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.

Adrian Stern, Inna Kruchakov, Eitan Yoavi, and Norman S. Kopeika Appl. Opt. 41(11) 2164-2171 (2002)

References

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Rule for the Angle Θ = (½)tan^{−1} (A/B) as a Function of the Signs of A and B^{a}

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 Θ_{s,m} (m = 2) for the Noisy Image of Fig. 2(a)^{a}

k

Θ_{2,2}

Θ_{4,2}

Θ_{6,2}

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
$\overline{\mathrm{\Theta}}$ and Principal Axes Θ_{0} 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

$\overline{\mathrm{\Theta}}$

124.59

129.86

1.33

44.00

114.60

124.30

Θ_{0}

124.15

130.00

1.30

40.37

113.01

124.23

148

$\overline{\mathrm{\Theta}}$

125.39

130.90

1.33

44.25

115.32

124.42

Θ_{0}

127.77

134.50

1.30

49.89

114.77

127.39

146

$\overline{\mathrm{\Theta}}$

127.01

132.76

3.13

46.78

116.55

124.81

Θ_{0}

139.03

147.11

6.84

34.53

121.77

135.60

144

$\overline{\mathrm{\Theta}}$

131.17

137.67

5.82

38.58

119.19

116.14

Θ_{0}

172.61

173.47

10.98

30.97

7.61

156.24

When the threshold decreases, the background cloud noise increases.
$\overline{\mathrm{\Theta}}$ is more robust than Θ_{0} against background noise.

Table 4

Normalization Parameters (Orientation
$\overline{\mathrm{\Theta}}$, Scale Factor K, and Intensity Factor G) Obtained by Two Methods for a Segmented Image of the Space Shuttle^{a}

Threshold

Method

$\overline{\mathrm{\Theta}}$

K

G

152

M_{2,2}, M_{2,0}, M_{3,0}

43.91

2.04

1.10

M_{4,2}, M_{2,0}, M_{4,0}

36.66

1.91

1.26

150

M_{2,2}, M_{2,0}, M_{3,0}

40.95

1.79

1.39

M_{4,2}, M_{2,0}, M_{4,0}

26.22

1.45

2.13

148

M_{2,2}, M_{2,0}, M_{3,0}

33.90

1.51

1.85

M_{4,2}, M_{2,0}, M_{4,0}

22.16

1.15

3.14

For images without noise,
$\overline{\mathrm{\Theta}}$ = 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 M_{2,2}, M_{2,0}, M_{3,0} Method^{a}

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 Prototypes^{a}

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 B^{a}

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 Θ_{s,m} (m = 2) for the Noisy Image of Fig. 2(a)^{a}

k

Θ_{2,2}

Θ_{4,2}

Θ_{6,2}

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
$\overline{\mathrm{\Theta}}$ and Principal Axes Θ_{0} 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

$\overline{\mathrm{\Theta}}$

124.59

129.86

1.33

44.00

114.60

124.30

Θ_{0}

124.15

130.00

1.30

40.37

113.01

124.23

148

$\overline{\mathrm{\Theta}}$

125.39

130.90

1.33

44.25

115.32

124.42

Θ_{0}

127.77

134.50

1.30

49.89

114.77

127.39

146

$\overline{\mathrm{\Theta}}$

127.01

132.76

3.13

46.78

116.55

124.81

Θ_{0}

139.03

147.11

6.84

34.53

121.77

135.60

144

$\overline{\mathrm{\Theta}}$

131.17

137.67

5.82

38.58

119.19

116.14

Θ_{0}

172.61

173.47

10.98

30.97

7.61

156.24

When the threshold decreases, the background cloud noise increases.
$\overline{\mathrm{\Theta}}$ is more robust than Θ_{0} against background noise.

Table 4

Normalization Parameters (Orientation
$\overline{\mathrm{\Theta}}$, Scale Factor K, and Intensity Factor G) Obtained by Two Methods for a Segmented Image of the Space Shuttle^{a}

Threshold

Method

$\overline{\mathrm{\Theta}}$

K

G

152

M_{2,2}, M_{2,0}, M_{3,0}

43.91

2.04

1.10

M_{4,2}, M_{2,0}, M_{4,0}

36.66

1.91

1.26

150

M_{2,2}, M_{2,0}, M_{3,0}

40.95

1.79

1.39

M_{4,2}, M_{2,0}, M_{4,0}

26.22

1.45

2.13

148

M_{2,2}, M_{2,0}, M_{3,0}

33.90

1.51

1.85

M_{4,2}, M_{2,0}, M_{4,0}

22.16

1.15

3.14

For images without noise,
$\overline{\mathrm{\Theta}}$ = 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 M_{2,2}, M_{2,0}, M_{3,0} Method^{a}

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 Prototypes^{a}

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