recognition, image processing, medical signal processing, radar and sonar and
in ..... Proakis, J. G., Manolakis, D. G. Digital Signal Processing, Proakis,.
Chapter 1 1.1 Introduction to Digital Signal Processing Digital Signal Processing (DSP) is an area of science and engineering that has been developed rapidly over the past three decades. DSP field has its roots in th
th
17 and 18 century mathematics. The rapid development in DSP is the result of the significant advancements in digital computer technology and integrated circuit fabrication. DSP is a technique used to manipulate digital signals with help of mathematical tools implemented through algorithms. DSP has overlapping borders with many other areas of science and engineering such as mathematics, communication theory, analog electronics, digital electronics, probability and statistics, decision theory and numerical analysis. Recent developments in digital signal processing have brought revolutionary changes in the fields of communication, medical-imaging, radar & sonar, high fidelity music reproduction and oil exploration/1-3/. Digital signal processing is concerned with the representation of signals by sequences of numbers or symbols and the processing of these sequences. The purpose of such processing may be to estimate characteristic parameters of a signal or to transform a signal into a form, which is in some sense more desirable. For example, in electrocardiogram (ECG) analysis or in speech transmission and speech recognition systems, signal processing techniques are used to remove noise interference from the signal or to modify the signal to present it in a desired form, which can easily be interpreted.
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The signal processing problems are not limited only to one-dimensional signals. The analysis of satellite pictures for the detection of forest fires or crop damage, the enhancement of television transmissions from lunar and deep space probes, seismic data analysis as required in oil exploration, earthquake measurements and nuclear test monitoring require multidimensional signal processing techniques. The Signal processing provides the basic analysis, modeling and synthesis tools for a diverse area of technological fields, including telecommunication, artificial intelligence, biological computation and system identification. Signal processing is concerned with the modeling, detection, identification and utilization of patterns and structures in a signal process. The applications of signal processing methods include communication, audio, video, cellular phones, voice recognition, vision, radar, sonar, geographical exploration, medical electronics, bio medical signal processing and in general any system that is concerned with the communication or processing and retrieval of information. The techniques and applications of digital signal processing are expanding at a tremendous rate. With the advent of ultra large scale integration(ULSI) and the resulting reduction in cost and size of digital components, together with increasing speed, the class of applications of digital signal processing techniques is growing /1-8/.
1.1.1 Signals and Information A ‘signal’ can be defined as a function that conveys information, generally about the state or behavior of a physical system. Although signals can be 10
represented in many ways, in all the cases the information is contained in a pattern of variations of some form. Signals are represented mathematically as functions of one or more independent variables. The independent variable of the mathematical representation of a signal may be either continuous or discrete. ‘Continuous time signals’ are signals that are defined at a continuum of times and thus are represented by continuous variable functions. ‘Discrete time signals’ are defined at discrete times and thus the independent variable takes on only discrete values, i.e., discrete time signals are represented as sequences of numbers. Signal processing systems may be classified along the same lines as signals. That is, continuous time systems and discrete times systems.
1.1.2 Basic Elements of a DSP System The block diagram of a DSP system is shown figure 1.1. Before converting the analog signal to digital signal, the input signal is processed with a low-pass filter to remove all frequencies above the Nyquist frequency. This prevents aliasing during sampling, and the filter is correspondingly called as Antialiasing filter. The digital signal processor performs the operation specified by the instructions built into the processor. The digital signal is passed through a digitalto-analog converter and a low-pass filter set to the Nyquist frequency. This output filter is called a reconstruction filter.
Figure 1.1: Block diagram of DSP system
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1.1.3 Digital signal processing methods The signal processing methods have evolved in algorithmic complexity, aiming for optimal utilization of the information in order to achieve the best performance. In general, the computational requirement of signal processing methods increases, often exponentially, with the algorithmic complexity. However, implementation cost of advanced signal processing methods has been offset and made affordable by the consistent trend in recent years of a continuing increase in the performance, coupled with a simultaneous decrease in the cost, of signal processing hardware. Depending on the method used, digital signal processing algorithms can be categorized into one or a combination of four broad categories. These are transform-based signal processing, model-based signal processing, Bayesian statistical signal processing and neural networks, as illustrated in figure 1.2 /8/.
Figure 1.2: A broad categorization of the most commonly used signal processing methods
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1.2 Applications of digital signal processing In recent years, the development and commercial availability of increasingly powerful and affordable digital computers has been accompanied by the development of advanced digital signal processing algorithms for a wide variety of applications such as noise reduction, telecommunications, radar, sonar, video and audio signal processing, pattern recognition, geophysics explorations, data forecasting, and the processing of large databases for the identification, extraction and organization of unknown underlying structures and patterns. Figure 1.3 shows a broad categorization of some digital signal processing (DSP) applications. This section provides a review of several key applications of DSP methods
Figure 1.3: A classification of the applications of digital signal processing
1.2.1 Adaptive noise cancellation In speech communication from a noisy acoustic environment, such as a moving car or train or over a noisy telephone channel, the speech signal is observed in an additive random noise. In signal measurement systems the
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information-bearing signal is often contaminated by noise from its surrounding environment. The noisy observation, y ( m ) can be modeled as (1.1)
y (m ) x (m ) n (m )
where x ( m ) and n ( m ) are the signal and the noise, m is the discrete-time index. In some situations, for example, while using a mobile telephone in a moving car, or while using a radio communication device in an aircraft cockpit, it may be possible to measure and estimate the instantaneous amplitude of the ambient noise using a directional microphone. The signal x ( m ) , may then be recovered by subtraction of an estimate of the noise from the noisy signal /9/.
1.2.2 Adaptive noise reduction In many applications, for example, at the receiver of a telecommunication system, there is no access to the instantaneous value of the contaminated noise, and only the noisy signal is available. In such cases the noise cannot be cancelled out, but it may be reduced, in an average sense, using the statistics of the signal and the noise process /5,6,10,11/.
1.2.3 Blind channel equalization The channel equalization refers to the recovery of a signal distorted in
transmission through a communication channel with a non-flat magnitude or a nonlinear phase response. When the channel response is unknown, the process of digital recovery is called ‘blind equalization’. Blind equalization has a wide range of applications. For example, in digital telecommunications for removal of intersymbol interference due to non-ideal channel and multipath propagation, in 14
speech recognition for removal of the effects of the microphones and communication channels, in correction of distorted images, in the analysis of seismic data and in de-reverberation of acoustic gramophone recordings /12-14/.
1.2.4 Signal classification and Pattern recognition The signal classification is used in detection, pattern recognition and decision-making systems. For example, a simple binary-state classifier can act as the detector of the presence, or absence, of a known waveform in noise. In signal classification, the aim is to design a minimum-error system for labeling a signal with one of a number of likely classes of the signal/15/.
1.2.5 Linear prediction modeling of speech Linear predictive models are widely used in speech processing applications such as low bit-rate speech coding in cellular telephony, speech enhancement and speech recognition. Speech is generated by inhaling air into lungs, and then exhaling it through the vibrating through glottis cords and vocal tract. The random noise-like, air flow from the lungs is spectrally shaped and amplified by the vibrations of glottal cords and the response of the vocal tract. The effect of the vibrations of the glottal cords and the vocal tract is to introduce a measure of correlation and predictability to the random variations of the air from the lungs. This is modeled as a source-filter model. The source models the lung and emits a random excitation signal which is filtered, first by a pitch filter model of the glottal cords and then by a model of the vocal tract /16, 17/.
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1.2.6 Digital coding of audio signals In the digital audio, the memory required to record a signal, the bandwidth required for signal transmission and the signal-to-quantization noise ratio are all directly proportional to the number of bits per sample. The objective in the design of a coder is to achieve high fidelity with as few bits per sample as possible, at an affordable implementation cost. Audio signal coding schemes utilize the statistical structures of the signal and a model of the signal generation, together with information on the psychoacoustics and the masking effects of hearing. In general, there two main categories of audio coders: model-based coders, used low-bit-rate speech coding in applications such as cellular telephony, and transform-based coders used in high-quality coding of speech and digital hi-fi audio /18,19/.
1.2.7 Detection of signals in noise In the detection of signals in noise, the aim is to determine if the observation consists of noise alone, or if it contains a signal. The noisy observation, y ( m ) can be modeled as y (m ) b (m ) x(m ) n (m )
(1.2)
where x ( m ) is the signal to be detected, n ( m ) is the noise and b ( m ) is a binaryvalued state indicator such that b ( m ) 1 indicates the presence of the signal, and b ( m ) 0 indicates the absence of signal. If the signal x ( m ) , has a known shape,
then a correlator or a matched filter can be used to detect the signal /17, 18, 20/.
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1.2.8 Radar signal processing: Doppler frequency shift A radar system is used to estimate the range and speed of an object such as a moving car or airplane or missile. The radar system consists of transceiver that generates and transmits sinusoidal pulses at microwave frequencies. The signal travels with the speed of light and is reflected back from any object in its path. The analysis of the echo received provides information such as range, speed and acceleration. The shift in the frequency of the reflected wave is known as Doppler frequency. The frequency analysis of the reflected signal can reveal information on the direction and speed of the object/18,19.21/.
1.3 Noise The noise can be defined as an unwanted signal that interferes with the communication or measurement, perception or processing of an information bearing signal. Noise itself is a signal that conveys information regarding the source of noise. For example, the noise from a car engine conveys information regarding the state of the engine and how smoothly it is running. The sources of noise are many and varied and include thermal noise intrinsic to electric conductors, shot noise inherent in electric current flows, audio-frequency acoustic noise emanating from moving, vibrating or colliding sources such as revolving machines, moving vehicles, computer fans, keyboard clicks, wind, rain, etc. and radio-frequency electromagnetic noise that can interfere with the transmission and reception of voice, image and data over the radio-frequency spectrum/16, 20/.
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The noise is the main factor limiting the capacity of data transmission in the telecommunications and accuracy in signal measurement systems. Therefore, the modeling and removing the noise has been at the core of the theory and practice of communications and signal processing. Noise reduction is an important problem in applications such as cellular mobile communications, speech recognition, image processing, medical signal processing, radar and sonar and in any application where the signals cannot be isolated from noise/16-18/. The noise can cause transmission errors and may even disrupt a communication process or degrade an image. Hence, de-noising process is an important and integral part of modern telecommunications and signal and image processing systems. The success of a noise processing method depends on its ability to characterize and model the noise process, and to use the noise characteristics advantageously to differentiate the signal from the noise. Depending on its source, a noise can be classified into number of categories, indicating the broad physical nature of the noise, as follows: 1. Acoustic noise- emanates from moving, vibrating or colliding sources and it is most familiar type of noise present in various degrees in everyday environments. Acoustic noise is generated by such sources as moving cars, air-conditioners, computer fans, traffic, people talking in the background, wind, rain etc. 2. Thermal noise and shot noise- thermal noise is generated by the random movements of thermally energized particles in an electric conductor. Thermal noise is intrinsic to all conductors and is present without any applied voltage. 18
The shot noise consists of random fluctuations of the electric current in an electrical conductor and is intrinsic to current flow. Shot noise is caused by the fact that the current is carried by discrete charges (i.e. electrons) with random fluctuations and random arrival times. 3. Electromagnetic noise- present at all frequencies and in particular at the radio frequency range, where telecommunications take place. All electric devices such
as
radio
and
television
transmitters
and
receivers
generate
electromagnetic noise. 4. Electrostatic noise – generated by the presence of a voltage with or without current flow. Fluorescent lighting is one of the more common sources of electrostatic noise 5. Channel distortions, echo and fading – due to non ideal characteristics of communication channels. Radio channels, such as those at GHz frequencies used by cellular mobile phone operators, are particularly sensitive to the propagation characteristics of the channel environment and fading of signals. 6. Processing noise– the noise that results from the digital-to-analog processing of signals, e.g. quantization noise in digital coding of speech or image signals, or lost data packets in digital data communication systems. Depending on its frequency spectrum or time characteristics, a noise can be further classified into one of several categories as follows: White noise – purely random noise that has a flat power spectrum and theoretically contains all frequencies in equal intensity.
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Band-limited white noise – a noise with a flat spectrum and limited width that usually covers the limited spectrum of the device or the signal of interest. Narrowband noise – a noise with a narrow bandwidth such as a 50–60 Hz such as ‘hum’ from the electricity supply. Colored noise – non-white noise or any wideband noise whose spectrum has a non-flat shape; examples are pink noise, brown noise and autoregressive noise. Impulsive noise – consists of short duration pulses of random amplitude and random duration. Transient noise pulses – consists of relatively long duration noise pulses.
1.3.1 Noise Power Estimation When noise power is not known a priori, then it must be determined using noise power estimation techniques. Once the noise power is determined, then the data can be normalized. The two commonly used techniques for denoising the signal are time-domain technique and wavelet-domain technique and are briefly discussed in this section. The time-domain technique relies on determining a section of the data with relatively signal free characteristics.
This section of data, which contains
predominately noise energy, is used to form an estimate of the noise power for the data. Once an estimate of the noise power is determined it is normalized to unity, by dividing the entire signal by the square root of the noise power. Then the resulting signal can be passed as a whole or in smaller subsections through the proposed de-noising algorithm. 20
The second method used to measure the noise power utilizes knowledge about the wavelet coefficients.
Mallat /24/ shows that any additive white
Gaussian noise (AWGN) at the input to the wavelet transform is transformed “at the finest scale” to AWGN of the same variance. If it can be assumed that signal lies predominantly in the lower half of the signal spectrum, then the detail coefficients at the first level of decomposition will be primarily AWGN. Consequently, measuring the variance of these coefficients and normalizing the signal leads to noise normalized data. In either case, the result is a signal combined with AWGN with a variance of one /20, 25/.
1.4 Time-frequency analysis The exposition and exploration of time-frequency methods of signal analysis is at present, an area of vigorous research and development area. Signal processing plays a central role in the development of digital telecommunication and automation systems, and in the efficient transmission, reception and decoding of information. The observed signals are often distorted, incomplete and noisy. Hence, noise reduction and removal of channel distortion and interference are important parts of a signal processing system /8/. The time-frequency analysis is a form of local Fourier analysis that treats time and frequency simultaneously and symmetrically. Time-frequency analysis originates in the early development of quantum mechanics by E Wigner /22/, and J. von Neumann/23/ around 1930. In the decades following Gabor’s article “Theory of communication” /26/ time-frequency analysis was considered to be the domain of engineers, while mathematics played only an ancillary role. In 1980, 21
the pioneering work by Guido (A.J.E.M) Janssen covered all aspects of timefrequency analysis. And he may be considered as father of mathematical timefrequency analysis. Since 1990 the development of time-frequency analysis has benefitted from the rise of wavelet theory. The wavelet theory is highlighted in Ingrid Daubechies’s article “The wavelet transform, time-frequency localization and signal analysis”/27/ and her book Ten Lectures on Wavelets/28/. Today, time-frequency analysis presents itself as a vibrant interdisciplinary area of research. On the applied side, time-frequency analysis deals with problems in signal analysis, communication theory, and image processing.
1.4.1 Time-frequency transforms The Fourier transform has become one of the most widely used signalanalysis tools across many areas of science and engineering. The basic idea of the Fourier transform is that any arbitrary signal can always be decomposed into a set of sinusoids of different frequencies. The Fourier transform is generated by the process of projecting the signal onto a set of basis functions, each of which is a sinusoid with a unique frequency. The resulting projection values form the Fourier transform (or the frequency spectrum) of the original signal. Its value at a particular frequency is a measure of the similarity of the signal to the sinusoidal basis at that frequency. Therefore, the frequency attributes of the signal can be revealed via the Fourier transform. In many engineering applications, this has proven to be extremely useful in the characterization, interpretation, and identification of signals. /29/ 22
The Fourier transform is the most popular and widely used tool in the analysis of stationary signals, but it cannot be used in the case of non-stationary signals as it projects the signal on infinite sinusoids, which are completely delocalized in time. The concepts of instantaneous frequency and group delay are also inherently un-adapted to a large number of non-stationary signals and in particular noisy signals. Thus, in such cases one-dimensional solution is not sufficient, and one has to consider two-dimensional functions, i.e. functions of variable time and frequency. Joint time-frequency transforms were developed for the purpose of characterizing the time-varying frequency content of a signal. The well-known time-frequency representation of time signal dates back to Gabor and is known as the short-time Fourier transform (STFT) /30/. It is basically a moving window Fourier transform. By examining the frequency content of the signal as the time window is moved, a 2D time-frequency distribution called the spectrogram is generated, this provides a time-localized frequency content of the transformed signal. The time-frequency transforms are broadly divided into two classes: linear time-frequency transforms and bilinear (or quadratic) transforms. In the following sections an overview of linear time-frequency transforms is presented with an emphasis on application perspective.
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1.4.2 Linear time-frequency transforms The Fourier transform is a widely used transformation and is defined for a continuous time signal f(t) as
F ( )
f ( t ) e j t dt .
(1.3)
The complex basis function, also referred to as a kernel, e-jωt , operates on f(t) and generates a one-dimensional representation of signal in the frequency domain. Here, it is often desirable to localize the time at which specific characteristics of a signal occur; however, such localization is not available in the Fourier transform. Thus, in the case of a non-stationary (or time varying) signal, it is not possible for Fourier transform to reflect the time-varying nature of the signal frequency spectrum accurately /31-33/.
1.4.3 Short time Fourier transform (STFT) The most standard approach to analyze a signal with time-varying frequency content is to decompose the time-domain signal into many segments and then take the Fourier transform of each segment. This is known as the short time Fourier transform (STFT) method and is defined as
F ( , )
f ( ) w( t )e
24
jt
d .
(1.4)
The operation (1.4) differs from the Fourier transform only by the presence of window function w(t). The signal f(t) and Fourier transform F(ω) have one dimensional nature, whereas the STFT transform F(ω, ) is represented in a twodimensional space. The magnitude squared of the STFT is called the spectrogram as shown in the figure 1.4, which provides time-localized frequency content of the transformed signal f(t). In figure 1.4, red colour indicates presence of signal blue colour indicates the absence of signal. Spectrum is obtained for the following constant amplitude linear chirp:
f ( t ) sin( 2 / F S f 0 ( t ))
,
(1.5)
where FS is set to 8000Hz, and frequency f 0(t) varies from 1 to 2000Hz The frequency resolution with a constant duration sliding window w(t- ) is fixed. In order to get a perfect spectrogram, it would be desirable to have infinite time and frequency resolution. But, STFT time and frequency resolutions are limited by the Heisenberg Uncertainty Principle /22/.
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Figure 1.4: Linear chirp spectrogram
Thus the well-known drawback of STFT is, the resolution limit imposed by the window function. A large window results in better frequency resolution, but leads to worse time resolution, and vice versa. Note that time-frequency characteristics may change with time when a signal is transmitted through various channels. As a result, a small time window is necessary but frequency resolution is degraded when the frequency content of a signal or channel characteristics change rapidly with time. A larger window may be sufficient (resulting in better frequency resolution), when the signal frequency characteristics change slowly. For unknown characteristics, an array of window sizes is required in order to find the ideal time and frequency resolution. Therefore, the fixed window size of the STFT limits its ability to span both time and frequency of unknown signals with resolution well matched to the signal characteristics/34/. 26
Multi-Resolution Analysis Multi-resolution analysis (MRA), as the name itself suggests, analyzes the signal at different frequencies with different resolutions. In much the same way as the STFT, MRA requires an operator to project the signal f(t) into another domain or vector space. Since MRA operator does not use a fixed window size, every spectral component is not resolved equally as was the case in the STFT. The MRA provides an alternative approach to analyze any signal, although the time and frequency resolution problems are results of Heisenberg’s Uncertainty Principle and exist irrespective of the transform used. MRA 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. This approach makes sense especially when the signal at hand has high frequency components for short durations and low frequency components for long durations /35,36/. The MRA can be implemented with minimum redundancy by employing orthonormal filters consisting of low pass filter and multiple band pass filters/37, 38/. The approximation coefficients are provided by low-pass filter and detail coefficients are provided by band-pass filters. The main difference between MRA and the STFT is in the filter bandwidths as shown in figure 1.5. The other promising attribute of MRA is that, many different conventional filters can be used to implement it.
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(a)
(b)
Figure 1.5: Frequency spectrums partitioning for: (a) MRA; (b) STFT
An additional advantage of MRA is that the transform operator itself can be real, and thus the coefficients will be real, resulting in a real-valued transform. Simple operations such as the application of a finite impulse response (FIR) filter can be applied to obtain real-valued transform coefficients if needed. The filters are chosen to form the detail and approximation coefficients. The same concept can be extended to a two-dimensional spatial image where the localized frequency components may be determined from the windowed transform. This is one of the bases of the conceptual understanding of wavelet transforms. Hence, wavelet transforms have been kept as the main consideration in the present work. It is well known that while receiving the input image some aberrations get introduced along with it and hence a noisy image is left with, for further processing. The denoising of an image naturally corrupted by noise is a classical problem in the field of signal or image processing. Additive random noise can easily be removed using simple threshold methods. Denoising of natural images corrupted by noise using wavelet techniques is very effective because of its ability to capture the energy of a signal in few energy transform values. The wavelet denoising scheme thresholds the wavelet coefficients arising 28
from the wavelet transform. The wavelet transform yields a large number of small coefficients and a small number of large coefficients. Simple denoising algorithms using wavelet transforms consist of three steps. Find the wavelet transform of the noisy signal. Modify the noisy wavelet coefficients according to some rule. Compute the inverse wavelet transform using the modified coefficients.
1.5 Applications of time-frequency analysis The time-frequency representation of signal has been extensively used in many applications to process (analyze, estimate, detect, classify, characterize or model) signals and systems. Some of the major applications are: 1. Industrial Automation: Oil and gas exploration, seismography, process control applications. 2. Biomedical signal processing: ECG, EEG, EMG analysis, cytological, histological
and stereological
applications, automated radiology
and
pathology, X-ray image analysis, mass screening of medical images such as chromosome slides for detection of various diseases, monograms, cancer smears, CAT, MRI, PET, SPECT, USG and other tomography images, routine screening of plant samples, 3D reconstruction and analysis etc. 3. Remote Sensing: Natural resources survey and management, estimation related to agriculture, hydrology, forestry, mineralogy, urban planning, environment and pollution control, cartography, registration of satellite images with terrain maps, monitoring traffic along roads, docks and airfields etc.
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4. Military Applications: Missile guidance and detection, target identification, navigation of pilotless vehicle, range finding in radar & sonar applications etc. 5. Communication System: Channel characterization, time-frequency receivers and channel diversity, modulation schemes, jamming interference mitigation, signal reconstruction, image denoising, data compression etc. In the present work, a new algorithm InterpolatedShrink using discrete wavelets, has been proposed
and compared with the traditional
methods like
BayesShrink/39/, VisuShrink/40/, SureShrink/41/ and Weiner filter/42/, using peak signal to noise ratio (PSNR). Various test images have been used for analysis. The performance of the applied image denoising algorithms has been investigated using different wavelet families and also at different levels of wavelet decomposition. Also, antother technique called Interpolate Median Filter (IMF) for image denoising is developed and is compared with conventional Median filter and other algorithms like minimum-maximum exclusive mean filter (MMEM), Adaptive median filtering (AMF) based on peak signal to noise ratio (PSNR).
1.6 Thesis outline The thesis is organized into five chapters. The chapter 2, gives a complete idea of the fundamentals of wavelet transforms and brief overview about signal and image denoising. The entire literature survey related with present study is also presented.
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In the chapter 3, a new approach called InterpolatedShrink technique for image denoising is presented. The effectiveness of the proposed method is demonstrated by comparing its performance with some of the existing methods. Also, the performance of the new technique is analyzed with the different wavelet families and at different levels of decomposition. In the chapter 4, a novel technique called Interpolate Median Filter (IMF) for image denoising is presented and its performance is compared with some popular methods like Median filter and other traditional methods. The conclusions and the scope of research work in future are presented in chapter 5.
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