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TitleLifting Scheme Cores for Wavelet Transform
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Table of Contents
Discrete Wavelet Transform
	Lifting Scheme
	2-D Decomposition
	Non-Separable Lifting Scheme
	Capabilities of Lifting Scheme
Computation Schedules
	Field-Programmable Gate Arrays
	Graphics Processing Units
Lifting Vectorization
	Horizontal Vectorization
	Vertical Vectorization
	Diagonal Vectorization
Lifting Core
	Core Reorganization
	Treatment of Signal Boundaries
	Lifting Scheme Choice
	Parallel Cores
Multi-Dimensional Cores
	2-D Core Reorganization
	Parallel 2-D Cores
	Extension to Multiple Dimensions
	Image Processing
	JPEG 2000
	3-D Decomposition
	Parallel Processing on GPU
	FPGA Devices
Document Text Contents
Page 1

Lifting Scheme Cores for Wavelet Transform
Jádra schématu lifting pro vlnkovou transformaci

Ph.D. thesis / dizertační práce

Ing. David Bařina

supervised by / školitel
prof. Dr. Ing. Pavel Zemčík

Brno, 2015

Page 2


The thesis focuses on efficient computation of the two-dimensional discrete wavelet trans-
form. The state-of-the-art methods are extended in several ways to perform the transform
in a single loop, possibly in multi-scale fashion, using a compact streaming core. This
core can further be appropriately reorganized to target the minimization of certain plat-
form resources. The approach presented here nicely fits into common SIMD extensions,
exploits the cache hierarchy of modern general-purpose processors, and is suitable for
parallel evaluation. Finally, the approach presented is incorporated into the JPEG 2000
compression chain, in which it has proved to be fundamentally faster than widely used


Práce se zaměřuje na efektivní výpočet dvourozměrné diskrétní vlnkové transformace.
Současné metody jsou v práci rozšířeny v několika směrech a to tak, aby spočetly tuto
transformaci v jediném průchodu, a to případně víceúrovňově, použitím kompaktního
jádra. Tohle jádro dále může být vhodně přeorganizováno za účelem minimalizace užití
některých prostředků. Představený přístup krásně zapadá do běžně používaných rozšíření
SIMD, využívá hierarchii cache pamětí moderních procesorů a je vhodný k paralelnímu
výpočtu. Prezentovaný přístup je nakonec začleňen do kompresního řetězce formátu
JPEG 2000, ve kterém se ukázal být zásadně rychlejší než široce používané implementace.


discrete wavelet transform, lifting scheme, Cohen-Daubechies-Feauveau wavelet, SIMD,
CPU cache, parallelization, JPEG 2000

Klíčová slova

diskrétní vlnková transformace, schéma lifting, vlnka Cohen-Daubechies-Feauveau, SIMD,
cache CPU, paralelizace, JPEG 2000


Please cite this work as: D. Barina. Lifting Scheme Cores for Wavelet Transform. PhD
thesis, Brno University of Technology, Brno, 2015.

Page 59


1 2 3
a dh v

Figure 6.5: Proposed non-separable lifting core of CDF 5/3 with three stages. The input
coefficients of active core are in bright box. The output ones are in dark one.

d v

h a

d v

h a

d v

h a

d v

h a

d v

h a

d v

h a

d v

h a

d v

h a

d v

h a

d v

h a

d v

h a

d v

h a

Figure 6.6: Shapes of the individual lifting steps of the newly proposed CDF 5/3 scheme
with three steps.

calculation of these coefficients is subdivided into separate parts. The sum of these parts
gives the desired result.

I will now explain this approach in detail. The starting point of the solution is the non-
separable lifting scheme of CDF 5/3 transform as described in [15]. The operations inside
the core were further appropriately reorganized. After some rewriting of expressions,
the following scheme has appeared. First, the scheme requires only 8 coefficients to
be passed between core iterations. This means that this new scheme has the same
memory demands as in the original separable case. Second, the number of the non-trivial
arithmetic operations was reduced from 24 to 22 compared to the original non-separable
scheme. The trick is that the h, v and a coefficients are not calculated at once. Instead,
these are calculated separately, split in two parts each.

For better understanding, the new scheme is graphically illustrated in Figure 6.5. The


, a

, v

) → (v(0)
m−1,m, a

m−1,n, v

m−1,n) coefficients have to be passed through the

buffer horizontally. Additional (h(0)

, a

, h

) → (h(0)
m,n−1, a

m,n−1, h

m,n−1) coefficients

have to be passed vertically. Lastly, (a(0)

, a

) → (a(0)
m−1,n−1, a

m−1,n−1) coefficients

have to be passed in diagonal direction. This gives 8 coefficients per core iteration in
total. It should be emphasized that this newly formed scheme cannot be derived using
instruments in [15]. This is caused by the fact that the authors of [15] do not specify

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a sequence of the operations. For comparison, shapes of the individual lifting steps are
shown in Figure 6.6.

Using the matrix notation, the proposed transform core can be described as

y = Aβ Tα,β Dα x, (6.14)


x =
am−1,n−1 am,n−1 am−1,n am,n hm,n−1 hm,n vm−1,n vm,n dm,n

y =

am−1,n−1 am,n−1 am−1,n am,n hm,n−1 hm,n vm−1,n vm,n dm,n

]T (6.15)

Page 118


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