The Design of 3-Dimensional Discrete Wavelet Transform for Image Processing Applications

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The Design of 3-Dimensional Discrete Wavelet

Transform for Image Processing Applications

G. Ravi

Assistant Professor Sagar Institute of Technology

K. Srishylam

Assistant Professor Sagar Institute of Technology

G. Sunil Kumar

Assistant Professor Trinity College of Engineering

Abstract - The digital data can be transformed using

Discrete Wavelet Transform (DWT). The Discrete Wavelet Transform (DWT) was based on time-scale representation, which provides efficient multi-resolution. The lifting based scheme (9, 7) (Here 9 Low Pass filter coefficients and the 7 High Pass filter coefficients) filter give lossy mode of information. The lifting based DWT are lower computational complexity and reduced memory requirements. Since Conventional convolution based DWT is area and power hungry which can be overcome by using the lifting based scheme.

Keywords – Discrete Wavelet Transform (DWT), Image Processing Applications, Filter Coefficients.

I. I

NTRODUCTION

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. This idea is not new. Approximation using superposition of functions has existed since the early 1800's, when Joseph Fourier discovered that he could superpose sines and cosines to represent other functions. However, in wavelet analysis, the scale that we use to look at data plays a special role. 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.Wavelet Transform uses a set of damped oscillating functions known as wavelet basis. WT in its continuous (analog) form is represented as CWT. CWT with various deterministic or non-deterministic basis is a more effective representation of signals for analysis as well as characterization. Continuous wavelet transform is powerful in singularity detection. A discrete and fast implementation of CWT (generally with real valued basis) is known as the standard DWT (Discrete Wavelet Transform).With standard DWT, signal has a same data size in transform domain and therefore it is a non-redundant transform. A very important property was Multi-resolution Analysis (MRA) allows DWT to view and process.

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.

Why wavelet

Fourier transform based spectral analysis is the dominant analytical tool for frequency domain analysis. However, Fourier transform cannot provide any information of the spectrum changes with respect to time. Fourier transform assumes the signal is stationary, but PD signal is always non-stationary. To overcome this deficiency, a modified method-short time Fourier transform allows representing the signal in both time and frequency domain through time windowing functions. The window length determines a constant time and frequency resolution. Thus, a shorter time windowing is used in order to capture the transient behavior of a signal; we sacrifice the frequency resolution. The nature of the real PD signals is non-periodic and transient as shown in such signals cannot easily be analyzed by conventional transforms. So, an alternative mathematical tool- wavelet transform must be selected to extract the relevant time-amplitude information from a signal. In the meantime, we can improve the signal to noise ratio based on prior knowledge of the signal characteristics. A continuous-time wavelet transform of f(t) is defined.

Here a, b R, a≠ 0 and they are dilating and translating

coefficients, respectively. The asterisk denotes a complex conjugate. This multiplication of [ a ]^1/2 is for energy normalization purposes so that the transformed signal will have the same energy at every scale. The analysis function

ψ (t) , the so-called mother wavelet, is scaled by a, so a wavelet analysis is often called a time-scale analysis rather than a time-frequency analysis. The wavelet transform decomposes the signal into different scales with different levels of resolution by dilating a single prototype function, the mother wavelet. Furthermore, a mother wavelet has to satisfy that it has a zero net area, which suggest that the transformation kernel of the wavelet transform is a compactly support function (localized in time), thereby offering the potential to capture the PD spikes which normally occur in a short period of time.

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The Continuous Wavelet Transform (CWT) is provided below, where x(t) is the signal to be analyzed. ψ(t) is the

mother wavelet or the basis function. All the wavelet functions used in the transformation are derived from the mother wavelet through translation (shifting) and scaling (dilation or compression).

The mother wavelet used to generate all the basis functions is designed based on some desired characteristics associated with that function. The

translation parameter τrelates to the location of the wavelet function as it is shifted through the signal. Thus, it corresponds to the time information in the Wavelet Transform. The scale parameter s is defined as |1/frequency| and corresponds to frequency information. Scaling either dilates (expands) or compresses a signal. Large scales (low frequencies) dilate the signal and provide detailed information hidden in the signal, while small scales (high frequencies) compress the signal and provide global information about the signal. Notice that the Wavelet Transform merely performs the convolution operation of the signal and the basis function. The above analysis becomes very useful as in most practical applications, high frequencies (low scales) do not last for a long duration, but instead, appear as short bursts, while low frequencies (high scales) usually last for entire duration of the signal.

The Wavelet Series is obtained by discretizing CWT. This aids in computation of CWT using computers and is obtained by sampling the time-scale plane. The sampling rate can be changed accordingly with scale change without violating the Nyquist criterion. Nyquist criterion states that, the minimum sampling rate that allows reconstruction

of the original signal is 2ω radians, where ω is the highest

frequency in the signal. Therefore, as the scale goes higher (lower frequencies), the sampling rate can be decreased thus reducing the number of computations.

II. W

ORKING

P

RINCIPLE

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.

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).

Types of Transforms

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 signal’s

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.

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(τ , f) using that sliding window g(t) at different times τ . Thus the FT of windowed signal

x(t) g*(t-τ) yields STFT.The time-frequency resolution is fixed over the entire time-frequency plane because the same window is used at all frequencies. There is always a trade off between time resolution and frequency resolution in STFT.

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 behaviour. 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 situation.

Comparative Visualisation

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III. S

IMULATION

R

ESULTS

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Simulation Result of DWT-2 Block with Both High and Low Pass Coefficients

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Simulation Result of 3D-DWT(TOP MODULE) Block with High and Low Pass Coefficients LLL, LLH, LHL, LHH, HLL, HLH, HHL and HHH.

IV. C

ONCLUSION

Basically the medical images need more accuracy without loosing of information. The Discrete Wavelet Transform (DWT) was based on time-scale representation,

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coefficients of the input image. The simulation results of DWT were verified with the appropriate test cases.Once the functional verification is done.

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A

UTHOR

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S

P

ROFILE

K. Srishylam

Received AMIE in 2007 from the Institution of Engineers (INDIA), M.Tech. from JNTU (Jawaharlal Nehru Technical University). His areas of interest includes very large scale integration, Astro physics, Micro electronics, Low power VLSI. Presently he is working as a Asst. Prof. in SITECH (Sagar Institute of Technology), Hyderabad. He guided many research projects in VLSI area. Currently he is doing research on implementing low power VLSI techniques in VMFU, FIR filters.

G. Ravi

Received B.E. in 2007 from Deccan College Of Engineering (Osmania University) and M.Tech. from Shadan College Of Engineering and Technology (Jawaharlal Nehru Technical University), Hyderabad. His areas of interest includes very large scale integration, Micro electronics, Low power VLSI. He guided many research projects for Engineering students .He is working as Assistant Professor in Sagar Institute of Technology (SITECH) Hyderabad.