Timur E. Gureyev,
Timothy J. Davis,
Andrew Pogany,
Sheridan C. Mayo,
and Stephen W. Wilkins
The authors are with Commonwealth Scientific and Industrial Research Organisation-Manufacturing and Infrastructure Technology, PB 33, Clayton South, VIC 3169, Australia.
Timur E. Gureyev, Timothy J. Davis, Andrew Pogany, Sheridan C. Mayo, and Stephen W. Wilkins, "Optical phase retrieval by use of first Born- and Rytov-type approximations," Appl. Opt. 43, 2418-2430 (2004)
The first Born and Rytov approximations of scattering theory are introduced in their less familiar near-field versions. Two algorithms for phase retrieval based on these approximations are then described. It is shown theoretically and by numerical simulations that, despite the differences in their formulation, the two algorithms deliver fairly similar results when used for optical phase retrieval in the near and intermediate fields. The algorithms are applied to derive explicit solutions to four phase-retrieval problems of practical relevance to quantitative phase-contrast imaging and tomography. An example of successful phase reconstruction by use of the Born-type algorithm with an experimental x-ray image is presented.
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Relative Error E in the Reconstructed Phase by Use of Eq. (28) as a Function of the Phase Magnitude and Regularization Parametera
Maximum Phase Excursion
α = 0 (%)
α = 10-7 (%)
α = 10-5 (%)
α = 10-3 (%)
Born, ϕmin =-0.01 rad
1.3
1.1
17
64
Rytov, ϕmin =-0.01 rad
3.0
2.7
15
64
Born, ϕmin =-0.1 rad
11
8.5
17
64
Rytov, ϕmin =-0.1 rad
31
29
8.7
63
Born, ϕmin = -1 rad
108
79
44
67
Rytov, ϕmin =-1 rad
307
293
194
64
Best results are shown in bold.
Table 2
Relative Error E in the Reconstructed Phase by Use of the Regularized Version of Eq. (29) as a Function of the Phase Magnitude and Regularization Parametera
Maximum Phase Excursion
α = 0 (%)
α = 10-7 (%)
α = 10-5 (%)
α = 10-3 (%)
Born, ϕmin =-0.01 rad
2.9
2.1
1.8
8.2
Rytov, ϕmin =-0.01 rad
2.7
2.0
1.7
7.8
Born, ϕmin =-0.1 rad
29
21
8.8
8.9
Rytov, ϕmin =-0.1 rad
27
21
8.5
6.4
Born, ϕmin =-1 rad
261
190
76
27
Rytov, ϕmin =-1 rad
261
205
88
40
Best results are shown in bold.
Table 3
Relative Error E in the Reconstructed Phase by Use of the Regularized Version of Eq. (30) as a Function of the Phase Magnitude and Regularization Parametera
Maximum Phase Excursion
α = 0 (%)
α = 10-9 (%)
α = 10-7 (%)
α = 10-5 (%)
Born, ϕmin =-0.01 rad
3.5
3.4
6.0
60
Rytov, ϕmin =-0.01 rad
3.7
3.6
4.7
60
Born, ϕmin =-0.1 rad
35
34
14.5
56
Rytov, ϕmin =-0.1 rad
38
38
19
59
Born, ϕmin =-1 rad
298
292
126
66
Rytov, ϕmin =-1 rad
456
449
258
71
Best results are shown in bold.
Table 4
Relative Error E in the Reconstructed Phase by Use of the Regularized Version of Eq. (31) as a Function of the Phase Magnitude and Regularization Parametera
Maximum Phase Excursion
α = 0 (%)
α = 10-9 (%)
α = 10-7 (%)
α = 10-5 (%)
Born, ϕmin =-0.01 rad
3.7
3.0
13
66
Rytov, ϕmin =-0.01 rad
3.7
3.0
11
66
Born, ϕmin =-0.1 rad
37
30
15
66
Rytov, ϕmin = -0.1 rad
37
31
11
65
Born, ϕmin =-1 rad
362
290
118
69
Rytov, ϕmin =-1 rad
392
317
182
65
Best results are shown in bold.
Table 5
Relative Error E in the Reconstructed Phase by Use of Eq. (2) as a Function of Defocus Distance
Method/Defocus
z = 10-3 m (%)
z = 10-2 m (%)
z = 10-1
m (%)
z = 1 m (%)
z = 10 m (%)
Born
0.04
0.04
1.0
1.1
1.5
Rytov
1.0
6.9
7.6
2.7
1.8
Table 6
Relative Error E in the Reconstructed Phase by Use of the Regularized Version of Eq. (27) as a Function of Noise Level
Method/Noise
0.25% (%)
0.5% (%)
1% (%)
2% (%)
3% (%)
Born
12
18
34
56
70
Rytov
9.9
17
34
54
68
Tables (6)
Table 1
Relative Error E in the Reconstructed Phase by Use of Eq. (28) as a Function of the Phase Magnitude and Regularization Parametera
Maximum Phase Excursion
α = 0 (%)
α = 10-7 (%)
α = 10-5 (%)
α = 10-3 (%)
Born, ϕmin =-0.01 rad
1.3
1.1
17
64
Rytov, ϕmin =-0.01 rad
3.0
2.7
15
64
Born, ϕmin =-0.1 rad
11
8.5
17
64
Rytov, ϕmin =-0.1 rad
31
29
8.7
63
Born, ϕmin = -1 rad
108
79
44
67
Rytov, ϕmin =-1 rad
307
293
194
64
Best results are shown in bold.
Table 2
Relative Error E in the Reconstructed Phase by Use of the Regularized Version of Eq. (29) as a Function of the Phase Magnitude and Regularization Parametera
Maximum Phase Excursion
α = 0 (%)
α = 10-7 (%)
α = 10-5 (%)
α = 10-3 (%)
Born, ϕmin =-0.01 rad
2.9
2.1
1.8
8.2
Rytov, ϕmin =-0.01 rad
2.7
2.0
1.7
7.8
Born, ϕmin =-0.1 rad
29
21
8.8
8.9
Rytov, ϕmin =-0.1 rad
27
21
8.5
6.4
Born, ϕmin =-1 rad
261
190
76
27
Rytov, ϕmin =-1 rad
261
205
88
40
Best results are shown in bold.
Table 3
Relative Error E in the Reconstructed Phase by Use of the Regularized Version of Eq. (30) as a Function of the Phase Magnitude and Regularization Parametera
Maximum Phase Excursion
α = 0 (%)
α = 10-9 (%)
α = 10-7 (%)
α = 10-5 (%)
Born, ϕmin =-0.01 rad
3.5
3.4
6.0
60
Rytov, ϕmin =-0.01 rad
3.7
3.6
4.7
60
Born, ϕmin =-0.1 rad
35
34
14.5
56
Rytov, ϕmin =-0.1 rad
38
38
19
59
Born, ϕmin =-1 rad
298
292
126
66
Rytov, ϕmin =-1 rad
456
449
258
71
Best results are shown in bold.
Table 4
Relative Error E in the Reconstructed Phase by Use of the Regularized Version of Eq. (31) as a Function of the Phase Magnitude and Regularization Parametera
Maximum Phase Excursion
α = 0 (%)
α = 10-9 (%)
α = 10-7 (%)
α = 10-5 (%)
Born, ϕmin =-0.01 rad
3.7
3.0
13
66
Rytov, ϕmin =-0.01 rad
3.7
3.0
11
66
Born, ϕmin =-0.1 rad
37
30
15
66
Rytov, ϕmin = -0.1 rad
37
31
11
65
Born, ϕmin =-1 rad
362
290
118
69
Rytov, ϕmin =-1 rad
392
317
182
65
Best results are shown in bold.
Table 5
Relative Error E in the Reconstructed Phase by Use of Eq. (2) as a Function of Defocus Distance
Method/Defocus
z = 10-3 m (%)
z = 10-2 m (%)
z = 10-1
m (%)
z = 1 m (%)
z = 10 m (%)
Born
0.04
0.04
1.0
1.1
1.5
Rytov
1.0
6.9
7.6
2.7
1.8
Table 6
Relative Error E in the Reconstructed Phase by Use of the Regularized Version of Eq. (27) as a Function of Noise Level