Error diffusion procedure: theory and applications in optical signal processing

Severin Weissbach; Frank Wyrowski

doi:10.1364/AO.31.002518

Applied Optics
Vol. 31,
Issue 14,
pp. 2518-2534
(1992)
•https://doi.org/10.1364/AO.31.002518

Error diffusion procedure: theory and applications in optical signal processing

Severin Weissbach and Frank Wyrowski

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Severin Weissbach and Frank Wyrowski, "Error diffusion procedure: theory and applications in optical signal processing," Appl. Opt. 31, 2518-2534 (1992)

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Abstract

An analysis of the error diffusion procedure is presented that is based on the terminology of filter theory. It is demonstrated that the error diffusion procedure is a powerful means to avoid signal error caused by a nonlinear system. An appropriate filter design method is described. The theoretical results are applied to treat picture binarization as well as quantization and coding in diffractive optics–digital holography.

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u₀	d(1)	d(2)	$\| \frac{d H}{d u} (u_{0}) \|$
0	−2	1	0
1/8	$- \sqrt{2}$	1	2.83π
1/4	0	1	4π

u₀	d(1)	d(2)
0	1	0
±1/(2Δ)	−1	0
Else	−2 cos(2πu₀Δ)	1

m′	φ	r₀	d′(1)	d′(2)	Period (1/Δ_φ)	\|∇H(u₀) · ê_φ\|
(1, 0)	0	1/4	0	1	1	4π
(1, 1)	45	3/8	$\sqrt{2}$	1	$1 / \sqrt{2}$	4π
(−1, 2)	116.57	0	−1	0	$1 / \sqrt{5}$	4.47π

u₀	m_φ′	m_ϑ′	(r₀)_φ	d_φ′(1)	d_ϑ′(1)
(0, 0)	(1, 0)	(0, 1)	0	−1	−1
(0, 0)	(1, 1)	(−1, 1)	0	−1	−1
(1/4, 1/4)	(1, 1)	(−1, 1)	1/2	1	−1

u₀	d(−1, 1)	d(0, 1)	d(0, 2)	d(1, 0)	d(1, 1)
(0, 0)	–	−1	–	−1	1
(0, 0)	−1	–	1	–	−1
(1/4, 1/4)	−1	–	−1	–	1

m′	Nonzero Diffusion Weights	SNR	Fig.
—	Direct binarization [see Eq. (43)]	16	—
(−1, 1)	d(−1, 1) = −1	225	16
(1, 1)	d(1, 1) = 1	250	17
(1, 0)	d(2, 0) = 1	110	—
(1, 1)	d(1, 1) = 1, d(−1, 1) = −1, d(0, 2) = −1	605	18

u₀	m′	Nonzero Diffusion Weights	SNR	Fig.
0	—	Direct nonlinearity [see Eq. (44)]	22	—
0	—	d(1, 0) = −0.5, d(0, 1) = −0.5	2200	—
0	(1, 0)	d(1, 0) = −1, d(0, 1) = −1, d(1, 1) = 1	12,400	19(a)
0	(1, 1)	d(1, 1) = −1, d(−1, 1) = −1, d(0, 2) = 1	2700	—
(1/4, 0)	(1, 1)	d(1, 1) = j, d(−1, 1) = −j, d(0, 2) = 1	2500	19(b)

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Applied Optics