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ABSTRACT

This paper proposes an improved version of lifting based 3D Discrete Wavelet Transform (DWT) VLSI architecture which uses bi-orthogonal 9/7 filter processing. The whole architecture was optimized in efficient pipeline and parallel design way to speed up and achieve higher hardware utilization. The Discrete Wavelet Transform (DWT) was based on time-scale representation, which provides efficient multi-resolution. The lifting based DWT architecture has the advantage of lower computational complexities transforming signals with extension and regular data flow. This is suitable for VLSI implementation. It uses a cascade combination of three 1-D wavelet transform along with a set of in-chip memory buffers between the stages. The discrete wavelet transform (DWT) is being increasingly used for image coding. This is due to the fact that DWT supports features like progressive image transmission (by quality, by resolution), ease of compressed image manipulation, region of interest coding, etc. DWT has traditionally been implemented by convolution. Such an implementation demands both a large number of computations and a large storage features that are not desirable for either high-speed or low-power applications. Recently, a lifting-based scheme that often requires far fewer computations has been proposed for the DWT. The main feature of the lifting based DWT scheme is to break up the high pass and low pass filters into a sequence of upper and lower triangular matrices and convert the filter implementation into banded matrix multiplications. Such a scheme has several advantages, including in-place computation of the DWT, integer-to-integer wavelet transform (IWT), symmetric forward and inverse transform, etc. Therefore, it comes as no surprise that lifting has been chosen in the upcoming.

Keywords Descrete wavelet transform, image compression, lifting, video, VLSI architecture.

I. INTRODUCTION

The fundamental idea behind wavelets is to analyze according to scale. Indeed, some researchers in the wavelet field feel that, by using wavelets, one is adopting a perspective in processing data. Wavelets are functions that satisfy certain mathematical requirements and are used in representing data or other functions. Wavelet algorithms process data at different scales or resolutions. Fourier Transform (FT) with its fast algorithms (FFT) is an important tool for analysis and processing of many natural signals. FT has certain limitations to characterize many natural signals, which are non-stationary (e.g. speech). Though a time varying, overlapping window based FT namely STFT (Short Time FT) is well known for speech processing applications, a time-scale based Wavelet Transform is a powerful mathematical tool for non-stationary signals.

1.1 Introduction to Wavelet Transform:

The wavelet transform is computed separately for different segments of the time-domain signal at different frequencies. Multi-resolution analysis: analyzes the signal at different frequencies giving different resolutions. Multi-resolution analysis is designed to give good time resolution and poor frequency resolution at high frequencies and good frequency resolution and poor time resolution at low frequencies. Good for signal having high frequency components for short durations and low frequency components for long duration, e.g. Images and video frames.

1.1.1 Wavelet Definition A wavelet is a small wave which has its energy concentrated in time. It has an oscillating wavelike characteristic but also has the ability to allow simultaneous time and frequency analysis and it is a suitable tool for transient, non-stationary or time-varying phenomena.

AN EFFICIENT ARCHITECTURE FOR 3-D

LIFTING-BASED DISCRETE WAVELET TRANSFORM

M. JANARDAN, M.TECH (DECS)

Dr. K ASHOK BABU Professor & HOD of (ECE) SRI INDU COLLEGE OF ENGG&TECHNOLOGY

(Affiliated to JNTU, Hyderabad) Ibrahimpatnam Hyderabad, Andhra Pradesh, India -501510

M Janardan et al, Int. J. Comp. Tech. Appl., Vol 2 (5), 1439-1458

IJCTA | SEPT-OCT 2011 Available online@www.ijcta.com

1439

ISSN:2229-6093

(a) (b)

Figure1.1 Representation of a (a) wave (b) wavelet

1.1.2 Wavelet Characteristics The difference between wave (sinusoids) and wavelet is shown in figure 1.1. Waves are smooth, predictable and everlasting, whereas wavelets are of limited duration, irregular and may be asymmetric. Waves are used as deterministic basis functions in Fourier analysis for the expansion of functions (signals), which are time-invariant, or stationary. The important characteristic of wavelets is that they can serve as deterministic or non-deterministic basis for generation and analysis of the most natural signals to provide better time-frequency representation, which is not possible with waves using conventional Fourier analysis.

1.1.3 Wavelet Analysis The wavelet analysis procedure is to adopt a wavelet prototype function, called an analyzing wavelet or mother wavelet. Temporal analysis is performed with a contracted, high frequency version of the prototype wavelet, while frequency analysis is performed with a dilated, low frequency version of the same wavelet. Mathematical formulation of signal expansion using wavelets gives Wavelet Transform (WT) pair, which is analogous to the Fourier Transform (FT) pair. Discrete-time and discrete-parameter version of WT is termed as Discrete Wavelet Transform (DWT).

1.2 Types of Transforms : 1.2.1 Fourier Transform (FT) Fourier transform is a well-known mathematical tool to transform time-domain signal to frequency-domain for efficient extraction of information and it is reversible also. For a signal x(t), the FT is given by

Though FT has a great ability to capture signals frequency content as long as x(t) is composed of few stationary

components (e.g. sine waves). However, any abrupt change in time for non-stationary signal x(t) is spread out over the whole frequency axis in X(f). Hence the time-domain signal sampled with Dirac-delta function is highly localized in time but spills over entire frequency band and vice versa. The limitation of FT is that it cannot offer both time and frequency localization of a signal at the same time. 1.2.2 Short Time Fourier Transform (STFT) To overcome the limitations of the standard FT, Gabor introduced the initial concept of Short Time Fourier Transform (STFT). The advantage of STFT is that it uses an arbitrary but fixed-length window g(t) for analysis, over which the actual non-stationary signal is assumed to be approximately stationary. The STFT decomposes such a pseudo-stationary signal x(t) into a two dimensional time-frequency representation S( , f) using that sliding window g(t) at different times . Thus the FT of windowed signal x(t) g*(t-) yields STFT as

1.2.3 Wavelet Transform (WT) Fixed resolution limitation of STFT can be resolved by letting the resolution in time-frequency plane in order to obtain Multi resolution analysis. The Wavelet Transform (WT) in its continuous (CWT) form provides a flexible time-frequency, which narrows when observing high frequency phenomena and widens when analyzing low frequency behavior. Thus time resolution becomes arbitrarily good at high frequencies, while the frequency resolution becomes arbitrarily good at low frequencies. This kind of analysis is suitable for signals composed of high frequency components with short duration and low frequency components with long duration, which is often the case in practical situations. 1.3 Comparative Visualisation : The time-frequency representation problem is illustrated in figure1.2. A comprehensive visualization of various time-frequency representations shown in figure 1.2, demonstrates the time-frequency resolution for a given signal in various transform domains with their corresponding basis functions.

M Janardan et al, Int. J. Comp. Tech. Appl., Vol 2 (5), 1439-1458

IJCTA | SEPT-OCT 2011 Available online@www.ijcta.com

1440

ISSN:2229-6093

Figure1.2 Comparative visualizations of time-frequency representation of an arbitrary non-stationary signal in various transform

domains

1.4 Difference between Continuous Wavelet Transform and Discrete Wavelet Transform :

Wavelet transforms are classified into discrete wavelet transforms (DWTs) and continuous wavelet transforms (CWTs). Note that both DWT and CWT are continuous-time (analog) transforms. They can be used to represent continuous-time (analog) signals. CWTs operate over every possible scale and translation whereas DWTs use a specific subset of scale and translation values or representation grid. The Wavelet transform is in fact an infinite set of various transforms, depending on the merit function used for its computation. This is the main reason, why we can hear the term "wavelet transform" in very different situations and applications.

Orthogonal wavelets are used to develop the discrete wavelet transform

Non-orthogonal wavelets are used to develop the continuous wavelet transform

1.5 Applications of Discrete Wavelet Transform :

Generally, an approximation to DWT is used for data compression if signal is already sampled, and the CWT for signal analysis. Thus, DWT approximation is commonly used in engineering and computer science, and the CWT in scientific research.

One use of wavelet approximation is in data compression. Like some other transforms, wavelet transforms can be used to transform data and then encode the transformed data, resulting in effective compression. For example, JPEG 2000 is an image compression standard that uses biorthogonal wavelets. A related use is that of smoothing/denoising data