Parallel Implementation of the 2-D Discrete Wavelet Transform

2. What is the discrete wavelet transform (DWT)
and what is it good for?
Real-world applications
JPEG 2000 (in digital cinema), CineForm (a video codec),
DjVu (an alternative to PDF), edge-avoiding wavelets (Photoshop)

3. Background: 1-D DWT
L
H
=
G1
(o)
G1
(e)
G0
(o)
G0
(e)
L
H
L
H
=
k
1 U(k)
0 1
1 0
P(k) 1
L
H

4. Background: 2-D DWT
x = LL HL LH HH
T
y = LL HL LH HH
T

5. Existing Approaches: Separable Convolution
y = NV
NH
x

6. Existing Approaches: Separable Lifting
TH
P
SH
U
TV
P SV
U
y = SV
U SH
U TV
P TH
P x

7. Our Approach: Non-Separable Lifting
TP for LL TP for HH
SU for LL SU for HH
y = SU TP x

8. Our Approach: Non-Separable Lifting (optimized)
TH
P0
TV
P0
SH
U0
SV
U0
TP1
SU1
y = SV
U0
SH
U0
SU1
TV
P0
TH
P0
TP1
x

9. Properties of OpenCL schemes
for the CDF 9/7 wavelet
separability scheme fences operations
separable convolution 2 56
separable polyconvolution 2K 20
separable lifting 4K 32
non-separable convolution 1 152
non-separable polyconvolution 1K 46
non-separable lifting 2K 36

10. Results
for the CDF 9/7 wavelet
0
10
20
30
40
50
60
70
80
90
100kpel 1Mpel 10Mpel 100Mpel
GB/s
OpenCL (AMD 6970)
0
5
10
15
20
25
30
35
100kpel 1Mpel 10Mpel 100Mpel
GB/s
Pixel Shader (NVIDIA Titan X)
separable lifting
separable polyconvolution
separable convolution
non-separable lifting
non-separable polyconvolution
non-separable convolution

11. Architectures
HLSL OpenCL C (CPU)
input/output off chip off chip off chip
intermediate results off chip on chip on chip
on-chip memory no 32–96 KiB 3–35 MiB
concurrent threads thousands thousands 2–112
4-tuples / thread 1 1–4 thousands
view global block-based block-based
block size global 642 5122