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.

For a review of some of these filters, please see D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

For a review of some of these filters, please see D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

For a review of some of these filters, please see D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

For a review of some of these filters, please see D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

For a review of some of these filters, please see D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Input image and correlation outputs for the minimum-mean-square-error filter and the minimum-mean-square-error filter designed with no additive noise in the test of detector noise and color background noise: (a) Input image with color background noise (mean, m_{n} = 0.5; standard deviation, σ_{n} = 0.1; bandwidth, B_{n} = 20 pixels) and white detector noise (zero mean; standard deviation, σ_{a} = 0.2). (b) Three-dimensional (3-D) plot of the correlation output for the minimum-mean-square-error filter for the input image shown in (a). (c) 3-D plot of the correlation output for the minimum-mean-square-error filter designed with no additive noise for the input image shown in (a).

Input image and correlation outputs for the minimum-mean-square-error filter and the minimum-mean-square-error filter designed with no target noise in the test of the target noise and color background noise: (a) Input image with a color background noise (mean, m_{n} = 0.5; standard deviation, σ_{n} = 0.15; bandwidth B_{n} = 35 pixels) and white target noise (zero mean; standard deviation, σ_{r} = 0.35), with no detector noise. (b) A 3-D plot of the correlation output for the minimum-mean-square-error filter for the input image shown in (a). (c) A 3-D plot of the correlation output for the minimum-mean-square-error filter designed with no target noise for the input image shown in (a).

Input image and correlation outputs for the minimum-mean-square-error filter and the same filter but designed with no additive or target noise for the test of distortion tolerance: (a) Input images with real background noise (mean, m_{n} = 0.57; standard deviation, σ_{n} = 0.12) and white detector noise (zero mean; standard deviation, σ_{a} = 0.05). Clockwise (from upper left) the image comprises a car, a vehicle (carrier), a helicopter, an ambulance, a car image that is scaled up by 10%, and a car image that is rotated by 4°. In addition to scaling and rotation of the target, a zero-mean white target noise with a standard deviation of σ_{r} = 0.05 overlaps with the last two cars. The target to be identified comprises the car images, and the other objects are to be rejected. (b) Image-identification diagram for (a). (c) 3-D plot of the correlation output for the minimum-mean-square-error filter for the input image shown in (a). (d) 3-D plot of the correlation output for the same filter but designed with no additive or target noise for the input image shown in (a).

Input image and correlation outputs for the minimum-mean-square-error filter and the same filter but designed with no additive or target noise for the discrimination test. (a) Input image with real background-scene noise composed of clouds (mean, m_{n} = 0.45; standard deviation, σ_{n} = 0.135) and white, zero-mean detector noise (standard, deviation, σ_{a} = 0.3), containing (clockwise from upper left image) a lear jet, an Embraer airplane, a Falcon airplane, a Canadian Airline’s jet, and another Falcon airplane. A zero-mean white target noise with a standard deviation of σ_{r} = 0.1 overlaps the second Falcon airplane. The target to be identified is the Falcon airplane, and the other objects are to be rejected. (b) Image-identification diagram for (a). (c) 3-D plot of the correlation output for the minimum-mean-square-error filter for the input image shown in (a). (d) 3-D plot of the correlation output for the same filter but designed with no additive or target noise for the input image shown in (a).

Block diagram of the minimum-mean-square-error filter. The impulse response of the filter h(· · ·) is optimized to minimize the mean square error between y_{d}(· · ·) and y_{a}(· · ·), where y_{d}(· · ·) is the desired response and y_{a}(· · ·) the actual response for the input x(i). E(· · ·) is the error at each pixel.

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

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

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.