The author is a fellow of the Science and Technology Agency of Japan with the Kansai Advanced Research Center, Communications Research Laboratory, Ministry of Posts and Telecommunications, 588-2 Iwaoka Nishi-ku, Kobe 651-24, Japan.
Jean-Christophe Terrillon, "Image preprocessing for rotation-invariant pattern recognition in the presence of signal-dependent noise," Appl. Opt. 35, 1879-1893 (1996)
I propose a new method that ensures efficient rotation-invariant pattern recognition in the presence of signal-dependent noise by combining the application of rotation-invariant correlation filters with preprocessing of the noisy input images. The preprocessing uses local suboptimal estimators derived from estimation theory and implies an a priori knowledge of a model describing the noise source. The image noise sources considered are speckle and film-grain noise. Four different metrics are used to analyze the correlation performance of the circular-harmonic filter, the phase-only circular-harmonic filter, and the binary phase-only circular-harmonic filter, with and without a preprocessing. Computer simulations show that signal-dependent noise can seriously degrade the performance of the phase-only circular-harmonic filter and the binary phase-only circular-harmonic filter. The most severe indication of correlation-performance degradation is the occurrence of false alarms in 15% to 20% of noise realizations of the correlation. Preprocessing increases the correlation-peak signal-to-noise ratio significantly and reduces the false-alarm probability by one to two orders of magnitude.
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Correlation Results for Three of the Objects in Fig. 1 without Noise and with Speckle SDN (M = 1) for the CHF, the POCHF, and the BPOCHF (m = 2)a
Without Noise
With Speckle, M = 1
Filters
Ip
Coordinates
SNRI
〈SNRI〉
SNR
PFA
LE000
CHF
1.00
(33, 26)
9.49
9.35
510.7
0.0000
POCHF
8.27
(33, 26)
24.14
17.79
67.26
0.1494
BPOCHF
2.64
(97, 90)
26.27
16.72
62.78
0.1982
LE045
CHF
1.00
(32, 32)
10.09
9.96
500.0
0.0000
POCHF
6.15
(32, 32)
21.92
15.83
67.13
0.1653
BPOCHF
2.40
(96, 96)
21.43
15.85
64.62
0.1781
X29-000
CHF
1.00
(62, 48)
6.95
6.92
582.0
0.0000
POCHF
17.40
(62, 48)
39.57
22.20
72.90
0.1631
In each simulation the statistics are calculated over 104 noise realizations of the correlation. In the case of LE000 and LE045 the filters are generated from LE000. For the CHF, in units of intensity, Ip = 0.53455 × 105 for LE000, Ip = 0.53284 × 105 for LE045, and Ip = 0.61162 × 104 for X29-000. The coordinates are those of the correlation peak (or of the proper center) for m = 2.
Table 2
Correlation Results of the POCHF (m = 2) of the Letter E Degraded with Speckle SDN (N = 1), and Preprocessed with the LLMMSE Estimator Designed for Multiplicative Noise or the Homomorphic Processing (H.T. + J.–S.) with a Window of Dimensions 3 × 3 or 5 × 5 Pixelsa
Input Image
Correlation, POCHF
〈MSD〉
〈 SNRI〉n
SNR
PFA
〈Ip〉n
LE000
Noise realizations
14435.8
0.7370
67.26
0.1494
1.1130
H.T. + J.-S. 3 × 3
4876.6
0.8144
102.27
0.0184
1.0475
5 × 5
3847.2
0.8252
101.32
0.0138
0.8388
LLMMSE 3 × 3
2423.8
0.6744
154.87
0.0091
0.5583
5 × 5
1842.5
0.6210
206.79
0.0099
0.4057
LE045
Noise realizations
14831.9
0.7222
67.13
0.1653
1.2518
H.T. + J.-S. 3 × 3
5020.9
0.8517
98.00
0.0286
1.1590
5 × 5
3934.8
0.8818
100.74
0.0153
0.9556
LLMMSE 3 × 3
2316.3
0.7961
141.04
0.0103
0.7097
5 × 5
1713.4
0.7190
194.07
0.0033
0.5009
In each simulation the statistics are calculated over 104 noise realizations or preprocessed realizations of the input image and the correlation. The POCHF is generated from LE000. 〈SNRI〉 and 〈Ip〉 are normalized with respect to SNRI and Ip, respectively (SNRI = 24.14, Ip = 0.442189 × 106 for LE000, and SNRI = 21.92, Ip = 0.327574 × 106 for LE045).
Table 3
Correlation Results of the POCHF (m = 2) of the Object LE045 Degraded with Speckle SDN (M = 1) for Different Values of the Radial Cutoff Frequency ρ0 of the Filter, and Preprocessed (without a Reduction of ρ0) by the LLMMSE Estimator Designed for Multiplicative Noise or the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 or 5 × 5 Pixelsa
Correlation, POCHF
Input Image
ρ0
〈SNRI〉n
SNR
PFA
〈Ip〉n
LE045
Noise realizations
31
0.7222
67.13
0.1653
1.2518
8
0.5406
199.16
0.0224
0.4180
H.T. + J.-S., 5 × 5
31
0.8818
100.74
0.0153
0.9556
Noise realizations
6
0.4047
255.04
0.0074
0.2578
LLMMSE 3 × 3
31
0.7961
141.04
0.0103
0.7097
5 × 5
0.7190
194.07
0.0033
0.5009
In each simulation the statistics are calculated over 104 noise realizations or preprocessed realizations of the correlation. The POCHF is generated from LE000. 〈SNRI〉 and 〈Ip〉 are normalized with respect to SNRI and IP, respectively, measured when ρ0 = 31.
Table 4
Correlation Results of the BPOCHF (m = 2) of the Object LE000 Degraded with Speckle SDN (M = 1), and Preprocessed by the LLMMSE Estimator Designed for Multiplicative Noise or the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 Pixelsa
Input Image
Correlation, BPOCHF
〈MSD〉
〈SNRI〉n
SNR
PFA
〈Ip〉n
LE000
Noise realizations
3567.9
0.6365
62.78
0.1982
1.1909
H.T. + J.-S. 3 × 3
1216.0
0.7522
91.24
0.0455
1.0813
LLMMSE 3 × 3
604.7
0.6361
124.71
0.0231
0.5744
In each simulation the statistics are calculated over 104 noise realizations or preprocessed realizations of the input image and the correlation. 〈MSD〉 is calculated over noise realizations of LE000 of doubled dimensions 128 × 128 pixels. 〈SNRI〉 and 〈Ip〉 are normalized with respect to SNRI and IP, respectively (SNRI = 26.27 and Ip = 0.140193 × 106 units of intensity).
Table 5
Correlation Results of the POCHF (m = 2) of the X-29 Aircraft Degraded with Speckle SDN (M = 1), and Preprocessed by the LLMMSE Estimator Designed for Multiplicative Noise or the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 or 5 × 5 Pixelsa
Input Image
Correlation, POCHF
〈MSD〉
〈SNRI〉n
SNR
PFA
〈 Ip〉n
X29-000
Noise realizations
1293.5
0.5610
72.90
0.1631
1.0678
H.T. + J.-S. 3 × 3
334.1
0.6096
131.76
0.0176
0.8449
5 × 5
221.8
0.6240
148.23
0.0030
0.6412
LLMMSE 3 × 3
189.0
0.6272
176.57
0.0019
0.6524
5 × 5
139.9
0.5830
231.26
0.0005
0.5020
X29-090
Noise realizations
1318.3
0.5524
77.24
0.1654
1.0645
H.T. + J.-S. 3 × 3
340.1
0.6060
136.74
0.0187
0.8437
LLMMSE 3 × 3
193.7
0.6232
178.09
0.0029
0.6537
In each simulation the statistics are calculated over 104 noise realizations or preprocessed realizations of the input image and of the correlation. The POCHF is generated from X29-000. 〈MSD〉 is calculated over noise realizations of dimensions 128 × 128 pixels. 〈SNRI〉 and 〈 Ip〉 are normalized with respect to SNRI and Ip, respectively (SNRI = 39.57 and Ip = 0.106421 × 106 both for X29-000 and X29-090).
Table 6
Correlation Results of the POCHF (m = 2) of the Object LE000 Degraded with Film-Grain SDN p = 0.5,
for Different Values of the Radial Cutoff Frequency ρ0 of the Filter, and Preprocessed (without a Reduction of ρ0) by the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 Pixelsa
Correlation, POCHF
Input Image
ρ0
〈SNRI〉n
SNR
PFA
〈Ip〉n
LE000
Noise realizations
31
0.7403
72.22
0.1217
1.1113
12
0.5708
166.48
0.0142
0.4860
6
0.4613
366.41
0.0102
0.2754
H.T. + J.-S., 3 × 3
31
0.6380
212.29
0.0013
0.4727
In each simulation the statistics are calculated over 104 noise realizations or preprocessed realizations of the correlation. 〈SNRI〉 and 〈 = Ip〉 are normalized with respect to SNRI and IP, respectively, measured when ρ0 = 31.
Tables (6)
Table 1
Correlation Results for Three of the Objects in Fig. 1 without Noise and with Speckle SDN (M = 1) for the CHF, the POCHF, and the BPOCHF (m = 2)a
Without Noise
With Speckle, M = 1
Filters
Ip
Coordinates
SNRI
〈SNRI〉
SNR
PFA
LE000
CHF
1.00
(33, 26)
9.49
9.35
510.7
0.0000
POCHF
8.27
(33, 26)
24.14
17.79
67.26
0.1494
BPOCHF
2.64
(97, 90)
26.27
16.72
62.78
0.1982
LE045
CHF
1.00
(32, 32)
10.09
9.96
500.0
0.0000
POCHF
6.15
(32, 32)
21.92
15.83
67.13
0.1653
BPOCHF
2.40
(96, 96)
21.43
15.85
64.62
0.1781
X29-000
CHF
1.00
(62, 48)
6.95
6.92
582.0
0.0000
POCHF
17.40
(62, 48)
39.57
22.20
72.90
0.1631
In each simulation the statistics are calculated over 104 noise realizations of the correlation. In the case of LE000 and LE045 the filters are generated from LE000. For the CHF, in units of intensity, Ip = 0.53455 × 105 for LE000, Ip = 0.53284 × 105 for LE045, and Ip = 0.61162 × 104 for X29-000. The coordinates are those of the correlation peak (or of the proper center) for m = 2.
Table 2
Correlation Results of the POCHF (m = 2) of the Letter E Degraded with Speckle SDN (N = 1), and Preprocessed with the LLMMSE Estimator Designed for Multiplicative Noise or the Homomorphic Processing (H.T. + J.–S.) with a Window of Dimensions 3 × 3 or 5 × 5 Pixelsa
Input Image
Correlation, POCHF
〈MSD〉
〈 SNRI〉n
SNR
PFA
〈Ip〉n
LE000
Noise realizations
14435.8
0.7370
67.26
0.1494
1.1130
H.T. + J.-S. 3 × 3
4876.6
0.8144
102.27
0.0184
1.0475
5 × 5
3847.2
0.8252
101.32
0.0138
0.8388
LLMMSE 3 × 3
2423.8
0.6744
154.87
0.0091
0.5583
5 × 5
1842.5
0.6210
206.79
0.0099
0.4057
LE045
Noise realizations
14831.9
0.7222
67.13
0.1653
1.2518
H.T. + J.-S. 3 × 3
5020.9
0.8517
98.00
0.0286
1.1590
5 × 5
3934.8
0.8818
100.74
0.0153
0.9556
LLMMSE 3 × 3
2316.3
0.7961
141.04
0.0103
0.7097
5 × 5
1713.4
0.7190
194.07
0.0033
0.5009
In each simulation the statistics are calculated over 104 noise realizations or preprocessed realizations of the input image and the correlation. The POCHF is generated from LE000. 〈SNRI〉 and 〈Ip〉 are normalized with respect to SNRI and Ip, respectively (SNRI = 24.14, Ip = 0.442189 × 106 for LE000, and SNRI = 21.92, Ip = 0.327574 × 106 for LE045).
Table 3
Correlation Results of the POCHF (m = 2) of the Object LE045 Degraded with Speckle SDN (M = 1) for Different Values of the Radial Cutoff Frequency ρ0 of the Filter, and Preprocessed (without a Reduction of ρ0) by the LLMMSE Estimator Designed for Multiplicative Noise or the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 or 5 × 5 Pixelsa
Correlation, POCHF
Input Image
ρ0
〈SNRI〉n
SNR
PFA
〈Ip〉n
LE045
Noise realizations
31
0.7222
67.13
0.1653
1.2518
8
0.5406
199.16
0.0224
0.4180
H.T. + J.-S., 5 × 5
31
0.8818
100.74
0.0153
0.9556
Noise realizations
6
0.4047
255.04
0.0074
0.2578
LLMMSE 3 × 3
31
0.7961
141.04
0.0103
0.7097
5 × 5
0.7190
194.07
0.0033
0.5009
In each simulation the statistics are calculated over 104 noise realizations or preprocessed realizations of the correlation. The POCHF is generated from LE000. 〈SNRI〉 and 〈Ip〉 are normalized with respect to SNRI and IP, respectively, measured when ρ0 = 31.
Table 4
Correlation Results of the BPOCHF (m = 2) of the Object LE000 Degraded with Speckle SDN (M = 1), and Preprocessed by the LLMMSE Estimator Designed for Multiplicative Noise or the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 Pixelsa
Input Image
Correlation, BPOCHF
〈MSD〉
〈SNRI〉n
SNR
PFA
〈Ip〉n
LE000
Noise realizations
3567.9
0.6365
62.78
0.1982
1.1909
H.T. + J.-S. 3 × 3
1216.0
0.7522
91.24
0.0455
1.0813
LLMMSE 3 × 3
604.7
0.6361
124.71
0.0231
0.5744
In each simulation the statistics are calculated over 104 noise realizations or preprocessed realizations of the input image and the correlation. 〈MSD〉 is calculated over noise realizations of LE000 of doubled dimensions 128 × 128 pixels. 〈SNRI〉 and 〈Ip〉 are normalized with respect to SNRI and IP, respectively (SNRI = 26.27 and Ip = 0.140193 × 106 units of intensity).
Table 5
Correlation Results of the POCHF (m = 2) of the X-29 Aircraft Degraded with Speckle SDN (M = 1), and Preprocessed by the LLMMSE Estimator Designed for Multiplicative Noise or the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 or 5 × 5 Pixelsa
Input Image
Correlation, POCHF
〈MSD〉
〈SNRI〉n
SNR
PFA
〈 Ip〉n
X29-000
Noise realizations
1293.5
0.5610
72.90
0.1631
1.0678
H.T. + J.-S. 3 × 3
334.1
0.6096
131.76
0.0176
0.8449
5 × 5
221.8
0.6240
148.23
0.0030
0.6412
LLMMSE 3 × 3
189.0
0.6272
176.57
0.0019
0.6524
5 × 5
139.9
0.5830
231.26
0.0005
0.5020
X29-090
Noise realizations
1318.3
0.5524
77.24
0.1654
1.0645
H.T. + J.-S. 3 × 3
340.1
0.6060
136.74
0.0187
0.8437
LLMMSE 3 × 3
193.7
0.6232
178.09
0.0029
0.6537
In each simulation the statistics are calculated over 104 noise realizations or preprocessed realizations of the input image and of the correlation. The POCHF is generated from X29-000. 〈MSD〉 is calculated over noise realizations of dimensions 128 × 128 pixels. 〈SNRI〉 and 〈 Ip〉 are normalized with respect to SNRI and Ip, respectively (SNRI = 39.57 and Ip = 0.106421 × 106 both for X29-000 and X29-090).
Table 6
Correlation Results of the POCHF (m = 2) of the Object LE000 Degraded with Film-Grain SDN p = 0.5,
for Different Values of the Radial Cutoff Frequency ρ0 of the Filter, and Preprocessed (without a Reduction of ρ0) by the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 Pixelsa
Correlation, POCHF
Input Image
ρ0
〈SNRI〉n
SNR
PFA
〈Ip〉n
LE000
Noise realizations
31
0.7403
72.22
0.1217
1.1113
12
0.5708
166.48
0.0142
0.4860
6
0.4613
366.41
0.0102
0.2754
H.T. + J.-S., 3 × 3
31
0.6380
212.29
0.0013
0.4727
In each simulation the statistics are calculated over 104 noise realizations or preprocessed realizations of the correlation. 〈SNRI〉 and 〈 = Ip〉 are normalized with respect to SNRI and IP, respectively, measured when ρ0 = 31.