Parallel Wavelet Schemes for Images

Parallel Wavelet Schemes for Images

Filter Bank

H̃(z−1) a

d

↓ 2

+

G̃(z−1) ↓ 2

↑ 2 H(z)

↑ 2 G(z)

Lifting Scheme

a

d

split P̃ (z−1)T P (z) merge

P̃ (z)T =∏

K>k>0

[1 U (k)(z)0 1

] [1 0

P (k)(z) 1

]

CDF 9/7 Wavelet

α

β

γ

δ

input

output

steps

even samples

odd samples

P̃ (z) =

[1 α

(1 + z−1

)0 1

] [1 0

β (1 + z) 1

] [1 γ

(1 + z−1

)0 1

] [1 0

δ (1 + z) 1

] [ζ 00 1/ζ

]

2-D Discrete Wavelet Transform

Objective

→ →

State of the Art: Separable Convolution

NH

NH

NV NV

y = NV |NH | x

State of the Art: Separable Lifting

THP TV

P SHU SV

U

y = SVU∣∣ SHU ∣∣TVP ∣∣THP ∣∣ x

Our Solution: Separable Polyconvolution

NH

NV NH NV

y = NVP,U |NHP,U | x

Our Solution: Non-Separable Convolution

N N

y = N∣∣ x

Our Solution: Non-Separable Convolution

N N

y = N∣∣ x

Our Solution: Non-Separable Polyconvolution

NP,U NP,U

y = NP,U

∣∣ x

Our Solution: Non-Separable Polyconvolution

NP,U NP,U

y = NP,U

∣∣ x

Our Solution: Non-Separable Lifting

TP TP SU SU

y = SU∣∣TP

∣∣ x

Our Solution: Non-Separable Implosion

TIP RI

P,U SIU

y = SIU∣∣RIP,U ∣∣TIP ∣∣ x

Our Solution: Non-Separable Explosion

TEP RE

P,U SEU

y = SEU∣∣REP,U ∣∣TEP ∣∣ x

Platforms

feature shaders OpenCL

results exchanged off-chip memory on-chip memorypassing in registers no yesSIMD available no yes, SIMD-32transform view global view block-based

Results: Properties

Table : Arithmetic Operations / Steps for CDF 9/7

separability scheme shaders OpenCL

separable convolution 58 / 2 56 / 1separable polyconvolution 42 / 4 20 / 2separable lifting 32 / 8 32 / 4

non-separable convolution 200 / 1 152 / 1non-separable polyconvolution 62 / 2 46 / 2non-separable lifting 36 / 4 36 / 4non-separable implosion 39 / 6 36 / 6non-separable explosion 37 / 6 36 / 6

Results: Performance in Shaders

2

3

4

5

6

7

8

9

10

11

12

13

10.0kpx 100.0kpx 1.0Mpx 10.0Mpx 100.0Mpx 1.0Gpx

GB

/s

Separable LiftingNon-Separable Lifting

Non-Separable ImplosionNon-Separable Explosion

Non-Separable Polyconv.Separable Polyconv.

Non-Separable ConvolutionSeparable Convolution