Colm Lynch and Nicholas Devaney, "Registration for images in the presence of additive and multiplicative fixed-pattern noise," Appl. Opt. 57, 1824-1831 (2018)

Image registration under conditions of fixed-pattern noise is a difficult problem that has not been solved in the literature. While traditional registration methods are adequate for additive random noise, these are not suited to spatially invariant noise that is additive or multiplicative. We present a method for image registration using a difference operation in the frequency domain. Shift values are then computed by dividing by the object Fourier transform and inverse transforming. The method described is valid for both additive and multiplicative noise and determines shifts with sub-pixel accuracy. Additionally, minimal prior knowledge of the corrupting pattern is required. We compare our method with previous registration methods for varying amounts of noise. Results are presented for both simulated images and images recorded from a thermal camera with significant fixed-pattern noise.

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Mean-Absolute-Difference Comparisons of Registration Methods for Various Images with No FPN^{a}

Image

${\mathrm{Prop}}_{x,y}$

${\mathrm{PAC}}_{x,y}$

${\mathrm{DOCP}}_{x,y}$

${\mathrm{PC}}_{x,y}$

Barb

0.049, 0.047

0.183, 0.971

4.096, 4.543

0.256, 0.159

Mand

0.044, 0.053

0.091, 3.418

3.820, 4.276

0.180, 0.212

Cam

0.052, 0.053

2.972, 0.516

3.843, 4.222

0.179, 0.176

USAF

0.040, 0.044

0.060, 0.047

4.048, 4.373

0.219, 0.283

MAD values are listed for each method for $x$ and $y$ shifts, respectively. The Barbara, Mandrill, Cameraman, and USAF test images are used. Registration methods tested are the proposed, PAC, DOCP, and phase-correlation methods. The maximum MAD values measured for the proposed method were Barb = 0.757, Mandrill = 0.759, Cameraman = 0.754, and USAF = 0.760.

Table 2.

Standard Deviation of Absolute Differences for Various Images with No FPN^{a}

Image

${\mathrm{Prop}}_{x,y}$

${\mathrm{PAC}}_{x,y}$

${\mathrm{DOCP}}_{x,y}$

${\mathrm{PC}}_{x,y}$

Barb

0.039, 0.049

0.140, 0.292

5.127, 6.645

0.554, 0.184

Mand

0.038, 0.050

0.053, 1.967

4.947, 8.504

0.212, 0.222

Cam

0.044, 0.052

2.304, 0.155

4.734, 6.322

0.216, 0.219

USAF

0.035, 0.049

0.046, 0.022

4.804, 5.754

0.998, 2.166

Standard deviation values are listed for each method for $x$ and $y$ shifts, respectively. The associated MAD values are shown in Table 1.

Table 3.

Registration Results for Images with Additive Simulated Gaussian Term Using the Barbara Test Image^{a}

FWHM

${\mathrm{Prop}}_{x}$

${\mathrm{Prop}}_{y}$

${\mathrm{PAC}}_{x}$

${\mathrm{PAC}}_{y}$

${\mathrm{DOCP}}_{x}$

${\mathrm{DOCP}}_{y}$

25

0.0495

0.0491

0.693

0.550

4.285

4.993

50

0.0499

0.0493

0.716

0.617

4.282

4.972

75

0.0501

0.0494

0.725

0.636

4.285

4.977

100

0.0502

0.0495

0.717

0.693

4.283

4.978

MAD measured for increasing FWHM. The maximum amplitude of the Gaussian terms was fixed at 100, which corresponds to a peak-to-signal value of 0.922 at the image center. Higher precision is used for the proposed method to show an increasing MAD. The maximum MAD values recorded for increasing values of FWHM were 0.780, 0.799, 0.778, 0.778.

Table 4.

Standard Deviation of Absolute Differences for Images with Additive Gaussian FPN^{a}

FWHM

${\mathrm{Prop}}_{x}$

${\mathrm{Prop}}_{y}$

${\mathrm{PAC}}_{x}$

${\mathrm{PAC}}_{y}$

${\mathrm{DOCP}}_{x}$

${\mathrm{DOCP}}_{y}$

25

0.0398

0.0501

0.503

0.219

5.209

7.384

50

0.0398

0.0503

0.517

0.224

5.211

7.366

75

0.0398

0.0502

0.528

0.226

5.216

7.385

100

0.0399

0.0504

0.517

0.221

5.215

7.390

Standard deviations here correspond to values presented in Table 3.

Table 5.

Registration Results for Images with Random Gaussian Noise and Additive FPN Watermark^{a}

${\mu}_{n}$

${\sigma}_{n}$

${\mathrm{Prop}}_{x}$

${\mathrm{Prop}}_{y}$

${\mathrm{PAC}}_{x}$

${\mathrm{PAC}}_{y}$

${\mathrm{DOCP}}_{x}$

${\mathrm{DOCP}}_{y}$

4

1

0.051

0.053

3.763

3.695

5.811

5.774

8

2

0.054

0.062

3.763

3.697

6.692

6.550

12

3

0.055

0.068

3.763

3.699

7.344

6.959

16

4

0.061

0.081

3.763

3.701

7.682

7.008

20

5

0.070

0.089

3.764

3.704

8.198

7.545

Watermark amplitude was fixed at 100. Additive random noise is normally distributed with mean ${\mu}_{n}$ and standard deviation ${\sigma}_{n}$. The Barbara image was used for testing. The largest values of MAD recorded for increasing $\mu $ and $\sigma $ were 0.798, 0.804, 0.807, 0.928.

Table 6.

Standard Deviation of Absolute Difference for Images Corrupted by an Additive Watermark with Gaussian Noise^{a}

Registration Results for Simulated Images Corrupted by a Multiplicative Fixed Pattern of Ones and Zeros^{a}

${N}_{\text{frac}}$

${\mathrm{Prop}}_{x,y}$

$\mathrm{Prop}{\sigma}_{x,y}$

${\mathrm{PAC}}_{x,y}$

$\mathrm{PAC}{\sigma}_{x,y}$

0.025

0.043, 0.045

0.041, 0.053

1.611, 0.477

1.152, 0.380

0.050

0.045, 0.045

0.039, 0.041

1.684, 0.686

1.237, 0.648

0.100

0.056, 0.058

0.322, 0.498

1.877, 1.204

1.476, 1.148

0.200

0.073, 0.088

1.402, 0.791

2.228, 2.358

1.733, 1.786

0.300

0.089, 0.124

3.909, 4.524

2.446, 3.206

1.832, 1.959

0.400

0.153, 0.181

3.551, 3.427

2.664, 3.394

1.865, 1.983

MAD values measured for increasing value of ${N}_{\text{frac}}$ [Eq. (30)]. The Barbara image was used for testing. Standard deviations for this set of data are denoted by $\sigma $. For increasing values of ${N}_{\text{frac}}$, the maximum MAD values recorded were 3.107, 0.806, 0.88, 0.934, 5.685.

Table 8.

Registration Results for Simulated Images with Saturated Columns^{a}

${N}_{\text{frac}}$

${\mathrm{Prop}}_{x,y}$

$\mathrm{Prop}{\sigma}_{x,y}$

${\mathrm{PAC}}_{x,y}$

$\mathrm{PAC}{\sigma}_{x,y}$

0.050

0.074, 0.080

0.073, 0.085

3.479, 0.555

1.970, 0.276

0.074

0.081, 0.088

0.080, 0.098

3.484, 0.549

1.969, 0.278

0.099

0.087, 0.100

0.084, 0.117

3.486, 0.546

1.970, 0.280

${N}_{\text{frac}}$ is a measure of saturated columns. The Barbara image was used for testing. The standard deviation of the absolute difference is shown for both the proposed and PAC methods, denoted by $\sigma $. The maximum MAD values recorded were 1.153, 1.664, 2.105 for an increasing value of ${N}_{\text{frac}}$.

Tables (8)

Table 1.

Mean-Absolute-Difference Comparisons of Registration Methods for Various Images with No FPN^{a}

Image

${\mathrm{Prop}}_{x,y}$

${\mathrm{PAC}}_{x,y}$

${\mathrm{DOCP}}_{x,y}$

${\mathrm{PC}}_{x,y}$

Barb

0.049, 0.047

0.183, 0.971

4.096, 4.543

0.256, 0.159

Mand

0.044, 0.053

0.091, 3.418

3.820, 4.276

0.180, 0.212

Cam

0.052, 0.053

2.972, 0.516

3.843, 4.222

0.179, 0.176

USAF

0.040, 0.044

0.060, 0.047

4.048, 4.373

0.219, 0.283

MAD values are listed for each method for $x$ and $y$ shifts, respectively. The Barbara, Mandrill, Cameraman, and USAF test images are used. Registration methods tested are the proposed, PAC, DOCP, and phase-correlation methods. The maximum MAD values measured for the proposed method were Barb = 0.757, Mandrill = 0.759, Cameraman = 0.754, and USAF = 0.760.

Table 2.

Standard Deviation of Absolute Differences for Various Images with No FPN^{a}

Image

${\mathrm{Prop}}_{x,y}$

${\mathrm{PAC}}_{x,y}$

${\mathrm{DOCP}}_{x,y}$

${\mathrm{PC}}_{x,y}$

Barb

0.039, 0.049

0.140, 0.292

5.127, 6.645

0.554, 0.184

Mand

0.038, 0.050

0.053, 1.967

4.947, 8.504

0.212, 0.222

Cam

0.044, 0.052

2.304, 0.155

4.734, 6.322

0.216, 0.219

USAF

0.035, 0.049

0.046, 0.022

4.804, 5.754

0.998, 2.166

Standard deviation values are listed for each method for $x$ and $y$ shifts, respectively. The associated MAD values are shown in Table 1.

Table 3.

Registration Results for Images with Additive Simulated Gaussian Term Using the Barbara Test Image^{a}

FWHM

${\mathrm{Prop}}_{x}$

${\mathrm{Prop}}_{y}$

${\mathrm{PAC}}_{x}$

${\mathrm{PAC}}_{y}$

${\mathrm{DOCP}}_{x}$

${\mathrm{DOCP}}_{y}$

25

0.0495

0.0491

0.693

0.550

4.285

4.993

50

0.0499

0.0493

0.716

0.617

4.282

4.972

75

0.0501

0.0494

0.725

0.636

4.285

4.977

100

0.0502

0.0495

0.717

0.693

4.283

4.978

MAD measured for increasing FWHM. The maximum amplitude of the Gaussian terms was fixed at 100, which corresponds to a peak-to-signal value of 0.922 at the image center. Higher precision is used for the proposed method to show an increasing MAD. The maximum MAD values recorded for increasing values of FWHM were 0.780, 0.799, 0.778, 0.778.

Table 4.

Standard Deviation of Absolute Differences for Images with Additive Gaussian FPN^{a}

FWHM

${\mathrm{Prop}}_{x}$

${\mathrm{Prop}}_{y}$

${\mathrm{PAC}}_{x}$

${\mathrm{PAC}}_{y}$

${\mathrm{DOCP}}_{x}$

${\mathrm{DOCP}}_{y}$

25

0.0398

0.0501

0.503

0.219

5.209

7.384

50

0.0398

0.0503

0.517

0.224

5.211

7.366

75

0.0398

0.0502

0.528

0.226

5.216

7.385

100

0.0399

0.0504

0.517

0.221

5.215

7.390

Standard deviations here correspond to values presented in Table 3.

Table 5.

Registration Results for Images with Random Gaussian Noise and Additive FPN Watermark^{a}

${\mu}_{n}$

${\sigma}_{n}$

${\mathrm{Prop}}_{x}$

${\mathrm{Prop}}_{y}$

${\mathrm{PAC}}_{x}$

${\mathrm{PAC}}_{y}$

${\mathrm{DOCP}}_{x}$

${\mathrm{DOCP}}_{y}$

4

1

0.051

0.053

3.763

3.695

5.811

5.774

8

2

0.054

0.062

3.763

3.697

6.692

6.550

12

3

0.055

0.068

3.763

3.699

7.344

6.959

16

4

0.061

0.081

3.763

3.701

7.682

7.008

20

5

0.070

0.089

3.764

3.704

8.198

7.545

Watermark amplitude was fixed at 100. Additive random noise is normally distributed with mean ${\mu}_{n}$ and standard deviation ${\sigma}_{n}$. The Barbara image was used for testing. The largest values of MAD recorded for increasing $\mu $ and $\sigma $ were 0.798, 0.804, 0.807, 0.928.

Table 6.

Standard Deviation of Absolute Difference for Images Corrupted by an Additive Watermark with Gaussian Noise^{a}

Registration Results for Simulated Images Corrupted by a Multiplicative Fixed Pattern of Ones and Zeros^{a}

${N}_{\text{frac}}$

${\mathrm{Prop}}_{x,y}$

$\mathrm{Prop}{\sigma}_{x,y}$

${\mathrm{PAC}}_{x,y}$

$\mathrm{PAC}{\sigma}_{x,y}$

0.025

0.043, 0.045

0.041, 0.053

1.611, 0.477

1.152, 0.380

0.050

0.045, 0.045

0.039, 0.041

1.684, 0.686

1.237, 0.648

0.100

0.056, 0.058

0.322, 0.498

1.877, 1.204

1.476, 1.148

0.200

0.073, 0.088

1.402, 0.791

2.228, 2.358

1.733, 1.786

0.300

0.089, 0.124

3.909, 4.524

2.446, 3.206

1.832, 1.959

0.400

0.153, 0.181

3.551, 3.427

2.664, 3.394

1.865, 1.983

MAD values measured for increasing value of ${N}_{\text{frac}}$ [Eq. (30)]. The Barbara image was used for testing. Standard deviations for this set of data are denoted by $\sigma $. For increasing values of ${N}_{\text{frac}}$, the maximum MAD values recorded were 3.107, 0.806, 0.88, 0.934, 5.685.

Table 8.

Registration Results for Simulated Images with Saturated Columns^{a}

${N}_{\text{frac}}$

${\mathrm{Prop}}_{x,y}$

$\mathrm{Prop}{\sigma}_{x,y}$

${\mathrm{PAC}}_{x,y}$

$\mathrm{PAC}{\sigma}_{x,y}$

0.050

0.074, 0.080

0.073, 0.085

3.479, 0.555

1.970, 0.276

0.074

0.081, 0.088

0.080, 0.098

3.484, 0.549

1.969, 0.278

0.099

0.087, 0.100

0.084, 0.117

3.486, 0.546

1.970, 0.280

${N}_{\text{frac}}$ is a measure of saturated columns. The Barbara image was used for testing. The standard deviation of the absolute difference is shown for both the proposed and PAC methods, denoted by $\sigma $. The maximum MAD values recorded were 1.153, 1.664, 2.105 for an increasing value of ${N}_{\text{frac}}$.