Image preprocessing for rotation-invariant pattern recognition in the presence of signal-dependent noise

Jean-Christophe Terrillon

Jean-Christophe Terrillon

^{}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.

J. F. Walkup, R. C. Choens, “Image processing in signal-dependent noise,” Opt. Eng. 13, 258–266 (1974).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

H. Siedentopf, “Concerning granularity, density fluctuations and the enlargement of photographic negatives,” Phys. Z. 38, 454 (1937).

C. M. Lo, “Estimation of image signals with Poisson noise,” Rep. 890 (Image Processing Institute, University of California, Los Angeles, Calif., 1979).

E. I. Shubnikov, “Effect of additive and multiplicative noise in the correlation comparison of images,” Opt. Spectrosc. (USSR) 62, 389–392 (1987).

G. M. Morris, “Pattern recognition using photon-limited images,” in Optical processing and Computing, H. H. Arse-nault, T. Szoplik, B. Macukow, eds. (Academic, Boston, Mass., 1989).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-7, 165–177 (1985).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 373–382 (1987).

A. K. Jain, C. R. Christensen, “Digital processing of images in speckle noise,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 243, 46–50 (1980).

H. H. Arsenault, M. Denis, “Image processing in signal-dependent noise,” Can. J. Phys. 61, 309–317 (1983).

Research and Education Association, Handbook of Mathematical, Scientific, and Engineering Formulas, Tables, Functions, Graphs, Transforms, (Research and Education Association, N.J., 1994), Chap. 16, p. 595.

M. Denis, H. H. Arsenault, “On the accuracy of a method to make film-grain noise independent of the signal,” Opt. Commun. 38, 166–169 (1981).

R. Wallis, “An approach to the space variant restoration and enhancement of images,” in Proceedings of the Symposium on Current Mathematical Problems in Image Science (Naval Postgraduate School, Monterey, Calif., 1976).

Y. Sheng, H. H. Arsenault, “Method for determining expansion centers and predicting sidelobe levels for circular harmonic filters,” J. Opt. Soc. Am. A 1, 1793–1797 (1987).

E. I. Shubnikov, “Effect of additive and multiplicative noise in the correlation comparison of images,” Opt. Spectrosc. (USSR) 62, 389–392 (1987).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 373–382 (1987).

Y. Sheng, H. H. Arsenault, “Method for determining expansion centers and predicting sidelobe levels for circular harmonic filters,” J. Opt. Soc. Am. A 1, 1793–1797 (1987).

Y. Sheng, H. H. Arsenault, “Method for determining expansion centers and predicting sidelobe levels for circular harmonic filters,” J. Opt. Soc. Am. A 1, 1793–1797 (1987).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 373–382 (1987).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-7, 165–177 (1985).

J. F. Walkup, R. C. Choens, “Image processing in signal-dependent noise,” Opt. Eng. 13, 258–266 (1974).

Christensen, C. R.

A. K. Jain, C. R. Christensen, “Digital processing of images in speckle noise,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 243, 46–50 (1980).

Denis, M.

H. H. Arsenault, M. Denis, “Image processing in signal-dependent noise,” Can. J. Phys. 61, 309–317 (1983).

M. Denis, H. H. Arsenault, “On the accuracy of a method to make film-grain noise independent of the signal,” Opt. Commun. 38, 166–169 (1981).

A. K. Jain, C. R. Christensen, “Digital processing of images in speckle noise,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 243, 46–50 (1980).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 373–382 (1987).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-7, 165–177 (1985).

C. M. Lo, “Estimation of image signals with Poisson noise,” Rep. 890 (Image Processing Institute, University of California, Los Angeles, Calif., 1979).

G. M. Morris, “Pattern recognition using photon-limited images,” in Optical processing and Computing, H. H. Arse-nault, T. Szoplik, B. Macukow, eds. (Academic, Boston, Mass., 1989).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 373–382 (1987).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-7, 165–177 (1985).

Y. Sheng, H. H. Arsenault, “Method for determining expansion centers and predicting sidelobe levels for circular harmonic filters,” J. Opt. Soc. Am. A 1, 1793–1797 (1987).

Shubnikov, E. I.

E. I. Shubnikov, “Effect of additive and multiplicative noise in the correlation comparison of images,” Opt. Spectrosc. (USSR) 62, 389–392 (1987).

Siedentopf, H.

H. Siedentopf, “Concerning granularity, density fluctuations and the enlargement of photographic negatives,” Phys. Z. 38, 454 (1937).

Strand, T. C.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 373–382 (1987).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-7, 165–177 (1985).

J. F. Walkup, R. C. Choens, “Image processing in signal-dependent noise,” Opt. Eng. 13, 258–266 (1974).

Wallis, R.

R. Wallis, “An approach to the space variant restoration and enhancement of images,” in Proceedings of the Symposium on Current Mathematical Problems in Image Science (Naval Postgraduate School, Monterey, Calif., 1976).

H. H. Arsenault, M. Denis, “Image processing in signal-dependent noise,” Can. J. Phys. 61, 309–317 (1983).

IEEE Trans. Acoust. Speech Signal Process.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive restoration of images with speckle,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 373–382 (1987).

IEEE Trans. Pattern Anal. Mach. Intell.

J. S. Lee, “Digital image enhancement and noise filtering by use of local statistics,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 165–168 (1980).

Y. Sheng, H. H. Arsenault, “Method for determining expansion centers and predicting sidelobe levels for circular harmonic filters,” J. Opt. Soc. Am. A 1, 1793–1797 (1987).

E. I. Shubnikov, “Effect of additive and multiplicative noise in the correlation comparison of images,” Opt. Spectrosc. (USSR) 62, 389–392 (1987).

Phys. Z.

H. Siedentopf, “Concerning granularity, density fluctuations and the enlargement of photographic negatives,” Phys. Z. 38, 454 (1937).

Other

C. M. Lo, “Estimation of image signals with Poisson noise,” Rep. 890 (Image Processing Institute, University of California, Los Angeles, Calif., 1979).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

G. M. Morris, “Pattern recognition using photon-limited images,” in Optical processing and Computing, H. H. Arse-nault, T. Szoplik, B. Macukow, eds. (Academic, Boston, Mass., 1989).

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive noise smoothing filter for images with signal-dependent noise,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-7, 165–177 (1985).

A. K. Jain, C. R. Christensen, “Digital processing of images in speckle noise,” in Applications of Speckle Phenomena, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 243, 46–50 (1980).

R. Wallis, “An approach to the space variant restoration and enhancement of images,” in Proceedings of the Symposium on Current Mathematical Problems in Image Science (Naval Postgraduate School, Monterey, Calif., 1976).

Research and Education Association, Handbook of Mathematical, Scientific, and Engineering Formulas, Tables, Functions, Graphs, Transforms, (Research and Education Association, N.J., 1994), Chap. 16, p. 595.

Input images of the letter E of dimensions 64 × 64 pixels: LE000 (upper left) and LE045 (upper right). Input images of an X-29 aircraft of dimensions 128 × 128 pixels: X29-000 (lower left) and X29-090 (lower right).

Detection of LE045 with the POCHF (generated from LE000, m = 2): (a) the original input image and the corresponding correlation (three-dimensional view and top view); (b) particular noise realization of the input image degraded with speckle SDN (M = 1) and the corresponding noisy correlation showing one false alarm; (c) the input image restored by the homomorphic processing with a window of dimensions 5 × 5 pixels and the resulting correlation. The correlation-peak intensity is normalized in each case.

Detection of the aircraft X29-000 with the POCHF (m = 2): (a) original input image and corresponding correlation (three-dimensional view and top view); (b) particular noise realization of the input image degraded with speckle SDN (M = 1) and the corresponding noisy correlation showing two false alarms; (c) the input image restored with the LLMMSE estimator designed for multiplicative noise with a processing window of dimensions 5 × 5 pixels and the resulting correlation. The correlation-peak intensity is normalized in each case.

Correlation results as a function of the parameter M of the POCHF (m = 2) of the aircraft X29-000 degraded with speckle SDN and preprocessed by the LLMMSE estimator with a window of dimensions 3 × 3 pixels and the homomorphic processing with a window of dimensions 5 × 5 pixels. Each data point is generated from 10^{4} noise realizations or preprocessed realizations of the correlation. (a) Correlation-peak SNR and (b) false-alarm probability P_{FA}. SNR_{in} is the SNR measured in the input image.

Correlation results for the object LE000 degraded with film-grain SDN (p = 0.5) with the CHF, the POCHF, and the BPOCHF (m = 2). Each data point is generated from 10^{4} noise realizations of the correlation. (a) Correlation-peak SNR versus the square of the SNR in the input image and (b) false-alarm probability P_{FA} versus the SNR in the input image. Here,
${\text{SNR}}_{\text{in}}=\sqrt{{S}_{0}}/k$[please see Eq. (6)#x0005D; with S_{0} = 255.

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

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

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 ρ_{0} of the Filter, and Preprocessed (without a Reduction of ρ_{0}) 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

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

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

Table 6 Correlation Results of the POCHF (m = 2) of the Object LE000 Degraded with Film-Grain SDN p = 0.5,
${\text{SNR}}_{\text{in}}=\sqrt{{S}_{0}}/k=1.0$ for Different Values of the Radial Cutoff Frequency ρ_{0} of the Filter, and Preprocessed (without a Reduction of ρ_{0}) by the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 Pixelsa

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

I_{p}

Coordinates

SNRI

〈SNRI〉

SNR

P_{FA}

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 10^{4} 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, I_{p} = 0.53455 × 10^{5} for LE000, I_{p} = 0.53284 × 10^{5} for LE045, and I_{p} = 0.61162 × 10^{4} 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

P_{FA}

〈I_{p}〉_{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 10^{4} noise realizations or preprocessed realizations of the input image and the correlation. The POCHF is generated from LE000. 〈SNRI〉 and 〈I_{p}〉 are normalized with respect to SNRI and I_{p}, respectively (SNRI = 24.14, I_{p} = 0.442189 × 10^{6} for LE000, and SNRI = 21.92, I_{p} = 0.327574 × 10^{6} 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 ρ_{0} of the Filter, and Preprocessed (without a Reduction of ρ_{0}) 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

ρ_{0}

〈SNRI〉_{n}

SNR

P_{FA}

〈I_{p}〉_{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 10^{4} noise realizations or preprocessed realizations of the correlation. The POCHF is generated from LE000. 〈SNRI〉 and 〈I_{p}〉 are normalized with respect to SNRI and I_{P}, respectively, measured when ρ_{0} = 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

P_{FA}

〈I_{p}〉_{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 10^{4} 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 〈I_{p}〉 are normalized with respect to SNRI and I_{P}, respectively (SNRI = 26.27 and I_{p} = 0.140193 × 10^{6} 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

P_{FA}

〈 I_{p}〉_{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 10^{4} 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 〈 I_{p}〉 are normalized with respect to SNRI and I_{p}, respectively (SNRI = 39.57 and I_{p} = 0.106421 × 10^{6} 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,
${\text{SNR}}_{\text{in}}=\sqrt{{S}_{0}}/k=1.0$ for Different Values of the Radial Cutoff Frequency ρ_{0} of the Filter, and Preprocessed (without a Reduction of ρ_{0}) by the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 Pixelsa

Correlation, POCHF

Input Image

ρ_{0}

〈SNRI〉_{n}

SNR

P_{FA}

〈I_{p}〉_{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 10^{4} noise realizations or preprocessed realizations of the correlation. 〈SNRI〉 and 〈 = I_{p}〉 are normalized with respect to SNRI and I_{P}, respectively, measured when ρ_{0} = 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

I_{p}

Coordinates

SNRI

〈SNRI〉

SNR

P_{FA}

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 10^{4} 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, I_{p} = 0.53455 × 10^{5} for LE000, I_{p} = 0.53284 × 10^{5} for LE045, and I_{p} = 0.61162 × 10^{4} 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

P_{FA}

〈I_{p}〉_{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 10^{4} noise realizations or preprocessed realizations of the input image and the correlation. The POCHF is generated from LE000. 〈SNRI〉 and 〈I_{p}〉 are normalized with respect to SNRI and I_{p}, respectively (SNRI = 24.14, I_{p} = 0.442189 × 10^{6} for LE000, and SNRI = 21.92, I_{p} = 0.327574 × 10^{6} 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 ρ_{0} of the Filter, and Preprocessed (without a Reduction of ρ_{0}) 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

ρ_{0}

〈SNRI〉_{n}

SNR

P_{FA}

〈I_{p}〉_{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 10^{4} noise realizations or preprocessed realizations of the correlation. The POCHF is generated from LE000. 〈SNRI〉 and 〈I_{p}〉 are normalized with respect to SNRI and I_{P}, respectively, measured when ρ_{0} = 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

P_{FA}

〈I_{p}〉_{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 10^{4} 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 〈I_{p}〉 are normalized with respect to SNRI and I_{P}, respectively (SNRI = 26.27 and I_{p} = 0.140193 × 10^{6} 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

P_{FA}

〈 I_{p}〉_{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 10^{4} 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 〈 I_{p}〉 are normalized with respect to SNRI and I_{p}, respectively (SNRI = 39.57 and I_{p} = 0.106421 × 10^{6} 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,
${\text{SNR}}_{\text{in}}=\sqrt{{S}_{0}}/k=1.0$ for Different Values of the Radial Cutoff Frequency ρ_{0} of the Filter, and Preprocessed (without a Reduction of ρ_{0}) by the Homomorphic Processing (H.T. + J.-S.) with a Window of Dimensions 3 × 3 Pixelsa

Correlation, POCHF

Input Image

ρ_{0}

〈SNRI〉_{n}

SNR

P_{FA}

〈I_{p}〉_{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 10^{4} noise realizations or preprocessed realizations of the correlation. 〈SNRI〉 and 〈 = I_{p}〉 are normalized with respect to SNRI and I_{P}, respectively, measured when ρ_{0} = 31.