The two-dimensional discrete wavelet transform (DWT) can be applied in the heart of many image-processing algorithms. Until recently, several studies have compared the performance of such transform on parallel architectures, for example, on graphics processing units (GPUs). All these studies however considered only separable calculation schedules.

1. Parallel Wavelet Schemes for Images

2. Filter Bank
˜H(z−1
) a
d
↓ 2
+
˜G(z−1
) ↓ 2
↑ 2 H(z)
↑ 2 G(z)

3. Lifting Scheme
a
d
split ˜P(z−1
)T
P(z) merge
˜P(z)T
=
K>k>0
1 U(k)(z)
0 1
1 0
P(k)(z) 1

4. CDF 9/7 Wavelet
α
β
γ
δ
input
output
steps
even samples
odd samples
˜P(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 0
0 1/ζ

5. 2-D Discrete Wavelet Transform

6. Objective
→ →

7. State of the Art: Separable Convolution
NH
NH
NV
NV
y = NV
| NH
| x

8. State of the Art: Separable Lifting
TH
P TV
P SH
U SV
U
y = SV
U SH
U TV
P TH
P x

9. Our Solution: Separable Polyconvolution
NH
NV
NH
NV
y = NV
P,U | NH
P,U | x

10. Our Solution: Non-Separable Convolution
N N
y = N x

11. Our Solution: Non-Separable Convolution
N N
y = N x

12. Our Solution: Non-Separable Polyconvolution
NP,U NP,U
y = NP,U x

13. Our Solution: Non-Separable Polyconvolution
NP,U NP,U
y = NP,U x

14. Our Solution: Non-Separable Lifting
TP TP SU SU
y = SU TP x

15. Our Solution: Non-Separable Implosion
TI
P RI
P,U SI
U
y = SI
U RI
P,U TI
P x

16. Our Solution: Non-Separable Explosion
TE
P RE
P,U SE
U
y = SE
U RE
P,U TE
P x

17. Platforms
feature shaders OpenCL
results exchanged off-chip memory on-chip memory
passing in registers no yes
SIMD available no yes, SIMD-32
transform view global view block-based

18. Results: Properties
Table : Arithmetic Operations / Steps for CDF 9/7
separability scheme shaders OpenCL
separable convolution 58 / 2 56 / 1
separable polyconvolution 42 / 4 20 / 2
separable lifting 32 / 8 32 / 4
non-separable convolution 200 / 1 152 / 1
non-separable polyconvolution 62 / 2 46 / 2
non-separable lifting 36 / 4 36 / 4
non-separable implosion 39 / 6 36 / 6
non-separable explosion 37 / 6 36 / 6

19. 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 Lifting
Non-Separable Lifting
Non-Separable Implosion
Non-Separable Explosion
Non-Separable Polyconv.
Separable Polyconv.
Non-Separable Convolution
Separable Convolution