Yunlong Sheng and Henri H. Arsenault, "Noisy-image normalization using low-order radial moments of circular-harmonic functions," J. Opt. Soc. Am. A 4, 1176-1184 (1987)
Radial moments of circular-harmonic functions are used for image normalization. The moment orders are lower than those used in the classical method. The principal axes of image are replaced by a mean direction of image. The influence of random and correlated noise on moment-based image normalization is analyzed. The new method is more robust than the classical method against background noise. Experimental comparisons between the two methods are given. The complete series of the radial moments of circular-harmonic functions can be represented in the Cartesian coordinate system by modified complex moments whose orders are real valued. An application of the new method to gray-level noisy-image recognition is demonstrated that is invariant under changes of position, rotation, scale, and intensity.
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Rule for the Angle Θ = (½)tan−1 (A/B) as a Function of the Signs of A and Ba
B
A
Θ
Condition
0
0
0°
0
+
45°
0
−
−45°
+
0
0°
−
0
−90°
+
+
ξ/2
0°< Θ < 45°
+
−
ξ/2
−45° < Θ 0°
−
+
ξ/2 + 90°
45° < Θ < 90°
−
−
ξ/2 − 90°
−90° < Θ < −45°
From Ref. 2; let ξ = tan−1(A/B) and −90° < ξ < 90°.
Table 2
Mean Values of Image Characteristic Directions Θs,m (m = 2) for the Noisy Image of Fig. 2(a)a
k
Θ2,2
Θ4,2
Θ6,2
0.67
70.11
68.85
69.64
0.50
69.64
69.28
59.70
The true orientation of the aircraft without the noise is 70°.
Table 3
The Mean Direction
and Principal Axes Θ0 for the Segmented Space Shuttle Images Shown in Fig. 4(b)a
Value for the Following Orientation of the Space Shuttle
Threshold
Parameter
I
II
III
IV
V
VI
Θ (true value)
123.22
128.22
0.0
43.22
113.22
−56.78
150
124.59
129.86
1.33
44.00
114.60
124.30
Θ0
124.15
130.00
1.30
40.37
113.01
124.23
148
125.39
130.90
1.33
44.25
115.32
124.42
Θ0
127.77
134.50
1.30
49.89
114.77
127.39
146
127.01
132.76
3.13
46.78
116.55
124.81
Θ0
139.03
147.11
6.84
34.53
121.77
135.60
144
131.17
137.67
5.82
38.58
119.19
116.14
Θ0
172.61
173.47
10.98
30.97
7.61
156.24
When the threshold decreases, the background cloud noise increases.
is more robust than Θ0 against background noise.
Table 4
Normalization Parameters (Orientation
, Scale Factor K, and Intensity Factor G) Obtained by Two Methods for a Segmented Image of the Space Shuttlea
Threshold
Method
K
G
152
M2,2, M2,0, M3,0
43.91
2.04
1.10
M4,2, M2,0, M4,0
36.66
1.91
1.26
150
M2,2, M2,0, M3,0
40.95
1.79
1.39
M4,2, M2,0, M4,0
26.22
1.45
2.13
148
M2,2, M2,0, M3,0
33.90
1.51
1.85
M4,2, M2,0, M4,0
22.16
1.15
3.14
For images without noise,
= 43.22°, K = 2.0, and G = 1.0.
Table 5
Normalization Parameters for Input Images I–IV [Figs. 2(a)–2(d)] Obtained by the M2,2, M2,0, M3,0 Methoda
Image
Value Obtained
Θ
K
G
I
True
70.0
1.0
1.43
Experimental
68.7
0.98
1.52
II
True
210.0
1.33
1.25
Experimental
28.0
1.34
1.53
III
True
50.0
2.0
1.0
Experimental
44.3
1.99
1.23
IV
Experimental
123.2
1.28
0.73
True values are for images without noise.
Table 6
Cross-Correlation Peak Values among Six Prototypesa
Peak Value of Cross Correlation with the Following Prototype
Prototype
Pr. 1
Pr. 1′
Pr. 2
Pr. 2′
Pr. 3
Pr. 3′
Pr. 1
1.00
Pr. 1−
0.64
1.00
Pr. 2
0.73
0.67
1.00
Pr. 2−
0.67
0.73
0.71
1.00
Pr. 3
0.80
0.63
0.80
0.62
1.00
Pr. 3′
0.63
0.80
0.62
0.80
0.61
1.00
Pr. 1, Lightning; Pr. 2, F-106; Pr. 3, space shuttle. Pr. 1′, Pr. 2′, and Pr. 3′ are, respectively, the prototypes Pr. 1, Pr. 2, and Pr. 3 rotated 180°.
Table 7
Cross-Correlation Peak Values among the Normalized Input Images I–IV and the Prototypes Pr. 1–Pr. 3 and Pr. 1′–Pr. 3′
Peak Value of Cross Correlation with the Following Prototype
Image
Pr. 1
Pr. 1′
Pr. 2
Pr. 2′
Pr. 3
Pr. 3′
I
0.85
0.57
0.65
0.58
0.71
0.55
II
0.64
0.70
0.69
0.96
0.58
0.76
III
0.73
0.63
0.83
0.63
0.90
0.62
IV
0.82
0.66
0.76
0.64
0.90
0.70
Tables (7)
Table 1
Rule for the Angle Θ = (½)tan−1 (A/B) as a Function of the Signs of A and Ba
B
A
Θ
Condition
0
0
0°
0
+
45°
0
−
−45°
+
0
0°
−
0
−90°
+
+
ξ/2
0°< Θ < 45°
+
−
ξ/2
−45° < Θ 0°
−
+
ξ/2 + 90°
45° < Θ < 90°
−
−
ξ/2 − 90°
−90° < Θ < −45°
From Ref. 2; let ξ = tan−1(A/B) and −90° < ξ < 90°.
Table 2
Mean Values of Image Characteristic Directions Θs,m (m = 2) for the Noisy Image of Fig. 2(a)a
k
Θ2,2
Θ4,2
Θ6,2
0.67
70.11
68.85
69.64
0.50
69.64
69.28
59.70
The true orientation of the aircraft without the noise is 70°.
Table 3
The Mean Direction
and Principal Axes Θ0 for the Segmented Space Shuttle Images Shown in Fig. 4(b)a
Value for the Following Orientation of the Space Shuttle
Threshold
Parameter
I
II
III
IV
V
VI
Θ (true value)
123.22
128.22
0.0
43.22
113.22
−56.78
150
124.59
129.86
1.33
44.00
114.60
124.30
Θ0
124.15
130.00
1.30
40.37
113.01
124.23
148
125.39
130.90
1.33
44.25
115.32
124.42
Θ0
127.77
134.50
1.30
49.89
114.77
127.39
146
127.01
132.76
3.13
46.78
116.55
124.81
Θ0
139.03
147.11
6.84
34.53
121.77
135.60
144
131.17
137.67
5.82
38.58
119.19
116.14
Θ0
172.61
173.47
10.98
30.97
7.61
156.24
When the threshold decreases, the background cloud noise increases.
is more robust than Θ0 against background noise.
Table 4
Normalization Parameters (Orientation
, Scale Factor K, and Intensity Factor G) Obtained by Two Methods for a Segmented Image of the Space Shuttlea
Threshold
Method
K
G
152
M2,2, M2,0, M3,0
43.91
2.04
1.10
M4,2, M2,0, M4,0
36.66
1.91
1.26
150
M2,2, M2,0, M3,0
40.95
1.79
1.39
M4,2, M2,0, M4,0
26.22
1.45
2.13
148
M2,2, M2,0, M3,0
33.90
1.51
1.85
M4,2, M2,0, M4,0
22.16
1.15
3.14
For images without noise,
= 43.22°, K = 2.0, and G = 1.0.
Table 5
Normalization Parameters for Input Images I–IV [Figs. 2(a)–2(d)] Obtained by the M2,2, M2,0, M3,0 Methoda
Image
Value Obtained
Θ
K
G
I
True
70.0
1.0
1.43
Experimental
68.7
0.98
1.52
II
True
210.0
1.33
1.25
Experimental
28.0
1.34
1.53
III
True
50.0
2.0
1.0
Experimental
44.3
1.99
1.23
IV
Experimental
123.2
1.28
0.73
True values are for images without noise.
Table 6
Cross-Correlation Peak Values among Six Prototypesa
Peak Value of Cross Correlation with the Following Prototype
Prototype
Pr. 1
Pr. 1′
Pr. 2
Pr. 2′
Pr. 3
Pr. 3′
Pr. 1
1.00
Pr. 1−
0.64
1.00
Pr. 2
0.73
0.67
1.00
Pr. 2−
0.67
0.73
0.71
1.00
Pr. 3
0.80
0.63
0.80
0.62
1.00
Pr. 3′
0.63
0.80
0.62
0.80
0.61
1.00
Pr. 1, Lightning; Pr. 2, F-106; Pr. 3, space shuttle. Pr. 1′, Pr. 2′, and Pr. 3′ are, respectively, the prototypes Pr. 1, Pr. 2, and Pr. 3 rotated 180°.
Table 7
Cross-Correlation Peak Values among the Normalized Input Images I–IV and the Prototypes Pr. 1–Pr. 3 and Pr. 1′–Pr. 3′
Peak Value of Cross Correlation with the Following Prototype
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