Bahram Javidi,1
Farokh Parchekani,1,2
and Guanshen Zhang1
1When this work was performed, the authors were with the Department of Electrical and Systems Engineering, University of Connecticut, Storrs, Connecticut 06269-3157.
2F. Parchekani is now with U.S. Robotics Access Corporation, Massachusetts Research and Development Laboratory, 4 Technology Drive, West-borough, Massachusetts 01581.
A minimum-mean-square-error filter is proposed to detect a noisy target in spatially nonoverlapping background noise. In this model, both the background noise that is spatially nonoverlapping with the target and the noise that is additive to the target and the input image are considered. The criterion used to design the filter is to minimize the mean-square-error between the filter output and a delta function located at the target position in the presence of the noise. Computer-simulation results for a number of noisy input images are presented, and the performance of the filter is determined. We also test the filter discrimination against undesired objects and tolerance to target distortions, such as rotation and scaling.
You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
Correlation Results for the Minimum-Mean-Square-Error Filter Designed by the Use of Different Detector-Noise Standard Deviationsa
Detector-Noise Standard Deviation σa
Peak-to-Correlation Intensity PCI
Correlation-Peak Intensity Ip
Total Squared Error e
0.05
182.0
0.8683
1.041
0.10
243.5
0.4942
1.021
0.15
277.8
0.3554
0.9981
0.20
294.1
0.3215
0.9955
0.25
296.2
0.3227
0.9957
0.30
287.5
0.3420
0.9963
0.35
271.9
0.3711
0.9968
The background noise is color noise, with a mean of mn = 0.5, a standard deviation of σn = 0.1, and a bandwidth of Bn = 20 pixels. The detector noise is white, with a zero mean and a standard deviation of σa = 0.2. For each combination of the background-noise parameters used in the filter design 30 statistical trials are conducted and the output results averaged.
Table 2
Correlation Results for the Minimum-Mean-Square-Error Filter Designed by Use of Different Target-Noise Standard Deviationsa
Target-Noise Standard Deviation σr
Peak-to-Correlation Intensity PCI
Correlation-Peak Intensity Ip
Total Squared Error e
Peak-to-Sidelobe Ratio PSR
0.1
460.5
0.1246
1.3969
2.7558
0.2
548.9
0.1110
1.0425
3.0839
0.3
565.53
0.0788
0.9986
3.6560
0.35
581.4.
0.0760
0.9942
3.6325
0.4
600.33
0.0727
0.9955
3.6203
0.5
584.6
0.0650
0.9967
3.9657
0.6
511.1
0.0507
0.9979
3.6442
0.7
470.6
0.0489
0.9985
3.5720
The background noise is color, with a mean of mn = 0.5, a standard deviation of σn = 0.15, and a bandwidth of Bn = 30 pixels. The target noise is white, with a zero mean and a standard deviation of σr = 0.35. For each combination of the background-noise parameters used in the filter design 30 statistical trials are conducted and the output results averaged.
Table 3
Correlation Results for the Minimum-Mean-Square-Error (MMSE) Filter and the Minimum-Mean-Square-Error Filter Designed without Additive and Target Noise for the Tolerance Testa
The background noise is a real scene with a mean of mn = 0.57 and a standard deviation of σn = 0.12. The detector noise is white, with a zero mean and a standard deviation of σa = 0.05. The target noise over the rotated and scaled car images is white, with a zero mean and a standard deviation of σr = 0.05.
Table 4
Correlation Results for the Minimum-Mean-Square-Error (MMSE) Filter and the Minimum-Mean-Square-Error Filter Designed without Additive and Target Noise for the Discrimination Testa
The background noise is a real scene of clouds with a mean of mn = 0.45 and a standard deviation of σn = 0.135. The detector noise is white, with a zero mean and a standard deviation of σa = 0.3. The target noise overlapping target 2 has a zero mean and a standard deviation of σr = 0.1.
Tables (4)
Table 1
Correlation Results for the Minimum-Mean-Square-Error Filter Designed by the Use of Different Detector-Noise Standard Deviationsa
Detector-Noise Standard Deviation σa
Peak-to-Correlation Intensity PCI
Correlation-Peak Intensity Ip
Total Squared Error e
0.05
182.0
0.8683
1.041
0.10
243.5
0.4942
1.021
0.15
277.8
0.3554
0.9981
0.20
294.1
0.3215
0.9955
0.25
296.2
0.3227
0.9957
0.30
287.5
0.3420
0.9963
0.35
271.9
0.3711
0.9968
The background noise is color noise, with a mean of mn = 0.5, a standard deviation of σn = 0.1, and a bandwidth of Bn = 20 pixels. The detector noise is white, with a zero mean and a standard deviation of σa = 0.2. For each combination of the background-noise parameters used in the filter design 30 statistical trials are conducted and the output results averaged.
Table 2
Correlation Results for the Minimum-Mean-Square-Error Filter Designed by Use of Different Target-Noise Standard Deviationsa
Target-Noise Standard Deviation σr
Peak-to-Correlation Intensity PCI
Correlation-Peak Intensity Ip
Total Squared Error e
Peak-to-Sidelobe Ratio PSR
0.1
460.5
0.1246
1.3969
2.7558
0.2
548.9
0.1110
1.0425
3.0839
0.3
565.53
0.0788
0.9986
3.6560
0.35
581.4.
0.0760
0.9942
3.6325
0.4
600.33
0.0727
0.9955
3.6203
0.5
584.6
0.0650
0.9967
3.9657
0.6
511.1
0.0507
0.9979
3.6442
0.7
470.6
0.0489
0.9985
3.5720
The background noise is color, with a mean of mn = 0.5, a standard deviation of σn = 0.15, and a bandwidth of Bn = 30 pixels. The target noise is white, with a zero mean and a standard deviation of σr = 0.35. For each combination of the background-noise parameters used in the filter design 30 statistical trials are conducted and the output results averaged.
Table 3
Correlation Results for the Minimum-Mean-Square-Error (MMSE) Filter and the Minimum-Mean-Square-Error Filter Designed without Additive and Target Noise for the Tolerance Testa
The background noise is a real scene with a mean of mn = 0.57 and a standard deviation of σn = 0.12. The detector noise is white, with a zero mean and a standard deviation of σa = 0.05. The target noise over the rotated and scaled car images is white, with a zero mean and a standard deviation of σr = 0.05.
Table 4
Correlation Results for the Minimum-Mean-Square-Error (MMSE) Filter and the Minimum-Mean-Square-Error Filter Designed without Additive and Target Noise for the Discrimination Testa
The background noise is a real scene of clouds with a mean of mn = 0.45 and a standard deviation of σn = 0.135. The detector noise is white, with a zero mean and a standard deviation of σa = 0.3. The target noise overlapping target 2 has a zero mean and a standard deviation of σr = 0.1.