CURVED WAVELET TRANSFORM FOR IMAGE

DENOISING

A MINOR PROJECT PRESENTATION PREPARED BY-

NIKHIL KUMAR-0511EC111056

AMAN PRAKASH-0511EC111007

Guided By –Prof. Rajesh Kumar Rai

CONTENTS….

• Introduction.

• Fourier analysis.

• Wavelet analysis.

• Wavelet analysis cont.…..

• Algorithm for image denoising.

• Illustrations.

Software Detail MATLAB R2012a Version (7.14.0.739) 64 Bit License no.-161052

INTRODUCTION

• Image denoising refers to the recovery of a digital image that has been

contaminated by additive white Gaussian noise (AWGN).

• Wavelet transform enable us to represent signals with a high degree of

scarcity. This is the principle behind a non-linear wavelet based signal

estimation technique known as wavelet denoising.

• Curved wavelet transform is a new multi-scale representation most

suitable for objects with curves.

• It developed by candès and donoho in 1999. This technique is Still not

fully matured but seems promising however.

FOURIER ANALYSIS

• Breaks down a signal into constituent

sinusoids of different frequencies. In

other words: transform the view of the

signal from time-base to frequency-

base.

• By using Fourier transform , we loose

the time information : when did a

particular event take place. FT can not

locate drift, trends, abrupt changes,

beginning and ends of events, etc.

Calculating use complex numbers.

WAVELET ANALYSIS

• A wavelet is a waveform of effectively

limited duration that has an average

value of zero.

• The DWT is identical to a hierarchical

sub band system. In DWT ,the original

image is transformed into different level

say four pieces which is normally

labelled as A1,H1,V1 and D1.The A1

sub-band called the approximation, can

be further decomposed into four sub-

bands. The remaining bands are called

detailed components.

WAVELET ANALYSIS CONT.…..• The image de-noising is the process to remove the noise from the image naturally corrupted by the noise.

The wavelet method is one among. The wavelet techniques are very effective to remove the noise because

of its ability to capture the energy of a signal in few energy Transform values. The wavelet methods are

based on shrinking the wavelet Coefficients in the wavelet domain. The objective is to remove the noise

without affecting the important feature of the image. The most commonly used procedure to remove the

noise is wavelet shrinkage by non-linear method proposed by donoho and Johnston (1994, 1995). In

Statistical context this can be referred as the estimation of the true curve from the Data contaminated with

the noise usually assume to be Gaussian noise. The estimation of the true curve involves three steps.

• Apply DWT which transforms the discrete data from time domain into time-frequency Domain. The

values of the transformed data in time-frequency domain are called the coefficients. The coefficients

with small absolute values dominated by noise, While the coefficients with large absolute values carry

more data information than Noise.

• In the second step the wavelet coefficient are set to zero (hard threshold Rule) or shrink (soft threshold

rule), if they are not crossing certain threshold Level.

• The last step is to reconstruct the signal from the resultant coefficient using IDWT.

• The simplest example of wavelet basis is haar basis (haar, 1910) which uses scaling function and

mother wavelet given by.

• In case of two dimension, the scaling function and the wavelets are defined as follows

where s = h; v; d are horizontal, vertical and diagonal details respectively definedas

ALGORITHM FOR DENOISING

• Open Matlab and in command window type the function wavemenu.

• Select wavelet 2-D from the wavelet toolbox menu.

• Load the image in the wavelet 2-D window.

• Select Haar wavelet and set the decomposition level to 5 and analyse the image.

• Compress the image using level thresholding by thresholding at Scare high.

• Then denoised the image by Penalizing the image at high threshold level.

• Wavemenu : Wavemenu opens a menu for accessing the various graphical tools provided in the

Wavelet Toolbox™ software.

• Wavelet 2-D: Wavelet Toolbox™ provides wavelet 2-D functions and an app for developing

wavelet-based algorithms for the analysis, synthesis, denoising, and compression of signals and

images.

• Haar wavelet: The Haar transform is the simplest orthogonal wavelet transform. It is computed by

iterating difference and averaging between odd and even samples of the signal.

• Decomposition level: Iterating the decomposition process, breaks the input signal into many lower-

resolution components: Wavelet decomposition tree or Square wavelet Decomposition.

• Thresholding: Image thresholding is a simple, yet effective, way of partitioning an image into a

foreground and background. Image thresholding is most effective in images with high levels of

contrast. Common image thresholding algorithms include histogram and multi-level thresholding.

• Thresholding at Scare high: This is basically a level of thresholding to compress the image and

for smoothening of image.

• Penalizing: Threshold is obtained by a wavelet packet coefficients selection rule using a

penalization method provided by Birge-Massart.

ILLUSTRATIONS…..

AIM

Denoised Image

THANKS