Mathematics in Image Processing Georges KOEPFLER
[email protected]
MAP5, UMR CNRS 8145 ´ UFR de Mathematiques et Informatique Universite´ Rene´ Descartes - Paris 5 ` 45 rue des Saints-Peres 75270 PARIS cedex 06, France
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 1/30
Outline
•
Natural and Numerical Images
•
Mathematical Representation
•
Image Processing Tasks
•
Denoising
•
Segmentation
•
Compression
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 2/30
Natural vs. Numerical Images
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Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 3/30
Natural vs. Numerical Images
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•
The human eye depends on a finite number of cells;
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Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 3/30
Natural vs. Numerical Images
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•
The human eye depends on a finite number of cells;
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• 150
Classical camera photos have a natural grain i.e. resolution, approached by current numerical cameras;
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Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 3/30
Natural vs. Numerical Images
50
•
The human eye depends on a finite number of cells;
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• 150
Classical camera photos have a natural grain i.e. resolution, approached by current numerical cameras;
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250 50
100
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natural or digital image ?
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⇓ we process only digital images (correctly sampled)
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 3/30
Numerical Images discrete grid 75 80 85 90 95 100 105 110 115 120 125 70
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Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 4/30
Numerical Images discrete grid 75 80 85 90 95 100 105 110 115 120 125 70
95
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110
126 132 118 114 104 105 89
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64
68
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88
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85
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122 124 113 93
89
77
66
46
60
56
68
75
72
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71
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88
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96
84
117 107 102 88
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46
33
34
42
47
61
78
89
76
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99
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92
82
96
105 96
83
100
105
110
115 80
93
58
13
15
18
16
22
26
33
46
64
65
88 109 127 115 84
85 100 70
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8
11
11
13
10
13
18
24
34
44
61
88 123 144 103 84
61
94
38
4
5
8
14
15
27
48
41
30
40
49
60
69
52
89
32
6
8
15
24
30
42
66 139 42
72
59
96 101 64
38
73
36
10
14
13
22
30 104 88 204 91 132 56 127 140 99
39
61
56
13
20
31
39
38 128 95 144 104 147 77 143 167 128 100 90
37
55
46
7
22
33
62
56
94
59
77
75 146 150 191 170 152 133 114 92
26
30
27
12
28
53
73
64
78
64
78
84 120 172 232 194 194 152 128 116
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21
19
22
35
55
95
71
75
73
76
85 102 168 215 223 237 144 118 136
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44
44
44
45
55 102 101 68
76
70
93 128 171 203 195 221 121 120 146
44
43
47
76
50
61
95 127 83
77
72
97 143 142 148 137 145 109 127 152
32
43
54
87
68
56
74
98
86
71
73
95 116 119 128 121 120 121 139 157
28
37
61
87
83
61
58
70
76
81
95 110 123 126 128 120 125 127 146 164
25
29
56 105 97
80
70
85 103 108 122 125 127 121 125 110 115 120 154 169
26
37
63 123 111 92
29
43
65 135 115 97 107 121 125 121 118 108 104 107 112 115 121 139 153 161
24
45
87 143 128 107 115 117 126 119 116 113 113 113 123 123 141 138 167 163
82
84
86
120
130
140
150
88 129 132 99 90 117 122 75
86 119 95
96 109 123 124 124 115 110 114 109 107 110 131 156 167
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90
92
94
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100
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 4/30
Numerical Images discrete grid 75 80 85 90 95 100 105 110 115 120 125 70
95
80
90
100
110
126 132 118 114 104 105 89
71
70
64
68
65
65
65
73
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88
77
85
76
122 124 113 93
89
77
66
46
60
56
68
75
72
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71
76
88
78
96
84
117 107 102 88
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46
33
34
42
47
61
78
89
76
91
99
90
92
82
96
105 96
83
100
105
110
115 80
93
58
13
15
18
16
22
26
33
46
64
65
88 109 127 115 84
85 100 70
17
8
11
11
13
10
13
18
24
34
44
61
88 123 144 103 84
61
94
38
4
5
8
14
15
27
48
41
30
40
49
60
69
52
89
32
6
8
15
24
30
42
66 139 42
72
59
96 101 64
38
73
36
10
14
13
22
30 104 88 204 91 132 56 127 140 99
39
61
56
13
20
31
39
38 128 95 144 104 147 77 143 167 128 100 90
37
55
46
7
22
33
62
56
94
59
77
75 146 150 191 170 152 133 114 92
26
30
27
12
28
53
73
64
78
64
78
84 120 172 232 194 194 152 128 116
22
21
19
22
35
55
95
71
75
73
76
85 102 168 215 223 237 144 118 136
35
44
44
44
45
55 102 101 68
76
70
93 128 171 203 195 221 121 120 146
44
43
47
76
50
61
95 127 83
77
72
97 143 142 148 137 145 109 127 152
32
43
54
87
68
56
74
98
86
71
73
95 116 119 128 121 120 121 139 157
28
37
61
87
83
61
58
70
76
81
95 110 123 126 128 120 125 127 146 164
25
29
56 105 97
80
70
85 103 108 122 125 127 121 125 110 115 120 154 169
26
37
63 123 111 92
29
43
65 135 115 97 107 121 125 121 118 108 104 107 112 115 121 139 153 161
24
45
87 143 128 107 115 117 126 119 116 113 113 113 123 123 141 138 167 163
82
84
86
90
92
94
130
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150
88 129 132 99 90 117 122 75
96
grid of numbers or pixels
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250
86 119 95
96 109 123 124 124 115 110 114 109 107 110 131 156 167
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120
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150
100 120 50
115 110
0 80
105 85
90
100 95
100
105
95
relief or function
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 4/30
Numerical Images (cont.)
image data
relief
topographic map
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 5/30
Mathematical representation Although discrete it is useful to represent an image with an infinite resolution, as a function on real numbers.
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 6/30
Mathematical representation Although discrete it is useful to represent an image with an infinite resolution, as a function on real numbers. Basic analysis: a real valued function of one variable 2 2 x − x2β x ∈ [−2, 2] : f (x) = α 1 − . e β 1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6 −2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 6/30
Mathematical representation Although discrete it is useful to represent an image with an infinite resolution, as a function on real numbers. Basic analysis: a real valued function of one variable 2 2 x − x2β x ∈ [−2, 2] : f (x) = α 1 − . e β 1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6 −2
−1.5
−1
−0.5
0
0.5
1
1.5
2
But in an image we need columns and rows: thus we use functions of two variables. Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 6/30
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 .
2
x +y u(x, y) = α 1 − β
2
e
2 +y 2
−x
2β
.
1 z 0.8 z=f(x,y) z=f(x,y)
0.6
0.4
0.2
y x
0 2
(x,y) (x,y) O
1
2 1
0 0 −1
−1 −2
−2
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 7/30
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 . 2
x +y u(x, y) = α 1 − β
2
e
2 +y 2
−x
2β
.
1.4
1
1.2
z
1
0.8
0.8 0.6
z=f(x,y) z=f(x,y)
0.6
0.4
0.4
0.2
0.2
−0.2
0
y
−0.4
x 0 2
−0.6 2
(x,y) (x,y) O
1
2 1
0
1 0
0 −1
−1 −2
−2
−1 −2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 7/30
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 . 2
x +y u(x, y) = α 1 − β
2
e
2 +y 2
−x
.
2β
1.4
1
1.2 50
z
1
0.8
0.8 0.6
z=f(x,y) z=f(x,y)
0.6
100
0.4
0.4
0.2
0.2
−0.2
0
y
−0.4
x 0 2
150
−0.6 2
(x,y) (x,y) O
1
2 1
0
200
1 0
0 −1
−1 −2
−2
−1 −2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2 250 50
100
150
200
250
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 7/30
Image processing D : digital, A : analogic digital D
m
......... .. ... LL.
OBJECT
A
-
camera scanner
ACQUISITION
A
-
sampling+quantification A/D
DIGITALIZATION
D
-
output
computer DSP
PROCESSING
@
@ @ D @ R @
analogic D/A output
•
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. . .
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 8/30
Image processing D : digital, A : analogic digital D
m
......... .. ... LL.
OBJECT
A
-
camera scanner
ACQUISITION
A
-
sampling+quantification A/D
DIGITALIZATION
D
-
output
computer DSP
PROCESSING
@
@ @ D @ R @
analogic D/A output
•
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! Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 8/30
Edge/contour detection Detect an object thanks to its boundary and characterize boundaries by change of luminosity.
Idea:
S S L S S J J L
S L BB
AA
S L C C
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 9/30
Edge/contour detection Detect an object thanks to its boundary and characterize boundaries by change of luminosity.
Idea:
S S L S S J J L
S L BB
AA
S L C C
A discrete rectangular grid : directions or NW W SW
N S
NE E SE
(line,column) j−1 j j+1 i−1
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 9/30
Edge/contour detection Detect an object thanks to its boundary and characterize boundaries by change of luminosity.
Idea:
S S L S S J J L
S L BB
AA
S L C C
A discrete rectangular grid : directions or NW W SW
N S
NE E SE
i−1 i i+1
(line,column) j−1 j j+1 • • • •
Finite difference operators are used, for example: p k∇u(i, j)k = (u(i + 1, j) − u(i − 1, j))2 + (u(i, j + 1) − u(i, j − 1))2 Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 9/30
Edge/contour detection (cont.)
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u(l, c)
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 10/30
Edge/contour detection (cont.)
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u(l, c)
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k∇uk (inverse colormap: 0=white, 255=black)
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 10/30
Edge/contour detection (cont.)
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Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 11/30
Edge/contour detection (cont.)
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(inverse video)
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 11/30
Edge/contour detection (cont.)
un (l, c) = u(l, c) + n(l, c)
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 12/30
Edge/contour detection (cont.)
un (l, c) = u(l, c) + n(l, c)
k∇un k
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 12/30
Edge/contour detection (cont.)
un (l, c) = u(l, c) + n(l, c)
k∇un k
Need to smooth / regularize / denoise the data!
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 12/30
Denoising Additive noise: a noisy pixel is very different from its neighbors. Data
Mean Filter
0
5
5
0
5
5
7
150
5
7
20
5
1
10
2
1
10
2
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 13/30
Denoising Additive noise: a noisy pixel is very different from its neighbors. Data
Mean Filter
Median Filter
0
5
5
0
5
5
0
5
5
7
150
5
7
20
5
7
5
5
1
10
2
1
10
2
1
10
2
0 ≤ 1 ≤ 2 ≤ 5 ≤ 5 ≤ 5 ≤ 7 ≤ 10 ≤ 150 =⇒ median=5
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 13/30
Median Filter
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The Median Filter is non linear, gives quite good results but needs a sorting algorithm.
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 14/30
Denoising: a zoo of equations • The Mean Filter is replaced by weighted local means, this
can be written ∂u = ∆u ∂t
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 15/30
Denoising: a zoo of equations • The Mean Filter is replaced by weighted local means, this
can be written ∂u = ∆u ∂t • Non linear Partial Differential Equations:
∂u = |∇u| div ∂t
∇u |∇u|
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 15/30
Denoising: a zoo of equations • The Mean Filter is replaced by weighted local means, this
can be written ∂u = ∆u ∂t • Non linear Partial Differential Equations:
∂u = |∇u| div ∂t
∇u |∇u|
• Total Variation Minimization:
∂u = div ∂t
∇u |∇u|
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 15/30
Heat Equation t = 0.1
t=0
u(x, y, 0.1)
u(x, y, 0) 1
1
0
0
0.33 1.0
1
1.0
0.67 −1
−1 1.0
0.5
1.0
0.5 0.5
0.5
y
0
0
x
y
0
0
x
t=1
t = 0.5
u(x, y, 1)
u(x, y, 0.5)
1
1
0
0
0.2
1.0
1.0 0.4
0.346 0.461
0.6
−1
−1 0.5
1.0
0.230
0.115
1.0
0.5 0.5
0.5
x y
0
0
y
0
0
x
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 16/30
Heat Equation t = 0.1
t=0
u(x, y, 0.1)
u(x, y, 0) 1
1
0
0
0.33 1.0
1
1.0
0.67 −1
−1 1.0
0.5
1.0
0.5 0.5
0.5
y
0
0
x
y
0
0
x
t=1
t = 0.5
u(x, y, 1)
u(x, y, 0.5)
1
1
0
0
0.2
1.0
1.0 0.4
0.346 0.461
0.6
−1
−1 0.5
1.0
0.230
0.115
1.0
0.5 0.5
0.5
x y
0
0
y
0
0
x
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 16/30
Heat Equation: evolution
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 17/30
Heat Equation: evolution
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 17/30
Heat Equation: evolution
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 17/30
Heat Equation: evolution
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 17/30
Heat Equation: evolution
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 17/30
Heat Equation: evolution
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 17/30
Heat Equation (cont.)
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 18/30
Heat Equation (cont.)
Gσ ⋆ u0 = u(., σ 2 /2)
k∇u(., σ 2 /2)k
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 18/30
Mean Curvature Motion
∂u = |∇u| div ∂t
∇u |∇u|
original
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 19/30
Mean Curvature Motion
∂u = |∇u| div ∂t
∇u |∇u|
k∇uk
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 19/30
Total Variation Minimization
∂u = div ∂t
∇u |∇u|
original
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 20/30
Total Variation Minimization
∂u = div ∂t
∇u |∇u|
k∇uk
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 20/30
Total Variation Minimization (cont.)
noisy image
denoised image
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 21/30
Total Variation Minimization (cont.)
original image
denoised image
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 22/30
Inpainting Idea:
Complete the level lines. . .
occlusions
level lines (each 20)
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 23/30
Inpainting Idea:
Complete the level lines. . .
partial restoration
level lines (each 20)
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 23/30
Inpainting Idea:
Complete the level lines. . .
initial level lines
level lines (each 20)
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 23/30
Segmentation Analyze original image g and get a simple image u. Thus to segment an image is: • replace the original by a cartoon; Idea:
• partition the image in homogeneous regions;
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 24/30
Segmentation Analyze original image g and get a simple image u. Thus to segment an image is: • replace the original by a cartoon; Idea:
• partition the image in homogeneous regions; M UMFORD -S HAH
functional: E(K) =
Z
(u − g)2 + λℓ(K)
Ω\K
S Ω = O , O regions s.t. O ∩ O′ = ∅ S K = ∂O, set of boundaries
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 24/30
Segmentation Analyze original image g and get a simple image u. Thus to segment an image is: • replace the original by a cartoon; Idea:
• partition the image in homogeneous regions; M UMFORD -S HAH
functional: E(K) =
Z
(u − g)2 + λℓ(K)
Ω\K
S Ω = O , O regions s.t. O ∩ O′ = ∅ S K = ∂O, set of boundaries
Merge two regions O,O′ iff E(K \ (∂O ∩ ∂O′ )) < E(K) .
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 24/30
Segmentation
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 25/30
Segmentation: medical application
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 26/30
Segmentation: medical application
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 26/30
Texture Segmentation
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 27/30
Texture Segmentation
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 27/30
Texture Segmentation
Original data
gray level segmentation
wavelet coefficient segmentation
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 27/30
Color Segmentation
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 28/30
Color Segmentation
original
100 regions
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 28/30
Compression Idea:
In an image objects of different scale/size are present.
Get a pyramidal or multiscale representation of an image.
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 29/30
Compression Idea:
In an image objects of different scale/size are present.
Get a pyramidal or multiscale representation of an image.
(Images & Idea: Basarab M ATEI, University Paris 6)
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 29/30
Compression with wavelets u(x, y) =
X
cλ Ψλ (x, y)
λ
(Images & Idea: Basarab M ATEI, University Paris 6)
Georges KOEPFLER – MAP5 – University Rene´ D ESCARTES – Paris 5 Mathematics in Image Processing – p. 30/30
