Wiener filters are often used for the restoration of images embedded in additive noise. However, it is often desirable to have a simpler filter structure that is easily implemented and performs well. In this paper we present the necessary conditions for a filter with an optimum piecewise-constant frequency-response function. Such a filter has a simpler implementation than the Wiener filter, and, in general, it suffers only a small loss in mean-square-error performance, as is illustrated in numerical examples.
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Using mismatched two-level piecewise-constant filter (in decibels).
Using mismatched three-level piecewise-constant filter (in decibels).
Using mismatched Wiener filter (in decibels).
Table 7
Optimum Piecewise-Constant Filter (C1 = 0) and Performance Comparison with Optimum Filter (Example 4)
SNR (σs2/No)
SNR (dB)
ropc
C2opc
eopc/σs2
eopc/σs2 (dB)
dB Loss with Respect to (eo/σs2)
1
0
2.216
0.701
0.359
−4.4
0.6
5
7.0
2.715
0.893
0.130
−8.9
0.6
10
10.0
2.937
0.935
0.078
−11.1
0.7
20
13.0
3.146
0.962
0.045
−13.5
0.6
30
14.8
3.273
0.972
0.032
−14.9
0.7
40
16.0
3.399
0.978
0.026
−15.9
0.7
50
17.0
3.482
0.981
0.021
−16.7
0.7
Table 8
Optimum Quantization of Ho(ω) (C1 = 0) and Comparison with Optimum Piecewise-Constant Filter (Example 3)
SNR (σs2/No
SNR (dB)
roq
C2oq
eoq/σs2
eoq/σs2 (dB)
(eoq−eopc)/σs2 (dB)
1
0
2.308
0.609
0.369
−4.3
0.1
5
7.0
2.804
0.763
0.148
−8.3
0.6
10
10.0
3.014
0.802
0.096
−10.2
0.9
20
13.0
3.216
0.832
0.062
−12.1
1.4
30
14.8
3.332
0.846
0.049
−13.1
1.8
40
16.0
3.412
0.855
0.041
−13.9
2.0
50
17.0
3.473
0.861
0.036
−14.4
2.3
Tables (8)
Table 1
Optimum Piecewise-Constant Filter with C1 = O, C2 = 1a
SNR (σs2/No)
S (dB)
eo/σs2
eo/σs2 (dB)
eopc/σs2
eopc/σs2 (dB)
dB Loss with Respect to (eo/σs2)
1
0
0.316
−5.0
0.452
−3.5
1.5
5
7.0
0.111
−9.5
0.142
−8.5
1.0
10
10.0
0.066
−11.8
0.082
−10.9
0.9
20
13.0
0.039
−14.1
0.046
−13.3
0.8
30
14.8
0.028
−15.6
0.033
−14.8
0.8
40
16.0
0.022
−16.6
0.026
−15.9
0.7
50
17.0
0.018
−17.4
0.021
−16.7
0.7
See Example 1, Section 2.
Table 2
Optimum Piecewise-Constant Filter (C1 = 0) and Performance Comparison with Optimum Filtera
SNR (σs2/No)
SNR (dB)
r1
r2
C2opc
eopc/σs2
eopc/σs2 (dB)
eo/σs2
eo/σs2 (dB)
dB Loss with Respect to (eo/σs2)
1
0
0.128
3.059
0.511
0.602
−2.2
0.543
−2.7
0.5
2
3.0
0.087
3.439
0.639
0.467
−3.3
0.407
−3.9
0.6
5
7.0
0.047
4.060
0.774
0.305
−5.2
0.257
−5.9
0.7
7
8.5
0.036
4.316
0.812
0.255
−5.9
0.213
−6.7
0.8
10
10.0
0.027
4.601
0.847
0.208
−6.8
0.172
−7.6
0.8
15
11.8
0.019
4.940
0.880
0.163
−7.9
0.134
−8.7
0.8
20
13.0
0.015
5.188
0.900
0.137
−8.6
0.112
−9.5
0.9
30
14.8
0.011
5.547
0.922
0.105
−9.8
0.086
−10.7
0.9
40
16.0
0.008
5.808
0.936
0.087
−10.6
0.071
−11.5
0.9
50
17.0
0.007
6.014
0.945
0.074
−11.3
0.061
−12.2
0.9
See Example 2, Section 3.
Table 3
Constrained Optimum Piecewise-Constant Filter (C1 = 0) and Performance Comparison with Optimum Filtera
SNR (σs2/No)
SNR (dB)
ro
C2copc
ecopc/σs2
ecopc/σs2 (dB)
dB Loss with Respect to (eo/σs2)
1
0
3.062
0.511
0.602
−2.2
0.5
2
3.0
3.440
0.639
0.467
−3.3
0.6
5
7.0
4.062
0.774
0.305
−5.2
0.7
7
8.5
4.314
0.813
0.255
−5.9
0.8
10
10.0
4.601
0.847
0.208
−6.8
0.8
15
11.8
4.940
0.880
0.163
−7.9
0.8
20
13.0
5.189
0.900
0.137
−8.6
0.9
30
14.8
5.548
0.922
0.105
−9.8
0.9
40
16.0
5.808
0.936
0.087
−10.6
0.9
50
17.0
6.013
0.945
0.074
−11.3
0.9
Example 2, Section 3.
Table 4
Two-Level Constrained Optimum Piecewise-Constant Filter (C1 = 0) and Performance Comparison with Optimum Filtera
γ
SNR (σs2/No)
r0
C2
ecopc/σs2
ecopc/σs2 (dB)
eo/σs2
eo/σs2 (dB)
dB Loss with Respect to (eo/σs2)
0.5
1
1.295
0.927
0.716
−1.5
0.665
−1.8
0.3
5
1.587
1.359
0.557
−2.5
0.491
−3.0
0.6
10
1.713
1.475
0.514
−2.9
0.425
−3.7
0.8
20
1.837
1.552
0.486
−3.1
0.367
−4.4
1.3
30
1.906
1.583
0.474
−2.2
0.336
−4.7
1.5
40
1.955
1.601
0.468
−3.3
0.315
−5.0
1.7
50
1.991
1.612
0.464
−3.3
0.300
−5.2
1.9
2.0
1
1.841
0.816
0.499
−3.0
0.475
−3.2
0.2
5
2.266
1.127
0.264
−5.8
0.249
−6.0
0.2
10
2.450
1.206
0.205
−6.9
0.182
−7.4
0.5
20
2.628
1.258
0.166
−7.8
0.132
−8.8
1.0
30
2.728
1.279
0.151
−8.2
0.109
−9.6
1.4
40
2.798
1.290
0.142
−8.5
0.095
−10.2
1.7
50
2.850
1.297
0.137
−8.6
0.085
−10.7
2.1
4.0
1
2.011
0.770
0.433
−3.6
0.404
−3.9
0.3
5
2.476
1.033
0.191
−7.2
0.180
−7.5
0.3
10
2.680
1.098
0.132
−8.8
0.121
−9.2
0.4
20
2.876
1.139
0.094
−10.3
0.081
−10.9
0.6
30
2.987
1.156
0.079
−11.0
0.063
−12.0
1.0
40
3.064
1.165
0.071
−11.5
0.053
−12.8
1.3
50
3.123
1.171
0.065
−11.8
0.046
−13.4
1.6
Example 3, Section 3.
Table 5
Three-Level Constrained Optimum Piecewise-Constant Filter (C1 = 0) and Performance Comparison with Optimum Filtera
Using mismatched two-level piecewise-constant filter (in decibels).
Using mismatched three-level piecewise-constant filter (in decibels).
Using mismatched Wiener filter (in decibels).
Table 7
Optimum Piecewise-Constant Filter (C1 = 0) and Performance Comparison with Optimum Filter (Example 4)
SNR (σs2/No)
SNR (dB)
ropc
C2opc
eopc/σs2
eopc/σs2 (dB)
dB Loss with Respect to (eo/σs2)
1
0
2.216
0.701
0.359
−4.4
0.6
5
7.0
2.715
0.893
0.130
−8.9
0.6
10
10.0
2.937
0.935
0.078
−11.1
0.7
20
13.0
3.146
0.962
0.045
−13.5
0.6
30
14.8
3.273
0.972
0.032
−14.9
0.7
40
16.0
3.399
0.978
0.026
−15.9
0.7
50
17.0
3.482
0.981
0.021
−16.7
0.7
Table 8
Optimum Quantization of Ho(ω) (C1 = 0) and Comparison with Optimum Piecewise-Constant Filter (Example 3)