double look‑up table method for fast light propagation ... · lobaz / double look‑up table...

DEPARTMENT OFCOMPUTER SCIENCE AND ENGINEERING

CENTREOF COMPUTER GRAPHICSAND VISUALIZATION

PLZEŇCZECH REPUBLIC

http://graphics.zcu.cz

FACULTYOF APPLIED SCIENCESUNIVERSITYOF WEST BOHEMIA

Petr Lobaz

University of West Bohemia Pilsen, Czech Republic [email protected]

Double look‑up table method for fast light propagation calculations

Lobaz / Double look‑up table method for fast light propagation calculations 2 / 16

The problem

Radially symmetric functions in holography

• Rayleigh‑Sommerfeld convolution kernel

KRS(x, y; z0) = – 12π

j 2πλ –

1r

exp(j2πr / λ)

r z0

r

r= x2 + y2 + z02

• free space transfer function (angular spectrum kernel)

• lens phase shift function

• … re

im x

y


Radially symmetric functions in holography

• require complicated elementary functions evaluation (sin, cos, sqrt, …)

• sometimes hard to calculate (e.g. filtered kernels)• highly oscillatory ⇒ require high precision arithmetic

The problem

correctcalculation

Example:Rayleigh-Sommerfeldkernel

λ=500 nmz0=2.1 m

singleprecision


Function evaluation should be

• fast and precise, easy to parallelize• easy to implement on GPU or in hardware

GPU features

Many simple cores optimized for• single precision arithmetic• independent per‑pixel operations• linear code execution (no branching)• computer graphics calculations (texture look‑up)• fast cache memory access

The wish list


Two-step calculation

Example: Rayleigh‑Sommerfeld convolution kernel

1. ρ(x, y)= x2 + y2

2. KRS(ρ; z0) = – 12π

j 2πλ –

1r

exp(j2πr / λ)

r z0

r

r= ρ2 + z02

Proposed method

Precalculate ρ(x, y) and KRS(ρ; z0) to look‑up tables rhoLUT and waveLUT (double look‑up table method)

D‑LUT principle


D‑LUT principle

input: (�, �; �0) output: �RS(�; �0) = �RS(�, �; �0)

rhoLUT

�min � �max�min

�

�max

�min

�maxwaveLUT

�RS()

�min � �max�min

�

�max

re

im

|�| |�| �

�0


Definition and usage of the rhoLUT

• rhoLUT[m, n] = (mΔρ)2 + (nΔρ)2

ρ(x, y)≈rhoLUT[x / Δρ, y / Δρ]

– use nearest‑neighbour or bilinear interpolation for noninteger x / Δρ, y / Δρ

• sampling distance Δρ small ⇒ higher precision, larger table• nearest‑neighbour interpolation ⇒ faster, less precise ⇒ requires smaller Δρ

rhoLUT properties


• error of r = ρ2 + z02 should be small

• example: error of r for bilinear interp. in the rhoLUT

rhoLUT properties

x [mm]0

3

3

y

∆� = 100 �m2.67 nm

0

0.50

1.00

1.50

2.00

2.50

x [�m]0

200

200

y

∆� = 100 �m

00 2 4 6 8 10 x [m]

123

[nm

]

�0 = 0.5 m�(��( , �), �0) – �(�( , �), �0) for � = ∆� / 2 ∆� = 100 �m

x [mm]0

3

3

y

∆� = 100 �m2.67 nm

0

0.50

1.00

1.50

2.00

2.50

x [�m]0

200

200

y

∆� = 100 �m

00 2 4 6 8 10 x [m]

123

[nm

]

�0 = 0.5 m�(��( , �), �0) – �(�( , �), �0) for � = ∆� / 2 ∆� = 100 �m

x [mm]0

3

3

y

∆� = 100 �m2.67 nm

0

0.50

1.00

1.50

2.00

2.50

x [�m]0

200

200

y

∆� = 100 �m

00 2 4 6 8 10 x [m]

123

[nm

]

�0 = 0.5 m�(��( , �), �0) – �(�( , �), �0) for � = ∆� / 2 ∆� = 100 �m


Selection of Δρ• estimate of the maximum error

maxRError(Δρ; z0) = 1.2 ρBilinear

Δρ2

, Δρ2

2

+ z02

1/2

–

1.2 ρExact

Δρ2

, Δρ2

2

+ z02

1/2

(factor1.2resultsfromdeepererroranalysis)

rhoLUT properties


Selection of Δρ• maxRError looks like a line in a log‑log graph

for wide range of Δρ, z0 and acceptable error values

rhoLUT properties

10–610–14

10–12

10–10

10–8

10–6

10–4

10–2

1 �0 = 0.001 m

max

REr

ror(

∆�,

�0)

[m]

10–5 10–4 10–3 10–2 10–1

∆� [m]

�0 = 0.01 m�0 = 0.1 m�0 = 1 m�0 = 10 m


rhoLUT properties

Selection of Δρ• linear approximation in the log‑log graph leads to

Δρ ≈exp

1κ log

maxRError(z0)ξ1 exp(ξ0)

(where κ ≈ 1.9998, ξ0 ≈ –1.9886, ξ1 ≈ –0.9999)

Example

propagation area 50 × 50 mm2, z0 = 1 m• maxRError = λ / 100 ⇒ rhoLUT size 264 × 264• maxRError = λ / 10 ⇒ rhoLUT size 84 × 84


Definition and usage of the waveLUT

• waveLUT[q; z0] = KRS(qΔw; z0)

KRS(ρ; z0)≈waveLUT[ρ / Δw; z0]

– use nearest‑neighbour or linear interpolation for noninteger ρ / Δw

• Δw should be small enough to capture oscillations of KRS(ρ; z0)

• calculate maximum local frequency of KRS(ρ; z0)• set Δw ≤ 1 / (8 × maximum local frequency)

waveLUT properties


Accuracy

• look‑up tables precalculated in double precision• measured error according to theoretical analysis• no problems with single precision calculations

Look-up tables sizes

• example values for typical scenario in computer generated holography (hologram size ≈ 5 × 5 cm, resolution 1 μm ⇒ hologram size 2.5 GB)

• rhoLUT size ≈ 500 × 500 samples (≈ 1 MB)• waveLUT size ≈ 250 000 × Z samples (≈ 2Z MB)

Results


Results

CPU implemenation speed

• 10× to 200× faster than direct calculation for complicated filtered propagation kernels in realistic scenarios (just with waveLUT, ρ calculated directly)

• 1.7× faster than direct calculation for simple propagation kernel (just with waveLUT, ρ calculated directly)

• careful rhoLUT implementation 1.2–1.4× faster than calculating ρ directly


GPU implemenation

• look‑up tables precalculated on CPU in double precision

• incorporation of texture look‑up and built‑in texture interpolation

• no problems with GPU accuracy• execution speed not yet fully tested

Work in progress (almost done)

• careful testing of GPU performance• angular spectrum propagation method

Results

DEPARTMENT OFCOMPUTER SCIENCE AND ENGINEERING

CENTREOF COMPUTER GRAPHICSAND VISUALIZATION

PLZEŇCZECH REPUBLIC

http://graphics.zcu.cz

FACULTYOF APPLIED SCIENCESUNIVERSITYOF WEST BOHEMIA

Thank you for your attention!

Feel free to ask me for the manuscript of the full text and scripts for detailed error analysis.

(Petr Lobaz / [email protected])

Double look‑up table method for fast light propagation calculations

double look‑up table method for fast light propagation ... · lobaz / double look‑up table...

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