120
0
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200
250
grid of numbers or pixels relief or function
Mathematics in Image Processing – p. 4/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Numerical Images (cont.)
image data relief topographic map
Mathematics in Image Processing – p. 5/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Mathematical representation
Although discrete it is useful to represent an image with aninfinite resolution, as a function on real numbers.
2
Mathematics in Image Processing – p. 6/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Mathematical representation
Although discrete it is useful to represent an image with aninfinite resolution, as a function on real numbers.Basic analysis: a real valued function of one variable
x ∈ [−2, 2] : f(x) = α
(
1 −x2
β
)
e−x2
2β .
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Mathematics in Image Processing – p. 6/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Mathematical representation
Although discrete it is useful to represent an image with aninfinite resolution, as a function on real numbers.Basic analysis: a real valued function of one variable
x ∈ [−2, 2] : f(x) = α
(
1 −x2
β
)
e−x2
2β .
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
But in an image we need columns and rows:thus we use functions of two variables.
Mathematics in Image Processing – p. 6/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Mathematical representation (cont.)
Consider the function of two variables x ∈ [−2, 2], y ∈ [−2, 2]thus (x, y) ∈ [−2, 2] × [−2, 2] and z = u(x, y) ∈ R .
u(x, y) = α
(
1 −x2 + y2
β
)
e−x2+y2
2β .
−2
−1
0
1
2
−2
−1
0
1
2
0
0.2
0.4
0.6
0.8
1
xx
zz
OO
yy
(x,y)(x,y)
z=f(x,y)z=f(x,y)
Mathematics in Image Processing – p. 7/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Mathematical representation (cont.)
Consider the function of two variables x ∈ [−2, 2], y ∈ [−2, 2]thus (x, y) ∈ [−2, 2] × [−2, 2] and z = u(x, y) ∈ R .
u(x, y) = α
(
1 −x2 + y2
β
)
e−x2+y2
2β .
−2
−1
0
1
2
−2
−1
0
1
2
0
0.2
0.4
0.6
0.8
1
xx
zz
OO
yy
(x,y)(x,y)
z=f(x,y)z=f(x,y)
−2−1.5
−1−0.5
00.5
11.5
2
−2
−1
0
1
2
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Mathematics in Image Processing – p. 7/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Mathematical representation (cont.)
Consider the function of two variables x ∈ [−2, 2], y ∈ [−2, 2]thus (x, y) ∈ [−2, 2] × [−2, 2] and z = u(x, y) ∈ R .
u(x, y) = α
(
1 −x2 + y2
β
)
e−x2+y2
2β .
−2
−1
0
1
2
−2
−1
0
1
2
0
0.2
0.4
0.6
0.8
1
xx
zz
OO
yy
(x,y)(x,y)
z=f(x,y)z=f(x,y)
−2−1.5
−1−0.5
00.5
11.5
2
−2
−1
0
1
2
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
50 100 150 200 250
50
100
150
200
250
Mathematics in Image Processing – p. 7/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Image processing
D : digital, A : analogic
LL.......................................... ..........
- - -
@@
@@@R
OBJECT
m
PROCESSINGDIGITALIZATIONACQUISITION
camera
scanner
sampling+quantification
A/D
computer
DSP
digital
output
D/A
analogic
output
A A D
D
D
• Satellite images give information about natural resources, meteorological data, . . .
• Medical images help detect anatomical pathologies, give quantitative data and
functional informations. . .
• Video surveillance is an important issue for security in public transportation. . .
• Images and videos have to be stored/transmitted efficiently. . .
Mathematics in Image Processing – p. 8/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Image processing
D : digital, A : analogic
LL.......................................... ..........
- - -
@@
@@@R
OBJECT
m
PROCESSINGDIGITALIZATIONACQUISITION
camera
scanner
sampling+quantification
A/D
computer
DSP
digital
output
D/A
analogic
output
A A D
D
D
• Satellite images give information about natural resources, meteorological data, . . .
• Medical images help detect anatomical pathologies, give quantitative data and
functional informations. . .
• Video surveillance is an important issue for security in public transportation. . .
• Images and videos have to be stored/transmitted efficiently. . .
Process images as a discrete grid of numbers or a function but take
into account the visual content!
Mathematics in Image Processing – p. 8/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Edge/contour detection
Idea: Detect an object thanks to its boundaryand characterize boundaries by change of luminosity.
JJL
LSS
SSCCLL
BBAASS
Mathematics in Image Processing – p. 9/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Edge/contour detection
Idea: Detect an object thanks to its boundaryand characterize boundaries by change of luminosity.
JJL
LSS
SSCCLL
BBAASS
A discrete rectangular grid :
directions or (line,column)
NW N NE
W E
SW S SE
j − 1 j j + 1
i − 1
Mathematics in Image Processing – p. 9/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Edge/contour detection
Idea: Detect an object thanks to its boundaryand characterize boundaries by change of luminosity.
JJL
LSS
SSCCLL
BBAASS
A discrete rectangular grid :
directions or (line,column)
NW N NE
W E
SW S SE
j − 1 j j + 1
i − 1 •
i • •
i + 1 •
Finite difference operators are used, for example:
‖∇u(i, j)‖ =√
(u(i + 1, j) − u(i − 1, j))2 + (u(i, j + 1) − u(i, j − 1))2
Mathematics in Image Processing – p. 9/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Edge/contour detection (cont.)
50 100 150 200 250
50
100
150
200
250
u(l, c)
Mathematics in Image Processing – p. 10/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Edge/contour detection (cont.)
50 100 150 200 250
50
100
150
200
250
50 100 150 200 250
50
100
150
200
250
u(l, c) ‖∇u‖
(inverse colormap: 0=white, 255=black)
Mathematics in Image Processing – p. 10/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Edge/contour detection (cont.)
50 100 150 200 250
50
100
150
200
250
Mathematics in Image Processing – p. 11/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Edge/contour detection (cont.)
50 100 150 200 250
50
100
150
200
250
50 100 150 200 250
50
100
150
200
250
(inverse video)
Mathematics in Image Processing – p. 11/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Edge/contour detection (cont.)
un(l, c) = u(l, c) + n(l, c)
Mathematics in Image Processing – p. 12/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Edge/contour detection (cont.)
un(l, c) = u(l, c) + n(l, c) ‖∇un‖
Mathematics in Image Processing – p. 12/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Edge/contour detection (cont.)
un(l, c) = u(l, c) + n(l, c) ‖∇un‖
Need to smooth / regularize / denoise the data!
Mathematics in Image Processing – p. 12/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Denoising
Additive noise: a noisy pixel is very different from its neighbors.
Data Mean Filter
0 5 5
7 150 5
1 10 2
0 5 5
7 20 5
1 10 2
Mathematics in Image Processing – p. 13/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Denoising
Additive noise: a noisy pixel is very different from its neighbors.
Data Mean Filter
0 5 5
7 150 5
1 10 2
0 5 5
7 20 5
1 10 2
Median Filter
0 5 5
7 5 5
1 10 2
0 ≤ 1 ≤ 2 ≤ 5 ≤ 5 ≤ 5 ≤ 7 ≤ 10 ≤ 150 =⇒ median=5
Mathematics in Image Processing – p. 13/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Median Filter
50 100 150 200 250
50
100
150
200
250
50 100 150 200 250
50
100
150
200
250
The Median Filter is non linear, gives quite good resultsbut needs a sorting algorithm.
Mathematics in Image Processing – p. 14/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Denoising: a zoo of equations
• The Mean Filter is replaced by weighted local means, thiscan be written
∂u
∂t= ∆u
Mathematics in Image Processing – p. 15/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Denoising: a zoo of equations
• The Mean Filter is replaced by weighted local means, thiscan be written
∂u
∂t= ∆u
• Non linear Partial Differential Equations:
∂u
∂t= |∇u|div
(
∇u
|∇u|
)
Mathematics in Image Processing – p. 15/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Denoising: a zoo of equations
• The Mean Filter is replaced by weighted local means, thiscan be written
∂u
∂t= ∆u
• Non linear Partial Differential Equations:
∂u
∂t= |∇u|div
(
∇u
|∇u|
)
• Total Variation Minimization:
∂u
∂t= div
(
∇u
|∇u|
)
Mathematics in Image Processing – p. 15/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Heat Equation
1
1
0
−1
1.0
0.5
0.51.0
00 xy
u(x, y, 0)
t = 0
1
0
−1
1.0
1.0
0.33
0.67
00
0.5
0.5
xy
u(x, y, 0.1)
t = 0.1
1
0
−1
0.5
1.0
1.0
0.5
0.2
0.4 0.6
00
xy
u(x, y, 0.5)
t = 0.5
1
0
−1
1.0
0.5
0.5
1.00.461
0.1150.230
0.346
00 x
y
u(x, y, 1)
t = 1
Mathematics in Image Processing – p. 16/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Heat Equation
1
1
0
−1
1.0
0.5
0.51.0
00 xy
u(x, y, 0)
t = 0
1
0
−1
1.0
1.0
0.33
0.67
00
0.5
0.5
xy
u(x, y, 0.1)
t = 0.1
1
0
−1
0.5
1.0
1.0
0.5
0.2
0.4 0.6
00
xy
u(x, y, 0.5)
t = 0.5
1
0
−1
1.0
0.5
0.5
1.00.461
0.1150.230
0.346
00 x
y
u(x, y, 1)
t = 1
Mathematics in Image Processing – p. 16/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Heat Equation: evolution
Mathematics in Image Processing – p. 17/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Heat Equation: evolution
Mathematics in Image Processing – p. 17/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Heat Equation: evolution
Mathematics in Image Processing – p. 17/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Heat Equation: evolution
Mathematics in Image Processing – p. 17/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Heat Equation: evolution
Mathematics in Image Processing – p. 17/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Heat Equation: evolution
Mathematics in Image Processing – p. 17/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Heat Equation (cont.)
Mathematics in Image Processing – p. 18/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Heat Equation (cont.)
Gσ ⋆ u0 = u(., σ2/2) ‖∇u(., σ2/2)‖
Mathematics in Image Processing – p. 18/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Mean Curvature Motion
∂u
∂t= |∇u|div
(
∇u
|∇u|
)
original
Mathematics in Image Processing – p. 19/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Mean Curvature Motion
∂u
∂t= |∇u|div
(
∇u
|∇u|
)
‖∇u‖
Mathematics in Image Processing – p. 19/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Total Variation Minimization
∂u
∂t= div
(
∇u
|∇u|
)
original
Mathematics in Image Processing – p. 20/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Total Variation Minimization
∂u
∂t= div
(
∇u
|∇u|
)
‖∇u‖
Mathematics in Image Processing – p. 20/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Total Variation Minimization (cont.)
noisy image denoised image
Mathematics in Image Processing – p. 21/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Total Variation Minimization (cont.)
original image denoised image
Mathematics in Image Processing – p. 22/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Inpainting
Idea: Complete the level lines. . .
occlusions level lines (each 20)
Mathematics in Image Processing – p. 23/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Inpainting
Idea: Complete the level lines. . .
partial restoration level lines (each 20)
Mathematics in Image Processing – p. 23/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Inpainting
Idea: Complete the level lines. . .
initial level lines level lines (each 20)
Mathematics in Image Processing – p. 23/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Segmentation
Idea: Analyze original image g and get a simple image u.Thus to segment an image is:
• replace the original by a cartoon;
• partition the image in homogeneous regions;
Mathematics in Image Processing – p. 24/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Segmentation
Idea: Analyze original image g and get a simple image u.Thus to segment an image is:
• replace the original by a cartoon;
• partition the image in homogeneous regions;
MUMFORD-SHAH functional:
E(K) =
∫
Ω\K(u − g)2 + λℓ(K)
Ω =⋃
O , O regions s.t. O ∩ O′ = ∅
K =⋃
∂O, set of boundaries
Mathematics in Image Processing – p. 24/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Segmentation
Idea: Analyze original image g and get a simple image u.Thus to segment an image is:
• replace the original by a cartoon;
• partition the image in homogeneous regions;
MUMFORD-SHAH functional:
E(K) =
∫
Ω\K(u − g)2 + λℓ(K)
Ω =⋃
O , O regions s.t. O ∩ O′ = ∅
K =⋃
∂O, set of boundaries
Merge two regions O,O′ iff E(K \ (∂O ∩ ∂O′)) < E(K) .
Mathematics in Image Processing – p. 24/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Segmentation
Mathematics in Image Processing – p. 25/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Segmentation: medical application
Mathematics in Image Processing – p. 26/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Segmentation: medical application
Mathematics in Image Processing – p. 26/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Texture Segmentation
Mathematics in Image Processing – p. 27/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Texture Segmentation
Mathematics in Image Processing – p. 27/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Texture Segmentation
Original data gray level wavelet coefficient
segmentation segmentation
Mathematics in Image Processing – p. 27/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Color Segmentation
Mathematics in Image Processing – p. 28/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Color Segmentation
original 100 regions
Mathematics in Image Processing – p. 28/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Compression
Idea: In an image objects of different scale/size are present.
Get a pyramidal or multiscale representation of an image.
Mathematics in Image Processing – p. 29/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Compression
Idea: In an image objects of different scale/size are present.
Get a pyramidal or multiscale representation of an image.
(Images & Idea: Basarab MATEI, University Paris 6)
Mathematics in Image Processing – p. 29/30
Georges KOEPFLER – MAP5 – University Rene DESCARTES – Paris 5
Compression with wavelets
u(x, y) =∑
λ
cλΨλ(x, y)
(Images & Idea: Basarab MATEI, University Paris 6)
Mathematics in Image Processing – p. 30/30
