Error diffusion (ED) is a powerful tool for the generation of binary computer-generated holograms (CGH's). Several modifications of the original ED algorithm have been proposed to incorporate special requirements and assumptions present in CGH's. This paper compares different versions of the algorithm for their application to computer-generated holography with respect to reconstruction errors and the overall brightness of the reconstruction.
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Relative Reconstruction Errors and Brightnesses for the Three-Hologram Calculation Schemes for an Object That Is Spatially Separable from the dc Peak
Object 90 × 70
Relative MSE
Relative Brightness
Original error diffusion
1
1
Hauck's first method
0.22
0.77
Barnard's optimum method
2.17
0.62
Table II
Relative Reconstruction Errors and Brightnesses for the Three-Hologram Calculation Schemes but Calculated over a Larger Window Size
Object 128 × 128
Relative MSE
Relative Brightnes
Original error diffusion
1
1
Hauck's first method
0.82
0.81
Barnard's optimum method
2.05
0.95
Note that the reconstruction window now touches the dc peak for both the original ED and Barnard's method.
Table III
Relative MSE and Relative Brightness for the Original Error Diffusion Method, Varying the Edge Enhancement Introduced by a Threshold Modulationa
Original ED
Relative MSE (Original ED = 1)
Relative Brightness (Original ED = 1)
Enhancement 0.0
1.04
0.73
Enhancement 0.5
0.71
0.77
Enhancement 1.0
1
1
Enhancement 1.5
1.79
1.42
Enhancement 2.0
2.48
1.99
Enhancement 2.5
2.96
2.73
Enhancement 3.0
3.21
3.54
The minimum of the MSE at 0.5 is in agreement with the measurements done on the edge characteristics of error diffusion.
Table IV
Relative MSE and Relative Brightness for Hauck's Methoda
Hauck's First Method
Relative MSE (Original ED = 1)
Relative Brightness (Original ED = 1)
Enhancement 1
0.22
0.77
Enhancement 3
1.22
1.11
The modified error diffusion algorithm has less influence on this method, since the error is distributed along the interference fringes with minimal variation.
Table V
Relative MSE and Relative Brightness for Barnard's Methoda
Barnard's Optimum Method
Relative MSE (Original ED = 1)
Relative Brightness (Original ED = 1)
Enhancement 0.0
3.64
0.35
Enhancement 0.5
3.24
0.38
Enhancement 1.0
2.17
0.62
Enhancement 1.5
1.72
1.09
Enhancement 2.0
1.99
1.77
Enhancement 2.5
2.45
2.49
Enhancement 3.0
2.83
3.44
The minimum of the MSE at 1.5 suggests that the weights used in this case introduce an edge blurring that can be compensated for by a slight enhancement through the threshold modulation.
Table VI
Relative MSE and Relative Brightness for the Original Error Diffusion Method, Varying the Edge Enhancement Introduced by Threshold Modulationa
Original ED
Relative MSE (Original ED = 1)
Relative Brightness (Original ED = 1)
Enhancement 0.0
0.92
0.68
Enhancement 0.5
0.57
0.79
Enhancement 1.0
1
1
Enhancement 1.5
1.85
1.31
Enhancement 2.0
2.81
1.73
Enhancement 2.5
3.50
2.22
Enhancement 3.0
4.04
3.79
Note that the window used for the simulations touches the dc peak.
Table VII
Relative MSE and Relative Brightness for Hauck's Methoda
Hauck's First Method
Relative MSE (Original ED = 1)
Relative Brightness (Original ED = 1)
Enhancement 1
0.82
0.81
Enhancement 3
2.83
1.62
The modified error diffusion algorithm has less influence on this method, since the error is distributed along the interference fringes with minimal variation.
Table VIII
Relative MSE and Relative Brightness for Barnard's Method, Varying the Edge Enhancement Introduced by a Threshold Modulationa
Barnard's Optimum Method
Relative MSE (Original ED = 1)
Relative Brightness (Original ED = 1)
Enhancement 0.0
2.99
0.69
Enhancement 0.5
2.34
0.69
Enhancement 1.0
2.05
0.95
Enhancement 1.5
2.27
1.47
Enhancement 2.0
2.92
2.26
Enhancement 2.5
3.34
3.23
Enhancement 3.0
3.56
3.61
Note that the window used for the simulations touches the dc peak.
Table IX
Experimental Measurement of the Reconstruction Brightness for the Original Error Diffusion Method
Original Error Diffusion
Relative Brightness (Original ED = 1)
Enhancement 0.0
0.8
Enhancement 0.5
0.9
Enhancement 1.0
1
Enhancement 1.5
1.3
Enhancement 2.0
1.8
Enhancement 2.5
2.3
Enhancement 3.0
2.9
Table X
Experimental Measurement of the Reconstruction Brightness for Hauck's Method
Hauck's First Method
Relative Brightness (Original ED = 1)
Enhancement 1
0.8
Enhancement 3
1.0
Table XI
Experimental Measurement of the Reconstruction Brightness for Barnard's Method
Barnard's Optimum Method
Relative Brightness (Original ED = 1)
Enhancement 0.0
0.5
Enhancement 0.5
0.6
Enhancement 1.0
0.8
Enhancement 1.5
1.1
Enhancement 2.0
1.8
Enhancement 2.5
2.4
Enhancement 3.0
3.1
Tables (11)
Table I
Relative Reconstruction Errors and Brightnesses for the Three-Hologram Calculation Schemes for an Object That Is Spatially Separable from the dc Peak
Object 90 × 70
Relative MSE
Relative Brightness
Original error diffusion
1
1
Hauck's first method
0.22
0.77
Barnard's optimum method
2.17
0.62
Table II
Relative Reconstruction Errors and Brightnesses for the Three-Hologram Calculation Schemes but Calculated over a Larger Window Size
Object 128 × 128
Relative MSE
Relative Brightnes
Original error diffusion
1
1
Hauck's first method
0.82
0.81
Barnard's optimum method
2.05
0.95
Note that the reconstruction window now touches the dc peak for both the original ED and Barnard's method.
Table III
Relative MSE and Relative Brightness for the Original Error Diffusion Method, Varying the Edge Enhancement Introduced by a Threshold Modulationa
Original ED
Relative MSE (Original ED = 1)
Relative Brightness (Original ED = 1)
Enhancement 0.0
1.04
0.73
Enhancement 0.5
0.71
0.77
Enhancement 1.0
1
1
Enhancement 1.5
1.79
1.42
Enhancement 2.0
2.48
1.99
Enhancement 2.5
2.96
2.73
Enhancement 3.0
3.21
3.54
The minimum of the MSE at 0.5 is in agreement with the measurements done on the edge characteristics of error diffusion.
Table IV
Relative MSE and Relative Brightness for Hauck's Methoda
Hauck's First Method
Relative MSE (Original ED = 1)
Relative Brightness (Original ED = 1)
Enhancement 1
0.22
0.77
Enhancement 3
1.22
1.11
The modified error diffusion algorithm has less influence on this method, since the error is distributed along the interference fringes with minimal variation.
Table V
Relative MSE and Relative Brightness for Barnard's Methoda
Barnard's Optimum Method
Relative MSE (Original ED = 1)
Relative Brightness (Original ED = 1)
Enhancement 0.0
3.64
0.35
Enhancement 0.5
3.24
0.38
Enhancement 1.0
2.17
0.62
Enhancement 1.5
1.72
1.09
Enhancement 2.0
1.99
1.77
Enhancement 2.5
2.45
2.49
Enhancement 3.0
2.83
3.44
The minimum of the MSE at 1.5 suggests that the weights used in this case introduce an edge blurring that can be compensated for by a slight enhancement through the threshold modulation.
Table VI
Relative MSE and Relative Brightness for the Original Error Diffusion Method, Varying the Edge Enhancement Introduced by Threshold Modulationa
Original ED
Relative MSE (Original ED = 1)
Relative Brightness (Original ED = 1)
Enhancement 0.0
0.92
0.68
Enhancement 0.5
0.57
0.79
Enhancement 1.0
1
1
Enhancement 1.5
1.85
1.31
Enhancement 2.0
2.81
1.73
Enhancement 2.5
3.50
2.22
Enhancement 3.0
4.04
3.79
Note that the window used for the simulations touches the dc peak.
Table VII
Relative MSE and Relative Brightness for Hauck's Methoda
Hauck's First Method
Relative MSE (Original ED = 1)
Relative Brightness (Original ED = 1)
Enhancement 1
0.82
0.81
Enhancement 3
2.83
1.62
The modified error diffusion algorithm has less influence on this method, since the error is distributed along the interference fringes with minimal variation.
Table VIII
Relative MSE and Relative Brightness for Barnard's Method, Varying the Edge Enhancement Introduced by a Threshold Modulationa
Barnard's Optimum Method
Relative MSE (Original ED = 1)
Relative Brightness (Original ED = 1)
Enhancement 0.0
2.99
0.69
Enhancement 0.5
2.34
0.69
Enhancement 1.0
2.05
0.95
Enhancement 1.5
2.27
1.47
Enhancement 2.0
2.92
2.26
Enhancement 2.5
3.34
3.23
Enhancement 3.0
3.56
3.61
Note that the window used for the simulations touches the dc peak.
Table IX
Experimental Measurement of the Reconstruction Brightness for the Original Error Diffusion Method
Original Error Diffusion
Relative Brightness (Original ED = 1)
Enhancement 0.0
0.8
Enhancement 0.5
0.9
Enhancement 1.0
1
Enhancement 1.5
1.3
Enhancement 2.0
1.8
Enhancement 2.5
2.3
Enhancement 3.0
2.9
Table X
Experimental Measurement of the Reconstruction Brightness for Hauck's Method
Hauck's First Method
Relative Brightness (Original ED = 1)
Enhancement 1
0.8
Enhancement 3
1.0
Table XI
Experimental Measurement of the Reconstruction Brightness for Barnard's Method