One-bit sigma-delta modulation on a closed loopSara Krause-Solberg, Olga Graf, Felix Krahmer
Technical University of Munich, Department of Mathematics
Overview of the problem
Our task:I For K -bandlimited signals whose domain is T = {z ∈ C : |z | = 1}, perform
1-bit Σ∆ modulation
Why is it challenging?I Σ∆ modulator requires causality, hence we have mismatch at the
initialization point and consequently large reconstruction error — need tofix that!
Why is it useful?I If domain is extended from 1D to 2D closed manifolds, 1-bit Σ∆ modulation
can be used as a digital halftoning technique for 3D color printing
1-bit Σ∆ modulation
First order 1-bit Σ∆ scheme:
vn = vn−1 + yn − qnqn = sign(vn−1 + yn)
initialize v−1 := 0
yn - samples (input)qn - quantized values (output) (1)vn - state variables
Assumptions for signal f :I f ∈ L2(T), i.e., f is 2π-periodicI ‖f ‖∞ ≤ 1
I f is K -bandlimited, i.e., f ∈ [−K ,K ]I samples yn = f
(2πnN
), n = 0, . . . ,N − 1, where
N = 2λK + 1 with oversampling parameter λ > 1
Signal reconstruction
Sampling theorem & reconstructed signal:
f (t) =1
N
N−1∑n=0
ynϕK
(t − 2πn
N
), frec(t) =
1
N
N−1∑n=0
qnϕK
(t − 2πn
N
),
where the reconstruction kernel is the Dirichlet kernel
ϕK(ξ) =sin((2K + 1)ξ2)
sin(ξ2)
Reconstruction error:
|f (t)− frec(t)| =1
N
∣∣∣∣∣∣N−1∑n=0
(yn − qn)ϕK
(t − 2πn
N
)∣∣∣∣∣∣≤ 1
N‖ϕ′K‖L1 +
|vN−1|N︸ ︷︷ ︸≤ 1
N
‖ϕK‖∞︸ ︷︷ ︸=2K+1
Can we get rid of the second term in the above estimate? Yes!
Proposed modification of Σ∆ scheme
I Aim: get rid of additional error term |vN−1|N ‖ϕK‖∞
I Idea: run Σ∆ for updated samples yn := yn + δ, where δ = −vN−1
N (2)and reconstruct from resulting output qn using the Dirichlet kernel
Theorem (Krause-Solberg, Graf, Krahmer)
Using 1-bit Σ∆ modulation scheme (1) on updated samples (yn)n leads to(vn)n with vN−1 = 0.
Reconstruction error after modification
From updated samples we get
|f (t)− frec(t)| =1
N
∣∣∣∣∣∣N−1∑n=0
(yn − qn)ϕK
(t − 2πn
N
)∣∣∣∣∣∣≤ 2
N‖ϕ′K‖L1 +
|vN−1|N︸ ︷︷ ︸=0
‖ϕK‖∞ + δ︸︷︷︸≤ 1
N
Numerical experiments & discussion
Consider signal f (t) = 0.1 sin(5t) cos(10t) + 0.2 with bandwidth 2K = 30.
Reconstruction error for first order Σ∆ scheme
N = 302 N = 3002 N = 30002
Figure 1. Reconstruction error for classical first order scheme (dashed) and for modifiedscheme (solid)
I Error is now distrubuted evenly, no large error at initialization pointI L∞ norm of the error is reduced
Outlook: Reconstruction error for higher order Σ∆ schemes [2]
f (t)− frec(t), classical scheme f (t)− frec(t), modified scheme
Figure 2. Reconstruction error behavior for classical and for modified schemes
I Updates are higher order variants of (2)I Error keeps oscillating around δ, constant shifts of O( 1
N) cannot be avoidedI Modified scheme optimizes oscillation amplitude around a constant shift,
thus leading to less visible artifacts
Long-term goal: Digital halftoning in 3D printing
Figure 3. Representation of colors on a surface using digital halftoning
I 1-bit Σ∆ modulation as digital halftoning technique: works well becauseerror is a high-pass sequence while human eye acts as a low-pass filter [3]
I Major challenge in 3D digital halftoning: surfaces to be colored are closedI 3D digital halftoning can be utilized in 3D printing tecnhologies which
rely on voxel-wise printing (e.g. HP Multi Jet Fusion) [4]
References
[1] I. Daubechies and R. DeVore.
Approximating a bandlimited function using very coarsely quantized data: a family of stable sigma-delta
modulators of arbitrary order.
Ann. Math., 158(2):679–710, 2003.
[2] P. Deift, C. S. Gunturk, and F. Krahmer.
An optimal family of exponentially accurate one-bit sigma-delta quantization schemes.
Comm. Pure Appl. Math., 64(7):883–919, 2011.
[3] T. Kite, B. Evans, A. Bovik, and T. Sculley.
Digital halftoning as 2-d delta-sigma modulation.
In Proc. Intl. Conf. Image Proc., 1:799–802, 1997.
[4] R. Mao, U. Sarkar, R. Ulichney, and J. Allerbach.
3d halftoning.
In Proc. Color Imaging XXII, IS&T Electronic Imaging, 2017.
Top Related