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DENOISING OF IMAGE USING DENOISING OF IMAGE USING WAVELETSWAVELETS
By- Asim Sagheer, Ravi Bhushan, Qutub Zeeshan
CONTENTS INTRODUCTION WHY WAVELET TRANSFORM WAVELETS CONTINUOUS WAVELET TRANSFORM DISCRETE WAVELET TRANSFORM ANALYSISAPPLICATIONS CONCLUSION
INTRODUCTION WHAT IS TRANSFORM?
- Transform of a signal is just another form of representing the signal. It does not change the information content present.
WHY TRANSFORM?- Mathematical transform are applied to signal to obtain further information which is not present in raw signal
WHY WAVELET TRANSFORMFOURIER TRANSFORMSHORT TIME FOURIER TRANSFORM WAVELET TRANSFORM
FOURIER TRANSFORM:Fourier Transform of a time domain signal gives frequency domain representation.
LIMITATION OF FOURIER TRANSFORM:When we are in time domain fourier transform will not give information regarding frequency and when we are in frequency domain it will not provide information regarding time.
ULTIMATE SOLUTION:
WAVELET TRANSFORM Wavelet transform provides time frequency
representation simultaneously.
It provides variable resolution as follows:“At high frequency wavelet transform gives good time
resolution and poor frequency resolution”
“At low frequency wavelet transform gives good frequency resolution and poor time resolution”.
SHORT TIME FOURIER TRANSFORM Short time fourier transform provides time frequency representation of a signal.
UNCERTAINITY PRINCIPLE “Which states that we cannot exactly know what frequency exist at what time instance but we can know only what frequency band exists at what time.”
DECOMPOSITION OF SIGNAL DECOMPOSITION OF SIGNAL
0-500 Hz
500-1000Hz
250-500 Hz0-250 Hz
0-125 Hz 125-250 Hz
WAVE
Demonstration of wave
A wave is a periodic oscillating function that travels through space and matter accompanied by a transfer of energy.
Wavelets are localized waves they have finite energy. They are suited for analysis of transient signal.
WAVELETS
Image PyramidsApproximation pyramidsPredictive residual pyramids
MRA
10N*N
N/2*N/2
N/4*N/4
N/8*N/8
N*N
N/2*N/2
N/4*N/4
NN/8*N/8/8*N/8
Noise ModelImage Noising:- Image noise is the random variation of brightness or
color information in images produced by the sensor and circuitry of a scanner or digital camera. Image noise can also originate in film grain and in the unavoidable shot noise of an ideal photon detector
Types of Noising ModelAmplifier Noise(Gaussian Noise)Salt and paper NoiseShot Noise(Poisson Noise)Speckle Noise
February 14, 201513
Denoising DenosingDenosing is the process with which we reconstruct a signal
from a noisy one.
original
denoised
Denoising
"Before" and "after" illustrations of a nuclear magnetic resonance signal.
Denoising an image
The top left image is the original. At top right is a close-up image of her left eye. At bottom left is a close-up image with noise added. At bottom right is a close-up image, denoised.
BLOCK DIAGRAM
PROPERTIES OF WAVELETS:Consider a real or complex value continuous time
function (t) with the following properties
---- (1)In equation (1) ( ) stands for Fourier transform of (t) .
The admissibility condition implies that the Fourier transform of (t) vanishes at the zero frequency i.e
A zero at the zero frequency also means that the average value of the wavelet in the time domain must be zero
(t) must be oscillatory. In other words (t) must be a wave.
Shifting operation gives time represntation of the spectral component.
Scaling operation gives frequency.
WAVELET FAMILIES
(a) Haar Wavelet (b) Daubechies4 Wavelet
(c)Coiflet1 Wavelet
(d) Symlet2 Wavelet (e) MexicanHat Wavelet
(f) Meyer Wavelet (g) Morlet Wavelet
THE CONTINUOUS WAVELET TRANSFORM
where * denotes complex conjugation of f(t) is the signal to be analyzed
S is the scaling factor
is the translation factor
Inverse wavelet transform is given by
DISCRETE WAVELET TRANSFORMSUB BAND CODING
MULTIRESOLUTION ANALYSIS USING FILTER BANK
Three-level wavelet decomposition tree
Three-level wavelet reconstruction tree.
CONDITION FOR PERFECT RECONSTRUCTIONTo achieve perfect reconstruction analysis and synthesis filter have to satisfy following conditions:G0 (-z) G1 (z) + H0 (-z). H1 (z) = 0 -------- (1)
G0 (z) G1 (z) + H0 (z). H1 (z) = 2z-d ------- (2)
Where G0(z) be the low pass analysis filter,
G1(z) be the low pass synthesis filter,
H0(z) be the high pass analysis filter,
H1(z) be the high pass synthesis filter.
First condition implies that reconstruction is aliasing free
Second condition implies that amplitude distortion has amplitude of unity
1D fast wavelet transforms Due to the separable properties, we can apply 1D FWT.
DWT IN 1D
26
[1]
2D fast wavelet transforms Due to the separable properties, we can apply 1D FWT to
do 2D DWTs.
DWT IN 2D
27
[1]
DWT IN 1D
– An example
DWT IN 2D
LL LH
HL HH
ANALYSIS The PSNR block computes the peak signal-to-noise ratio, in decibels, between two images. This ratio is often used as a
quality measurement between the original and a compressed image. The higher the PSNR, the better the
quality of the compressed, or reconstructed image.The Mean Square Error (MSE) and the Peak Signal to Noise
Ratio (PSNR) are the two error metrics used to compare image compression quality. The MSE represents the
cumulative squared error between the compressed and the original image, whereas PSNR represents a measure of the
peak error. The lower the value of MSE, the lower the error.
To compute the PSNR, the block first calculates the mean-squared error using the following equation:
In the previous equation, M and N are the number of rows and columns in the input images, respectively. Then the block computes the PSNR using the following equation:
In the previous equation, R is the maximum fluctuation in the input image data type. For example, if the input image has a double-precision floating-point data type, then R is 1. If it has an 8-bit unsigned integer data type, R is 255, etc.
EXAMPLE
APPLICATIONS
APPLICATIONS1. Numerical Analysis
2. Signal Analysis
3. Control Applications
4. Audio Applications
CONCLUSIONFourier transform provided information regarding
frequency.Short time fourier transform gives only constant
resolution. So, wavelet transform is preferred over fourier transform
and short time fourier transform since it provided multiresolution.
REFERENCES Wavelet Transform, Introduction to Theory and
Applications, By Raghaveer M.Rao Ajit.S., Bopardikar Digital Image Processing, 2nd edition,
Rafael.C.Gonzalez , Richard E.Woods. http://www.amara.com/ieeewave/iw_ref.html#ten. http://www.thewavelet tutorial by ROBI Polikar.htm
