A VLSI Architecture for the 2-D Discrete Wavelet Transform

Zhiyu LiuXin Zhou

May 2004

Motivation

To apply the knowledge learned in ECE734 to practical problems

Popular Methods Folding

Parallel Pipeline

Systolic …

Possible Application Image Processing (Discrete Wavelet Transform) Speech Processing Communication Systems …

Goal

To propose a new VLSI architecture to implement 2-D Discrete Wavelet Transform (DWT).

Based on RPA (Recursive Pyramid Algorithm) Parallel Architecture Systolic Architecture to Deal with Borders

Wavelets: functions that cut up data into different frequency components, and then study each component with a resolution matched to its scale

2

2

2

f(n) 2

)(nh

)(nh

)(ng

)(ng

2

2

)(nh

)(ng

Wavelet Transform

Mallat’s Pyramid Algorithm

Mallat algorithm : Hierarchically perform DWT decomposition and reconstruction.

If the coefficients of two scale equation is looked as filter, then mallat algorithm is in reality two-channel filter banks. In the sense, scale function and wavelet are known as low-pass filter and high-pass filter,

HPF

LPFx(n)

2

2HPF

LPF

2

2HPF

LPF

2

2

y(1,n)y(2,n)

y(3,n)

x(1,n)

Recursive Pyramid Algorithm

RPA is a modification of Mallat’s PA In Mallat’s PA, each level is completely computed

before the next RPA rearranges the order of the N outputs such that

an output is scheduled at the `earliest' instance that it can be scheduled.

The earliest instance is based on the following precedence relation: if the earliest `instance' of the ith octave clashes with that of the (i + 1)th octave, then the ith octave gets scheduled first.

Recursive Pyramid Algorithm(Cont.)

Recursive Pyramid Algorithm(Cont.2)

Proposed Algorithm

Using Parallel Architecture to build up row-based RPA and using the Systolic Architecture to deal with borders

Proposed Algorithm(Cont.)

Row based RPA Intersperse the row operation of various octaves

in between the first octave After each row operation, we proceed the column

operation

Proposed Algorithm(Cont.1)

0 Octave1 Row + Octave1 Column

1 Octave2 Row + Octave2 Column

2 Octave1 Row + Octave1 Column

3 Octave3 Row + Octave3 Column

4 Octave1 Row + Octave1 Column

5 Octave2 Row + Octave2 Column

6 Octave1 Row + Octave1 Column

… …

Proposed Algorithm(Cont.2)--- Hardware Implementation Architecture

Memory

PEs

Memory

PEsCoefficients

Memory

PEs

Memory

PEs

Proposed Algorithm(Cont.3)-- Dealing with Borders Decomposition of Periodic Extension RPA

Reconstruction of Periodic Extension RPA

1

10

0

(2 1 ) ( ) 12

( )

( ) 22 2 2

L

j jm

j

j j j

Na i L m h m i

a iN N N

a i i L

0

( ) 1( )

( ) 2

X i i Na i

X i N N i N L

/ 2 1

1 1/ 2 ( 2) / 2

2( ) ( ) (2 ) ( ) (2 ) 1

2

i

j j jm i L

La i a m h m L i d m g m L i i

1 1 1( ) ( ) 0 log 1;0 1j j j ja i a N i j N i N

Proposed Algorithm(Cont.4)

Conclusion

Parallel To speed up the chip

Periodic extension Implement the perfect reconstruction Compute coefficients.

Future work Theoretic analysis