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.