WAVELET BASED IMAGE COMPRESSION

PERFORMANCE EVALUATION OF WAVELET BASED IMAGE COMPRESSION USING EZW & SPIHT ALGORITHMS

ABSTRACT

Digital images are widely used in computer applications. Uncompressed digital images require considerable storage capacity and transmission bandwidth. Efficient image compression solutions are becoming more critical with the recent growth of data intensive, multimedia-based web applications. This thesis studies image compression with wavelet transform as a necessary background. The basic concepts of graphical image storage and currently used compression algorithms are discussed.

Transform coding has become the defecto standard paradigm in image and video coding, where the discrete cosine transform is used because of its nice decorrelation and energy compaction properties. On the other hand DCT can also cause picture degradation due to features like block distortion. In recent years, much of the research activities in image coding have been focused on the discrete wavelet transform, while the good results obtained by wavelet coders like EZW, SPIHT are partly attributable to the wavelet transform. The performance gain is obtained by carefully designing quantizers that are tailored to the transform structure.

Embedded zerotree wavelet coding, introduced by J. M.Shapiro, is a very effective and computationally simple technique for image compression. We present a new and different implementation based on set partitioning in hierarchical trees by Said and Pearlman which provides even better performance than that of the EZW.

The SPIHT algorithm is based on two key concepts one is a discrete wavelet transform or hierarchical subband decomposition, and the other is Prediction of the absence of significant information across scales by exploiting the self-similarity inherent in images.

1 INTRODUCTION

Digital imagery has an enormous impact on industrial, scientific and computer applications. It is no surprise that image coding has been a subject of great commercial interest in today’s world. Uncompressed digital images require considerable storage capacity and transmission bandwidth. Efficient image compression solutions are becoming more critical with the recent growth of data intensive, various research works were carried out on both lossy and lossless image compression. The JPEG committee released a new image-coding standard, JPEG 2000 that serves the enhancement to the existing JPEG system. The JPEG 2000 implements a new way of compressing images based on the wavelet transform in contrast to the Discrete Cosine Transformation used in JPEG standard.

The problem of image compression is more important in many applications, particularly for progressive transmission, image browsing, multimedia applications, and compatible transcoding for multiple bit rates. A majority of today’s Internet bandwidth is estimated to be used for images and video transmission. Recent multimedia applications for handheld and portable devices place a limit on the available wireless bandwidth. Wavelet based image compression techniques such as JPEG2000 offers more compression than conventional methods in terms of compression ratio. Flexible energy-efficient implementations that can handle multimedia functions such as image processing, coding and decoding are of major importance.

Wavelet transform is arguably the most powerful, and most widely-used, tool to arise in the field of signal processing in the last several decades. Their inherent capacity for multiresolution representation akin to the operation of the human visual system motivated a quick adoption and widespread use of wavelets in Image processing applications.

2 DIGITAL IMAGE REPRESENTATION

A digital image is a two-dimensional discrete signal. Mathematically, such signals can be represented as functions of two independent variables, for example, a brightness function of two spatial variables. A monochrome digital image f(x, y) is a 2-D array of luminance values,

A color digital image is typically represented by a triplet of values, one for each of the color channels, as in the frequently used RGB color scheme. The individual color values are almost universally 8-bit values, resulting in a total of 3 bytes or 24 bits per pixel. This yields a threefold increase in the storage requirements for color versus monochrome images.

The term monochrome image or simply image, refers to a two-dimensional light intensity function f (x, y), where x and y denote spatial coordinates and the value of ‘f’ at any point (x, y) is proportional to the brightness or gray level of the image at that point. So a digital image is an image f (x, y) that has been discretized both in spatial coordinates

and brightness. A digital image can be considered as a matrix whose rows and columns indices identify a point in the image and the corresponding matrix element value identifies the gray picture elements, pixels or pels, with the last two being commonly used abbreviations of “picture elements”.

The RGB color scheme is just one of many color representation methods used in practice. The letters R, G, and B stand for red, green, and blue, the three primary colors used to synthesize any one of 2²⁴or approximately 16 million colors. Equal quantities of the three-color values result in shades of gray in the range of 0 to 255. Other supported color models include monochrome, HSV, which stands for hue, saturation, value and CMYK, which stands for cyan, magenta, yellow, black. The latter has found application primarily in the printing and graphics markets. HSV is useful in color image processing since it separates the color information from brightness.

1.3 IMAGE COMPRESSION

Uncompressed Image data requires considerable storage capacity and transmission bandwidth. Despite rapid progress in mass-storage density, processor speeds, and digital communication system performance, demand for data storage capacity and data-transmission bandwidth continues to outstrip the capabilities of available technologies. The recent growth of data intensive multimedia-based web applications has sustained the need for more efficient way to compress the images.

Image compression techniques, especially nonreversible or lossy ones, have been known to grow computationally more complex as they grow more efficient, confirming the tenets of source coding theorems in information theory that a code for a stationary source approaches optimality in the limit of infinite computation source length. Notwithstanding, the image coding technique called embedded zerotree wavelet, introduced by Shapiro, interrupted the simultaneous progression of efficiency and complexity. This technique not only was competitive in performance with the most complex techniques, but was extremely fast in execution and produced an embedded bit stream. With an embedded bit stream, the reception of code bits can be stopped at any point and the image can be decompressed and reconstructed.

1.4 IMAGE CODING

Various types of coding techniques were proposed for the Image processing Application. The two techniques used are:

1. Predictive coding
2. Transform coding

Predictive coding is a technique where information already sent or available is used to predict future values, and the difference is coded. Since this is done in the image or spatial domain, it is relatively simple to implement and is readily adapted to local image characteristics. Differential Pulse Code Modulation (DPCM) is one particular example of predictive coding.

Transform coding, on the other hand, first transforms the image from its spatial domain representation to a different type of representation using some well-known transform and then codes the transformed coefficient values. This method provides greater data compression compared to predictive methods, although at the expense of greater computation.

1.5 IMAGE COMPRESSION MODEL

Figure 1.1 shows a fundamental JPEG Image compression system consists of two distinct structural blocks: an Encoder and Decoder. An input image is fed to encoder, which created a set of symbols from the input data. After transmission over the channel, the encoded representation is fed to the decoder where a reconstructed output image is generated. In general, the reconstructed image may or may not be replica of original image. If it is, the system is error free if not some level of distortion is present in the reconstructed image.

SOURCE ENCODER

A source Encoder reduces or eliminates redundancies in the input image. An overall Block Diagram of source encoder is shown in figure1.2.

The Mapper unit transforms the input image into a format designed to reduce inter pixel redundancy. The mapper’s transformed image can be restored. Quantizer fixes the coefficients of the mapper’s output to some pre-determined values, which does not affect the image for reconstruction. Symbol encoder assigns a variable length code to the Quantized coefficients. Here by assigning a short code word to high probable coefficients, coding redundancy can be removed.

FOURIER ANALYSIS

Signal analysts already have at their disposal an impressive arsenal of tools. Perhaps the most well-known of these is Fourier analysis, which breaks down signal into constituent sinusoids of different frequencies. Another way to think of Fourier analysis is as a mathematical technique fortransforming our view of the signal from time-based to frequency-based.

For many signals, Fourier analysis is extremely useful because the signal’s frequency content is of great importance. So why do we need other techniques, like wavelet analysis. Why because Fourier analysis has a serious drawback. In transforming to the frequency domain, time information is lost. When looking at a Fourier transform of a signal, it is impossible to tell when a particular event took place. If the signal properties do not change much over time that is, if it is what is called astationary signal this drawback isn’t very important. However, most interesting signals contain numerous nonstationary or transitory characteristics i.e. drift, trends, abrupt changes, and beginnings and ends of events. These characteristics are often the most important part of the signal, and Fourier analysis is not suited in detecting them.

2.2 SHORT-TIME FOURIER ANALYSIS

In an effort to correct this deficiency, Dennis Gabor in the year 1946 adapted the Fourier transform to analyze only a small section of the signal at a time, and the technique is called windowingthe signal. Gabor’s adaptation, called the Short-Time Fourier Transform (STFT) maps a signal into a two-dimensional function of time and frequency.

The STFT represents a sort of compromise between the time and frequency-based views of a signal. It provides some information about both when and at what frequencies a signal event occurs. However, you can only obtain this information with limited precision, and that precision is determined by the size of the window. While the STFT compromise between time and frequency information can be useful, the drawback is that once you choose a particular size for the time window, that window is the same for all frequencies. Many signals require a more flexible approach, where we can vary the window size to determine more accurately either time or frequency.

2.3 DCT

The Block diagram of DCT is as shown in the figure given below. Here a reversible linear transform such as Fourier Transform is used to map the image into a set of transform coefficients, which are then Quantized and encoded. For most natural images, a significant number of coefficients have small magnitude and can be coarsely quantized with little image distortion.

5 DCT versus DWT

2.5.1 DCT Advantages

1. The DCT is the most used transform for compression because it does the energy compaction better than other transforms.
2. Computationally simpler because of availability of fast algorithms

2.5.2 Disadvantages

A major problem related with the DCT techniques is that the decoded images, especially at low bit rates, exhibit visually annoying blocking effects, i.e., visible gray-level discontinuities along the block boundaries. This is due to the fact that transform coefficient blocks are quantized independently. Therefore, a new concept based on wavelets used to overcome blocking artifacts problem in DCT.

INTRODUCTION TO WAVELETS

A wavelet is a waveform of effectively limited duration that has an average value of zero. Compare wavelets with sine waves, which are the basis of Fourier analysis. Sinusoids do not have limited duration, they extend from minus to plus infinity and where sinusoids are smooth and predictable, wavelets tend to be irregular and asymmetric.

Fourier analysis consists of breaking up a signal into sine waves of various frequencies. Similarly, wavelet analysis is the breaking up of a signal into shifted and scaled versions of the original or mother wavelet. Just looking at pictures of wavelets and sine waves, we can see intuitively that signals with sharp changes might be better analyzed with an irregular wavelet than with a smooth sinusoid, just as some foods are better handled with a fork than a spoon. It also makes sense that local features can be described better with wavelets that have local extent.

One major advantage afforded by wavelets is the ability to perform local analysis that is, to analyze a localized area of a larger signal. Consider a sinusoidal signal with a small discontinuity, one so tiny as to be barely visible. Such a signal easily could be generated in the real world, perhaps by a power fluctuation or a noisy switch. Wavelet analysis is capable of revealing aspects of data that other signal analysis techniques miss such as aspects like trends, breakdown points, discontinuities in higher derivatives, and self-similarity. Furthermore, because it affords a different view of data than those presented by traditional techniques, wavelet analysis can often compress or de-noise a signal without appreciable degradation.

In wavelet analysis the use of a fully scalable modulated window solves the signal-cutting problem. The window is shifted along the signal and for every position the spectrum is calculated. Then this process is repeated many times with a slightly shorter or longer window for every new cycle. In the end the result will be a collection of time-frequency representations of the signal, all with different resolutions. Because of this collection of representations of image at multi-level a multi resolution analysis can be carried out on the image considered. This gives a time-scale representation of the image rather than time-frequency representation as in other transformation.

2.8 WAVELET CODERS

The important application of wavelets is separating the smooth variations and details of the image, which is done by wavelet decomposition of the image using a Discrete Wavelet Transform (DWT). This is done before the quantizing and entropy encoding of the signal.

Discrete wavelet transform is used instead of Continuous wavelet transform in all practical systems because we can compute DWT much easier than CWT using digital computers. That we will get identical results have been proven by Nyquist. Since images and videos are 2-dimensional data, 2-D DWT is used. 2-D DWT is a separable transform, which means that the transform of the data can be taken by first transforming the rows followed by transforming the columns. An important feature of DWT is the fact that the transform can be implemented using appropriately designed quadrature mirror filters (QMFs) A QMF consists of a pair of high-pass and low-pass filters. With each application of the filter, the original signal is successively decomposed into components of lower resolution, while the high frequency components are not analyzed any further. Using this approach, an efficient construction of forward and reverse DWTs can be implemented

THE WAVELET TRANSFORM

The Wavelet transform provides the time and frequency information simultaneously, hence giving a time-frequency representation of the signal. The time-domain signal is passed through various high pass and low pass filters, which filters out either high frequency or low frequency portions of the signal. The process is iterated, and every time some portion of the signal corresponding to some frequencies being removed from the signal.

For example a signal through which has frequencies up to 1000 Hz, will split up in to two parts by passing the signal from a high pass and a low pass filter in the first stage. Which results in two different versions of the same signal, those are portion of the signal corresponding to 0-500 Hz (low pass portion), and 500-1000 Hz (high pass portion). Considering low pass filtered portion, and iterating the operation the signal get split down to further sub-bands. This operation is called sub-banddecomposition.

The frequency and time information of a signal at certain point in the time-frequency plane cannot be predicted. i.e. it cannot be predicted what spectral component exists at any given time ofinstant. This give rise to the problem of resolution. Higher frequencies are better resolved in time, and lower frequencies are better resolved in frequency. This means that, a certain high frequency component can be located better in time with less relative error than a low frequency component. On the contrary, a low frequency component can be located better in frequency compared to high frequency component.

Wavelet Transforms were used to analyze non-stationary signals, i.e., whose frequency response varies in time. Although the time and frequency resolution problems are results of a physical phenomenon and exist regardless of the transform used, it is possible to analyze any signal by using an alternative approach called the multi resolution analysis (MRA). MRA, analyzes the signal at different frequencies with different resolutions. MRA, basically 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 is useful especially when the signal considered has high frequency components for short durations and low frequency components for long durations, which are basically used in practical applications.

2.10 WAVELET FILTER

With the redundancy removed, we still have two hurdles to take before we have the wavelet transform in a practical form. We continue by trying to reduce the number of wavelets needed in the wavelet transform and save the problem of the difficult analytical solutions for the end.

Even with discrete wavelets we still need an infinite number of scalings and translations to calculate the wavelet transform. The easiest way to tackle this problem is simply not to use an infinite number of discrete wavelets. Of course this poses the question of the quality of the transform. Is it possible to reduce the number of wavelets to analyze a signal and still have a useful result.

The translations of the wavelets are of course limited by the duration of the signal under investigation so that we have an upper boundary for the wavelets. This leaves us with the question of dilation: how many scales do we need to analyze our signal? How do we get a lower bound? It turns out that we can answer this question by looking at the wavelet transform in a different way.

If we look below we see that the wavelet has a band-pass like spectrum. From Fourier theory we know that compression in time is equivalent to stretching the spectrum and shifting it upwards:

SCALING FUNCTION

The careful reader will now ask him or herself the question how to cover the spectrum all the way down to zero. Because every time you stretch the wavelet in the time domain with a factor of 2, its bandwidth is halved. In other words, with every wavelet stretch you cover only half of the remaining spectrum, which means that you will need an infinite number of wavelets to get the job done.

The solution to this problem is simply not to try to cover the spectrum all the way down to zero with wavelet spectra, but to use a cork to plug the hole when it is small enough. This cork then is a low-pass spectrum and it belongs to the so-called scaling function. Mallat introduced the scaling function. Because of the low-pass nature of the scaling function spectrum it is sometimes referred to as theaveraging filter.

SUB-BAND ANALYSIS

A time-scale representation of a digital signal is obtained using digital filtering techniques. Recall that the CWT is a correlation between a wavelet at different scales and the signal with the scale or the frequency being used as a measure of similarity. The continuous wavelet transform was computed by changing the scale of the analysis window, shifting the window in time, multiplying by the signal, and integrating over all times. In the discrete case, filters of different cutoff frequencies are used to analyze the signal at different scales. The signal is passed through a series of high pass filters to analyze the high frequencies, and it is passed through a series of low pass filters to analyze the low frequencies.

The resolution of the signal, which is a measure of the amount of detail information in the signal, is changed by the filtering operations, and the scale is changed by up sampling and down sampling (sub sampling) operations. Sub sampling a signal corresponds to reducing the sampling rate, or removing some of the samples of the signal. For example, sub sampling by two refers to dropping every other sample of the signal. Sub sampling by a factor n reduces the number of samples in the signal n times.

Up sampling a signal corresponds to increasing the sampling rate of a signal by adding new samples to the signal. For example, up sampling by two refers to adding a new sample, usually a zero or an interpolated value, between every two samples of the signal. Up sampling a signal by a factor ofn increases the number of samples in the signal by a factor of n.

Although it is not the only possible choice, DWT coefficients are usually sampled from the CWT on a dyadic grid, i.e., s₀ = 2 and π ₀ = 1, yielding s=2^j and π =k*2^j, as described in Part 3. Since the signal is a discrete time function, the terms function and sequence will be used interchangeably in the following discussion. This sequence will be denoted by x[n], where n is an integer.

A half band low pass filter removes all frequencies that are above half of the highest frequency in the signal. For example, if a signal has a maximum of 1000 Hz component, then half band low pass filtering removes all the frequencies above 500 Hz.

The unit of frequency is of particular importance at this time. In discrete signals, frequency is expressed in terms of radians. Accordingly, the sampling frequency of the signal is equal to 2π radians in terms of radial frequency. Therefore, the highest frequency component that exists in a signal will be π radians, if the signal is sampled at Nyquist’s rate (which is twice the maximum frequency that exists in the signal); that is, the Nyquist’s rate corresponds to π rad/s in the discrete frequency domain. Therefore using Hz is not appropriate for discrete signals. However, Hz is used whenever it is needed to clarify a discussion, since it is very common to think of frequency in terms of Hz. It should always be remembered that the unit of frequency for discrete time signals is radians.

After passing the signal through a half band low pass filter, half of the samples can be eliminated according to the Nyquist’s rule, since the signal now has a highest frequency of π/2 radians instead of π radians. Simply discarding every other sample will sub sample the signal by two, and the signal will then have half the number of points. The scale of the signal is now doubled. Note that the low pass filtering removes the high frequency information, but leaves the scale unchanged. Only the sub sampling process changes the scale. Resolution, on the other hand, is related to the amount of information in the signal, and therefore, it is affected by the filtering operations. Half band low pass filtering removes half of the frequencies, which can be interpreted as losing half of the information. Therefore, the resolution is halved after the filtering operation. Note, however, the sub sampling operation after filtering does not affect the resolution, since removing half of the spectral components from the signal makes half the number of samples redundant anyway. Half the samples can be discarded without any loss of information. In summary, the low pass filtering halves the resolution, but leaves the scale unchanged. The signal is then sub sampled by 2 since half of the number of samples are redundant.

Advantages of wavelet based compression

1. Wavelet coding schemes at higher compressions avoid blocking artifacts.
2. Wavelet-based coding is more robust under transmission and decoding errors, and also facilitates progressive transmission of images.
3. They are better matched to the HVS (Human Visual System) characteristics.
4. Compression with wavelets is scalable as the transform process can be applied to an image as many times as wanted and hence very high compression ratios can be achieved.
5. They provide an efficient decomposition of signals prior to compression.
6. Wavelet based compression allows parametric gain control for image smoothening and sharpening.
7. Wavelet compression is very efficient at low bit rate

2.12.2 Disadvantages of wavelet based compression

Wavelet compression does require more computational power than compression based on other techniques such as Discrete Cosine Transform (DCT).

TWO MAJOR WAVELET CODING ALGORITHMS:-

n Embedded zero-tree wavelet (EZW)

1. Use DWT (discrete wavelet transform) for the decomposition of an image at each level.

2. Scans wavelet coefficients subband by subband using Morton scan order.

n Set partitioning in hierarchical trees (SPIHT)

1. Highly refined version of EZW.

2. Perform better at higher compression ratio for a wide variety of images than EZW.

3.2 EZW CODING

An EZW encoder was specially designed by Shapiro to use with wavelet transforms. In fact, EZW coding is more like a quantization method. It was originally designed to operate on 2D images, but it can also be used on other dimensional signals. The EZW encoder is based on progressive encoding to compress an image into a bit stream with increasing accuracy. This means that when more bits are added to the stream, the decoded image will contain more detail, a property similar to JPEGencoded images. Progressive encoding is also known as embedded encoding, which explains the E in EZW. Coding an image using the EZW scheme, together with some optimizations, results in a remarkably effective image compressor with the property that the compressed data stream can haveany bit rate desired. Any bit rate is only possible if there is information loss somewhere, so that the compressor is lossy. However, lossless compression is also possible with an EZW encoder, but of course with less spectacular results. The EZW encoder is based on two important observations:

1. Natural images in general have a low pass spectrum. When an image is wavelet transformed, the energy in the subbands decreases as the scale decreases low scale means high resolution, so the wavelet coefficients will, on average, be smaller in the higher subbands than in the lower subbands. This shows that progressive encoding is a very natural choice for compressing wavelet transformed images, since the higher subbands only add detail.
2. Large wavelet coefficients are more important than small wavelet coefficients.

These two observations are used by encoding the wavelet coefficients in decreasing order, in several passes. For every pass, a threshold is chosen against which all the wavelet coefficients are measured. If a wavelet coefficient is larger than the threshold, it is encoded and removed from the image; if it is smaller it is left for the next pass. When all the wavelet coefficients have been visited, the threshold is lowered, and the image is scanned again to add more detail to the already encoded image. This process is repeated until all the wavelet coefficients have been encoded completely or another criterion has been satisfied i.e. maximum bit rate for instance. A wavelet transform transforms a signal from the time domain to the joint time-scale domain. This means that the wavelet coefficients are two-dimensional. If we want to compress the transformed signal, we have to code not only the coefficient values, but also their position in time. When the signal is an image, then the position in time is better expressed as the position in space. After wavelet transforming an image, we can represent it using trees because of the sub sampling that is performed in the transform. A coefficient in a low subband will have four descendants in the next higher subband as shown in fig. below. One of each four descendants also has four descendants in the next higher subband and we see a quad-tree emerge i.e. every parent has four children.

Coding the Embedded Zerotree of the Wavelet Coefficients

The EZW algorithm is a quantization method for images which accommodate four different techniques:

3.2.2 Wavelet Transform

Application of discrete wavelet transform is to produces the multi resolution decomposition of the image along predetermined depth.

3.2.3 Zerotree coding

A compact presentation of the location of wavelet coefficients.

3.2.4 Successive approximation

Increase, during the production of the output code i.e. the compressed image, the precision of the quantized wavelet coefficients

CODING ALGORITHM

Since the order in which the subsets are tested for significance is important, in a practical implementation the significance information is stored in three ordered lists, called list of insignificant sets (LIS), list of insignificant pixels (LIP), and list of significant pixels (LSP). In all lists each entry is identified by a coordinate (2, j), which in the LIP and LSP represents individual pixels, and in the LIS represents either the set D(z, J ) or C(i, j ) . To differentiate between them, we say that a LIS entry is of type A if it represents D(z; j), and of type B if it represents L(i, j ) . During the sorting pass (see Algorithm I), the pixels in the LIP-which were insignificant in the previous pass-are tested, and those that become significant are moved to the LSP. Similarly, sets are sequentially evaluated following the LIS order, and when a set is found to be significant it is removed from the list and partitioned. The new subsets with more than one element are added back to the LIS, while the single-coordinate sets are added to the end of the LIP or the LSP, depending whether they are insignificant or significant, respectively. The LSP contains the coordinates of the pixels that are visited in the refinement pass. Below we present the new encoding algorithm in its entirety. It is essentially equal to Algorithm I, but uses the set partitioning approach in its sorting pass.

3.4.1ALGORITHM

1) Initialization: output n = [log ₂(max _{(i, j)} {|c _i,j |})]; set the LSP as an empty list, and add the coordinates (i, j) Є H to the LIP, and only those with descendants also to the LIS, as type A entries.

2) Sorting pass:

2.1) for each entry (i, j) in the LIP do:

2.1.1) output S_n (i, j);

2.1.2) if S_n (i, j) =1then move (i, j) to the LSP and output the Sign of c_{i, j;}

2.2) for each entry (i, j) in the LIS do:

2.2.1) if the entry is of type A then

· output S_n (D (i, j));

· if s_n (D (i, j)) =1 then

* for each (k, l) Є O (i, j);

· Output S_n (k, l);

· If S_n (k, l) =1 then add (k, l) to the LSP and output the sign of c_k,l;

· If S_n (k, l) =0 then add (k, l) to the end of LIP;

* if L (i, j) ≠ f then move (i, j) to the end of LIS, as an entry of type B and go to step 2.2.2); otherwise, remove entry (i,j) from the LIS;

2.2.2)if the entry is of type B then

· output S_n(L(i,j));

· if S_n(L(i,j))=1 then

* add each (k, l) c O (i, j) to the end of the LIS as entry of type A;

* remove (i, j) from the LIS.

3) Refinement Pass: for each entry (2, j) in the LSP, except those included in the last sorting pass (i.e., with same n), output the nth most significant bit of |c_i,j|;

4) Quantization-Step Update: decrement n by 1 and go to Step 2.

CODE:

Find the code in next post

CONCLUSION

After going through the results, out of the two image compression algorithms, those are EZW & SPIHT, SPIHT have shown a better results in terms of subjective fidelity criteria.

There is a small increase in PSNR due to the better exploitation of spatial correlation present across the sub bands in the image. And at the same threshold level we get better compression ratios in SPIHT, when compared to that of EZW.

So, we can come to a conclusion that SPIHT algorithm is giving better performance than that of the EZW.

6.2FUTURE SCOPE

The present thesis can be extended to video coding by exploiting temporal correlation present in the video and there by increasing the compression ratio. However the suitability of wavelets to video coding has to be analyzed in detail to really threaten the DCT based MPEG codecs in all multimedia and broadcasting applications.

REFERENCES

IEEE papers:

[1] Antonini.M , barlaud.M ,Mathieu.p , and Daubechies.I , “Image coding using

wavelet transform” ,IEEE Transactions on Image Processing , Vol . 1, No.2 (1992).

[2] Shapiro.J.M “Embedded Image Coding Using Zero trees of Wavelet Coefficients”,

IEEE Transactions on Signal Processing, Vol.41, No.12(1993) , pp.3445-3462.

BOOKS:

[1] “Digital Image processing using MATLAB “by Rafael C.Gonzalez, Richard E.Woods and Steven L.Eddins.

[2] “Wavelet Transforms Introductions to theory &applications “by RaghuveerM.Rao,S.BOpardikar.

WEB RESOURCES:

[1] http://wavelet.org

[2] http://www.amar.com/current/wavelet.html