Minimum-mean-square-error filters for detecting a noisy target in background noise

Bahram Javidi, Farokh Parchekani, and Guanshen Zhang

Bahram Javidi,^{1} Farokh Parchekani,^{1,}^{2} and Guanshen Zhang^{1}

^{1}When this work was performed, the authors were with the Department of Electrical and Systems Engineering, University of Connecticut, Storrs, Connecticut 06269-3157.

^{2}F. 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.

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Correlation Results for the Minimum-Mean-Square-Error Filter Designed by the Use of Different Detector-Noise Standard Deviationsa

Detector-Noise Standard Deviation σ_{a}

Peak-to-Correlation Intensity PCI

Correlation-Peak Intensity I_{p}

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 m_{n} = 0.5, a standard deviation of σ_{n} = 0.1, and a bandwidth of B_{n} = 20 pixels. The detector noise is white, with a zero mean and a standard deviation of σ_{a} = 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 σ_{r}

Peak-to-Correlation Intensity PCI

Correlation-Peak Intensity I_{p}

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 m_{n} = 0.5, a standard deviation of σ_{n} = 0.15, and a bandwidth of B_{n} = 30 pixels. The target noise is white, with a zero mean and a standard deviation of σ_{r} = 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 m_{n} = 0.57 and a standard deviation of σ_{n} = 0.12. The detector noise is white, with a zero mean and a standard deviation of σ_{a} = 0.05. The target noise over the rotated and scaled car images is white, with a zero mean and a standard deviation of σ_{r} = 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 m_{n} = 0.45 and a standard deviation of σ_{n} = 0.135. The detector noise is white, with a zero mean and a standard deviation of σ_{a} = 0.3. The target noise overlapping target 2 has a zero mean and a standard deviation of σ_{r} = 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 σ_{a}

Peak-to-Correlation Intensity PCI

Correlation-Peak Intensity I_{p}

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 m_{n} = 0.5, a standard deviation of σ_{n} = 0.1, and a bandwidth of B_{n} = 20 pixels. The detector noise is white, with a zero mean and a standard deviation of σ_{a} = 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 σ_{r}

Peak-to-Correlation Intensity PCI

Correlation-Peak Intensity I_{p}

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 m_{n} = 0.5, a standard deviation of σ_{n} = 0.15, and a bandwidth of B_{n} = 30 pixels. The target noise is white, with a zero mean and a standard deviation of σ_{r} = 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 m_{n} = 0.57 and a standard deviation of σ_{n} = 0.12. The detector noise is white, with a zero mean and a standard deviation of σ_{a} = 0.05. The target noise over the rotated and scaled car images is white, with a zero mean and a standard deviation of σ_{r} = 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 m_{n} = 0.45 and a standard deviation of σ_{n} = 0.135. The detector noise is white, with a zero mean and a standard deviation of σ_{a} = 0.3. The target noise overlapping target 2 has a zero mean and a standard deviation of σ_{r} = 0.1.