Vectorization of the 2D Wavelet Lifting Transform Using SIMD Extensions

1
Vectorization of the 2D Wavelet Lifting Transform
Using SIMD Extensions
D. Chaver, C. Tenllado, L. Piñuel, M. Prieto,
F. Tirado
2
Index

1. Motivation
2. Experimental environment
3. Lifting Transform
4. Memory hierarchy exploitation
5. SIMD optimization
6. Conclusions
7. Future work

3
Motivation
4
Motivation

• Applications based on the Wavelet Transform
• JPEG-2000 MPEG-4
• Usage of the lifting scheme
• Study based on a modern general purpose
  microprocessor
• Pentium 4
• Objectives
• Efficient exploitation of Memory Hierarchy
• Use of the SIMD ISA extensions

5
Experimental Environment
6
Experimental Environment
Intel Pentium4 (2,4 GHz)
Platform
DFI WT70-EC
Motherboard
NA
IL1
Cache
DL1
8 KB, 64 Byte/Line, Write-Through
512 KB, 128 Byte/Line
L2
1 GB RDRAM (PC800)
Memory
Operating System
RedHat Distribution 7.2 (Enigma)
Intel ICC compiler GCC compiler
Compiler
7
Lifting Transform
8
Lifting Transform
Original element
1st step
A
D
A
D
D
D
A
A
1st
1st
1st
1st
1st
1st
1st
1st
2nd step
9
Lifting Transform
Original element
Approximation
Horizontal Filtering (1D Lifting Transform)
1 Level
N Levels
Vertical Filtering (1D Lifting Transform)
10
Lifting Transform
Horizontal Filtering
Vertical Filtering
11
Memory Hierarchy Exploitation
12
Memory Hierarchy Exploitation

• Poor data locality of one component (canonical
  layouts)
• E.g. column-major layout ? processing image
  rows (Horizontal Filtering)

• Aggregation (loop tiling)

• Poor data locality of the whole transform

• Other layouts

13
Memory Hierarchy Exploitation
Horizontal Filtering
Vertical Filtering
14
Memory Hierarchy Exploitation
Aggregation
Horizontal Filtering
IMAGE
15
Memory Hierarchy Exploitation

• INPLACE
• Common implementation of the transform
• Memory Only requires the original matrix
• For most applications needs post-processing
• MALLAT
• Memory requires 2 matrices
• Stores the image in the expected order
• INPLACE-MALLAT
• Memory requires 2 matrices
• Stores the image in the expected order

Different studied schemes
16
Memory Hierarchy Exploitation
O
O
O
O
MATRIX 1
INPLACE
O
O
O
O
O
O
O
O
O
O
O
O
logical view
physical view
17
Memory Hierarchy Exploitation
O
O
O
O
MATRIX 1
MATRIX 2
MALLAT
O
O
O
O
O
O
O
O
O
O
O
O
logical view
physical view
18
Memory Hierarchy Exploitation
O
O
O
O
INPLACE- MALLAT
O
O
O
O
MATRIX 2
MATRIX 1
O
O
O
O
O
O
O
O
logical view
physical view
19
Memory Hierarchy Exploitation

• Execution time breakdown for several sizes
  comparing both compilers.
• I, IM and M denote inplace, inplace-mallat, and
  mallat strategies respectively.
• Each bar shows the execution time of each level
  and the post-processing step.

20
Memory Hierarchy Exploitation
CONCLUSIONS

• The Mallat and Inplace-Mallat approaches
  outperform the Inplace approach for levels 2 and
  above
• These 2 approaches have a noticeable slowdown for
  the 1st level
• Larger working set
• More complex access pattern
• The Inplace-Mallat version achieves the best
  execution time
• ICC compiler outperforms GCC for Mallat and
  Inplace-Mallat, but not for the Inplace approach

21
SIMD Optimization
22
SIMD Optimization

• Objective Extract the parallelism available on
  the Lifting Transform
• Different strategies
• Semi-automatic vectorization
• Hand-coded vectorization
• Only the horizontal filtering of the transform
  can be semi-automatically vectorized (when using
  a column-major layout)

23
SIMD Optimization

• Automatic Vectorization (Intel C/C Compiler)
• Inner loops
• Simple array index manipulation
• Iterate over contiguous memory locations
• Global variables avoided
• Pointer disambiguation if pointers are employed

24
SIMD Optimization
Original element
1st step
D
A
1st
1st
2nd step
25
SIMD Optimization
Vectorial Horizontal filtering
Horizontal filtering
a
x

a
a
x

a

a
Column-major layout

26
SIMD Optimization
Vectorial Vertical filtering
Vertical filtering
a
x

a
a
x

a

a
Column-major layout

27
SIMD Optimization
Horizontal Vectorial Filtering (semi-automatic)
for(j2,k1jlt(columns-4)j2,k) pragma
vector aligned for(i0iltrowsi)
/ 1st operation / col3col3 alfa(
col4 col2) / 2nd operation /
col2col2 beta( col3 col1) / 3rd
operation / col1col1 gama( col2
col0) / 4th operation / col0 col0
delt( col1 col-1) / Last step /
detail col1 phi_inv aprox col0
phi
28
SIMD Optimization

• Hand-coded Vectorization
• SIMD parallelism has to be explicitly expressed
• Intrinsics allow more flexibility
• Possibility to also vectorize the vertical
  filtering

29
SIMD Optimization
Horizontal Vectorial Filtering (hand)
/ 1st operation / t2 _mm_load_ps(col2) t4
_mm_load_ps(col4) t3 _mm_load_ps(col3) coeff
_mm_set_ps1(alfa) t4 _mm_add_ps(t2,t4) t4
_mm_mul_ps(t4,coeff) t3 _mm_add_ps(t4,t3) _mm_
store_ps(col3,t3) / 2nd operation / / 3rd
operation / / 4th operation / / Last step
/ _mm_store_ps(detail,t1) _mm_store_ps(aprox,t0)

a
a
x

a
a

t2
t3
t4
30
SIMD Optimization

• Execution time breakdown of the horizontal
  filtering (10242 pixels image).
• I, IM and M denote inplace, inplace-mallat and
  mallat approaches.
• S, A and H denote scalar, automatic-vectorized
  and hand-coded-vectorized.

31
SIMD Optimization
CONCLUSIONS

• Speedup between 4 and 6 depending on the
  strategy. The reason for such a high improvement
  is due not only to the vectorial computations,
  but also to a considerable reduction in the
  memory accesses.
• The speedups achieved by the strategies with
  recursive layouts (i.e. inplace-mallat and
  mallat) are higher than the inplace version
  counterparts, since the computation on the latter
  can only be vectorized in the first level.
• For ICC, both vectorization approaches (i.e.
  automatic and hand-tuned) produce similar
  speedups, which highlights the quality of the ICC
  vectorizer.

32
SIMD Optimization

• Execution time breakdown of the whole transform
  (10242 pixels image).
• I, IM and M denote inplace, inplace-mallat and
  mallat approaches.
• S, A and H denote scalar, automatic-vectorized
  and hand-coded-vectorized.

33
SIMD Optimization
CONCLUSIONS

• Speedup between 1,5 and 2 depending on the
  strategy.
• For ICC the shortest execution time is reached by
  the mallat version.
• When using GCC both recursive-layout strategies
  obtain similar results.

34
SIMD Optimization

• Speedup achieved by the different vectorial codes
  over the inplace-mallat and inplace.
• We show the hand-coded ICC, the automatic ICC,
  and the hand-coded GCC.

35
SIMD Optimization
CONCLUSIONS

• The speedup grows with the image size since.
• On average, the speedup is about 1.8 over the
  inplace-mallat scheme, growing to about 2 when
  considering it over the inplace strategy.
• Focusing on the compilers, ICC clearly
  outperforms GCC by a significant 20-25 for all
  the image sizes

36
Conclusions
37
Conclusions

• Scalar version We have introduced a new scheme
  called Inplace-Mallat, that outperforms both the
  Inplace implementation and the Mallat scheme.
• SIMD exploitation Code modifications for the
  vectorial processing of the lifting algorithm.
  Two different methodologies with ICC compiler
  semi-automatic and intrinsic-based
  vectorizations. Both provide similar results.
• Speedup Horizontal filtering about 4-6
  (vectorization also reduces the pressure on the
  memory system).
• Whole transform around 2.
• The vectorial Mallat approach outperforms the
  other schemes and exhibits a better scalability.
• Most of our insights are compiler independent.

38
Future work
39
Future work

• 4D layout for a lifting-based scheme
• Measurements using other platforms
• Intel Itanium
• Intel Pentium-4 with hiperthreading
• Parallelization using OpenMP (SMT)

For additional information http//www.dacya.ucm.
es/dchaver