IMAGE DATA COMPRESSION BASED ON DISCRETE WAVELET TRANSFORMATION


 Magnus McCarthy
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1 179 IMAGE DATA COMPRESSION BASED ON DISCRETE WAVELET TRANSFORMATION Marina ĐOKOVIĆ, Aleksandar PEULIĆ, Željko JOVANOVIĆ, Đorđe DAMNJANOVIĆ Technical faculty, Čačak, Serbia Key words: Discrete Wavelet Transformation, Image compression, Image coding schemes Marina ĐOKOVIĆ Ph. D. Aleksandar PEULIĆ Željko JOVANOVIĆ M.Sc. Đorđe DAMNJANOVIĆ, Abstract : Image compression research aims at reducing the number of bits needed to represent an image by removing the spatial and spectral redundancies as much as possible. New algorithms for image compression based on wavelets have been developed. These methods have resulted in practical advances such as: lossless and lossy compression, progressive transmission by pixel, accuracy and resolution, region of interest coding and others. The various wavelet based image coding schemes are discussed in this paper. Each of these schemes finds use in different applications owing to their unique characteristics. The methods of lossy compression that we concentrated on are the following: the EZW algorithm, the SPIHT algorithm, the WDR algorithm, and the ASWDR algorithm. These are relatively recent algorithms which achieve some of the lowest errors per compression rate and highest perceptual quality yet reported. After describing these algorithms in detail, we show and discuss the experimental results obtained for three different types of images. We also showed that some important features of image, such as standard deviation and mean pixel intensity values, only slightly change after compression. This fact is very important in medical image compression. 1. INTRODUCTION Wavelet image coding has been fertile area of research in the image processing community in recent years particurarly in relation to image compression. It does not only provide a good compression result, but it is also suitable for progressive transmissions and provides a multi resolution capability. However, applying the wavelet transform on images for compression alone does not reduce the amount of data to be compressed, since it may remove some of the redundancy and decorrelate the neighbour pixels [1]. Typically, compression scheme can be categorized into two major categories: lossless and lossy compressions. Lossless image compression can be achieved if the original input image can be perfectly recovered from the compressed data while lossy image compression cannot regenerate the original image data. Lossy image compression, however, is able to maintain most details of the original image that is useful for diagnosis. The precise detail preservation of an image is not usually strictly required because the degraded part of the image is often not visible to a human observer. But the lossy image compression is not very commonly used in clinical practice and diagnosis because even with a slight data loss, it is possible that physicians and radiologists missed the critical diagnostic information that could be a decisive element for the diagnosis of a patient and the following treatment [2]. In general, there are three essential stages in a transformbased image compression system: transformation, quantization, and lossless entropy coding. Fig. 1. depicts the encoding and decoding processes in which the reversed stages are performed to compose a decoder. The only different part in the decoding process is that the dequantization takes place and it is followed by an inverse transform in order to approximate the original image. The purpose of transformation stage is to convert the image into a transformed domain in which correlation and entropy can be lower and the energy can be concentrated in a small part of the transformed image. Quantization stage results in loss of data because it reduces the number of bits of the transform coefficients. Coefficients that do not make significant contributions to the total energy or visual appearance of the image are represented with a small number of bits or discarded while the coefficients in the opposite case are quantized in a finer fashion. Such operations reduce the visual redundancies of the input image. The entropy coding takes place at the end of the whole encoding process. It assigns the fewest bit code words to the most frequently occurring output values and most bit code words to the unlikely outputs. This reduces the coding redundancy and thus reduces the size of the resulting bitstream [2].
2 180 Fig. 1. Block diagram of the general compression and decompression processes Since more and more medical images are in digital format, more economical and effective data compression technologies are required to minimize mass volume of digital image data produced in the hospitals [2]. Medical image compression based on wavelet decomposition has become a stateoftheart compression technology since it can produce notably better medical image results compared to the compression results that are generated by Fourier transform based methods such as the discrete cosine transform [2]. Image coding utilizing scalar quantization on hierarchical structures of transformed images has been a very effective and computationally simple technique [3]. Shapiro was the first to introduce such a technique with his Embedded Zerotree Algorithm (EZW) algorithm. The EZW algorithm, was one of the first algorithms to show the full power of waveletbased image compression. An EZW encoder is an encoder specially designed to use with wavelet transforms. The EZW encoder was originally designed to operate on images (2Dsignals) but it can also be used on other dimensional signals. Said and Pearlman successively improved the EZW algorithm by extending this coding scheme, and succeeded in presenting a different implementation based on a set partitioning sorting algorithm. This new coding scheme, called the Set Partitioning in Hierarchical Trees (SPIHT), provided an even better performance than the improved version of EZW. The problem in SPIHT is that it only implicitly locates the position of significant coefficients. This makes it difficult to perform operations, such as region selection on compressed data. By region selection, selecting a portion of a compressed image which requires increased resolution. Compressed data operations are possible with the Wavelet Difference Reduction (WDR) algorithm of Tian and Wells. Though WDR produces better perceptual results than SPIHT, there is still room for improvement. One such algorithm is the Adaptively Scanned Wavelet Diﬀerence Reduction (ASWDR) algorithm of Walker. The adjective adaptively scanned refers to the fact that this algorithm modifies the scanning order used by WDR in order to achieve better performance. All of these scalar quantized schemes employ some kind of significance testing of sets or groups of pixels, in which the set is tested to determine whether the maximum magnitude in it is above a certain threshold [4]. The results of these significance tests determine the path taken by the coder to code the source samples. These significance testing schemes are based on some very simple principles which allow them to exhibit excellent performance. An interesting thing to note in these schemes is that all of them have relatively low computational complexity, considering the fact that their performance is comparable to the bestknown image coding algorithms. An important characteristic that this class of coders possesses is the property of progressive transmission and embedded nature. Progressive transmission refers to the transmission of information in decreasing order of its information content [5]. In other words, the coefficients with the highest magnitudes are transmitted first. Since all of these coding schemes transmit bits in decreasing bit plane order, this ensures that the transmission is progressive. Such a transmission scheme makes it possible for the bit stream to be embedded, i.e., a single coded file can used to decode the image at various rates less than or equal to the coded rate, to give the best reconstruction possible with the particular coding scheme. With these desirable features of excellent performance and low complexity, along with others such as embedded coding and progressive transmission, these scalar quantized significance testing schemes have recently become very popular in the search for practical, fast and efficient image coders, and in fact, have become the basis for serious consideration for future image compression standards. 2. TRANSFORMATION Wavelet transform exploits both the spatial and frequency correlation of data by dilations (or contractions) and translations of mother wavelet on the input data. It supports the multiresolution analysis of data i.e. it can be applied to different scales according to the details required, which allows progressive transmission and zooming of the image without the need of extra storage [6]. Another encouraging feature of wavelet transform is its symmetric nature that is both the forward and the inverse transform has the same complexity, building fast compression and decompression routines [7]. The implementation of wavelet compression scheme is very similar to that of subband coding scheme: the signal is decomposed using filter banks. The output of the filter banks is downsampled, quantized, and encoded. The decoder decodes the coded representation, upsamples and recomposes the signal [8]. Details of multiresolution analysis are explained below. 2.1 Multiresolution analysis Multiresolution 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. This approach makes sense especially when the signal at hand has high frequency components for short durations and low frequency components for long durations. The continuous wavelet transform was developed as an alternative approach to the short time Fourier transform (STFT) to overcome the resolution problem. The wavelet analysis is done in a similar way to the STFT analysis, in the sense that the signal is multiplied with a function, (it the wavelet), similar to the window function in the STFT, and the transform is computed separately for different segments of the timedomain signal. The continuous wavelet transform is defined as follows: (, )= 1 ( ) ( ) (1) As seen in the above equation, the transformed signal is a function of two variables, τ and s, the translation and scale parameters, respectively. (t) is the transforming function, and it is called the mother wavelet.
3 The term translation is used in the same sense as it was used in the STFT; it is related to the location of the window, as the window is shifted through the signal. This term, obviously, corresponds to time information in the transform domain. However, we do not have a frequency parameter, as we had before for the STFT. Instead, we have scale parameter which is defined as 1/frequency. The term frequency is reserved for the STFT. The parameter scale in the wavelet analysis is similar to the scale used in maps. As in the case of maps, high scales correspond to a nondetailed global view (of the signal), and low scales correspond to a detailed view. Similarly, in terms of frequency, low frequencies (high scales) correspond to a global information of a signal (that usually spans the entire signal), whereas high frequencies (low scales) correspond to a detailed information of a hidden pattern in the signal (that usually lasts a relatively short time). In today's world, computers are used to do most computations. It is apparent that neither the FT, nor the STFT, nor the CWT can be practically computed by using analytical equations. It is therefore necessary to discretize the transforms. First consider the discretization of the scale axis. Among that infinite number of points, only a finite number are taken, using a logarithmic rule. The most common value for the base of the logarithmis 2 because of its convenience. If 2 is chosen, only the scales 2, 4, 8, 16, 32, 64,...etc. are computed. The time axis is then discretized according to the discretization of the scale axis. Since the discrete scale changes by factors of 2, the sampling rate is reduced for the time axis by a factor of 2 at every scale. Expressing the above discretization procedure in mathematical terms, the scale discretization is S=S0j, and translation discretization is τ=k S0j τ0 where S0>1 and τ0>0. When discretized dilation and tanslacija replace the continuous wavelet function : 1, =, ( )= obtain the discretized wavelet function: (2) (3) Although the discretized continuous wavelet transform enables the computation of the continuous wavelet transform by computers, it is not a true discrete transform. As a matter of fact, the wavelet series is simply a sampled version of the CWT, and the information it provides is highly redundant as far as the reconstruction of the signal is concerned. This redundancy, on the other hand, requires a significant amount of computation time and resources. The discrete wavelet transform (DWT), on the other hand, provides sufficient information both for analysis and synthesis of the original signal, with a significant reduction in the computation time. The main idea is the same as it is in the CWT. A timescale 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 181 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 upsampling and downsampling (subsampling) operations. Subsampling a signal corresponds to reducing the sampling rate, or removing some of the samples of the signal. The DWT analyzes the signal at different frequency bands with different resolutions by decomposing the signal into a coarse approximation and detail information. DWT employs two sets of functions, called scaling functions and wavelet functions, which are associated with low pass and highpass filters, respectively. The decomposition of the signal into different frequency bands is simply obtained by successive highpass and lowpass filtering of the time domain signal. The original signal x[n] is first passed through a halfband highpass filter g[n] and a lowpass filter h[n]. After the filtering, half of the samples can be eliminated according to the Nyquist s rule, since the signal now has a highest frequency of p /2 radians instead of p. The signal can therefore be subsampled by 2, simply by discarding every other sample. This constitutes one level of decomposition and can mathematically be expressed as follows: [ ]= [ ]= [ ] [2 ] (4) [ ] [2 ] where yhigh[k] and ylow[k] are the outputs of the highpass and lowpass filters, respectively, after subsampling by 2. This decomposition halves the time resolution since only half the number of samples now characterizes the entire signal. However, this operation doubles the frequency resolution, since the frequency band of the signal now spans only half the previous frequency band, effectively reducing the uncertainty in the frequency by half. The above procedure, which is also known as the subband coding, can be repeated for further decomposition. At every level, the filtering and subsampling will result in half the number of samples (and hence half the time resolution) and half the frequency band spanned (and hence double the frequency resolution) [4]. The difference between this transformation and Fourier transformation is that it is known time of occurrence frequency. However, the time during which these frequencies will have a resolution which depends on the level at which they appear. If the main information signal contained in very low frequencies, their temporal location will not be very accurate, because only a few samples used to represent the signals at these frequencies. This algorithm provides good temporal resolution for high frequencies and good frequency resolution for low frequency signals. Frequencies that are not highlighted in the original signal will have very low amplitude and that part of the signal obtained by the discrete wavelet transformation can be discarded without significant loss of information, which significantly reduces the amount of data. In the case of twodimensional signals such as images, in this way is formed multiresolution pyramid, shown in Fig. 2. At each higher level is kept image two times less
4 182 resolution than it was at the previous level and image details needed for the reconstruction of the signal. In the EZW s algorithm, the information on which the coefficients are significant is generated and then encoded via quantization. The significance map determines whether a DWT coefficient is to be quantized as zero or not. A wavelet coefficient x is considered. insignificant with respect to a given threshold T if x T Otherwise a coefficient is called significant. Since the wavelet decomposition has the hierarchical structure in which each coefficient can be related to a set of coefficients that is at the next finer resolution level, a tree structure depicted in Fig. 5 can be defined as the concept of descendants and ancestors. The coefficient at the coarse scale is called the parent, and all coefficients corresponding to the same spatial location at the next finer scale of similar orientation are called children. For a given parent, the set of all coefficients at all finer scales of similar orientation corresponding to the same location are called descendants. Similarly, for a given child, the set of coefficients at all coarser scales of similar orientation corresponding to the same location are called ancestors. Fig. 4. shows that parents must be scanned before children. Also note that all positions in a given subband are scanned before the scan moves to the next subband [10]. Fig. 2. Multiresolution pyramid Multiresolution representation of images in each level of decomposition consists of a discrete image approximation at lower resolution, and three image detail. Approximation corresponds to part of the spectrum which is obtained by lowpass filtering (LP) in both directions in the frequency plane. One of the image detail is obtained with horizontal LP and vertical highpass filtering (HP), the second vertical LP and horizontal HP filtering, while the diagonal detail images obtained by HP filtering in both directions. Multiple repeat leads to images with worsening resolution, which corresponds to a pyramidal decomposition, Fig. 3. Fig. 4. Scanning order of the subbands for encoding a significance map Fig. 3. Pyramidal decomposition of image 3. QUANTIZATION 3.1 Embedded Zerotree Wavelet The embedded zerotree wavelet (EZW) is an effective algorithm employed in quantization stage. At a given compression ratio in bit rate, EZW is able to achieve the best image quality and encode the image so that all lower bit rate encodings are embedded at the beginning of the final bitstream. An embedded coding is a process of encoding the transform magnitudes that allows for progressive transmission of the compressed image. Zerotrees are a concept that allows for a concise encoding of the positions of signi_cant values that result during the embedded coding process. The EZW algorithm is based on four key concepts: 1) a discrete wavelet transform or hierarchical subband decomposition, 2) prediction of the absence of significant information across scales by exploiting the selfsimilarity inherent in images, 3) entropycoded successiveapproximation quantization, and 4) universal lossless data compression which is achieved via adaptive arithmetic coding [10]. Given a threshold T to determine whether or not a coefficient is significant, a coefficient x is said to be an element of a zerotree root (ZRT) for the threshold T if itself and all of its descendents are insignificant with respect to the threshold T. An element of a zerotree for threshold T is a Zerotree root if it is not the descendant of a previously found zerotree root for threshold T, i.e., it is not predictably insignificant from the discovery of a zerotree root at a coarser scale at the same threshold. For the case which not all the descendants are insignificant, the coefficients are encoded as isolated zero (IZ). For encoding a significant coefficient, the symbol POS and NEG are used. Therefore, given a threshold T, the wavelet coefficients can be represented by the four symbols: zerotree root (ZRT), isolated zero (IZ), positive significant (POS) and negative significant (NEG) [2]. To perform the embedded coding, successiveapproximation quantization (SAQ) is applied. SAQ is related to bitplane encoding of the magnitudes. The SAQ sequentially applies a sequence of thresholds To,, TN1, to determine significance, where the thresholds are chosen so that Ti = Ti1 /2. The initial threshold To is chosen so that Xj <2T0 for all transform coefficients xj [10]. During the encoding (and decoding), two separate lists of wavelet coefficients are maintained. At any point in the process, the dominant list contains the coordinates of those coefficients that have not yet been found to be significant in the same relative order as the initial scan. The subordinate list contains the magnitudes of those coefficients that have been
5 found to be significant. For each threshold, each list is scanned once. During a dominant pass, coefficients with coordinates on the dominant list, i.e., those that have not yet been found to be significant, are compared to the threshold T, to determine their significance, and if significant, their sign. This significance map is then zerotree coded. Each time a coefficient is encoded as significant, (positive or negative), its magnitude is appended to the subordinate list, and the coefficient in the wavelet transform array is set to zero so that the significant coefficient does not prevent the occurrence of a zerotree on future dominant passes at smaller thresholds. A dominant pass is followed by a subordinate pass in which all coefficients on the subordinate list are scanned and the specifications of the magnitudes available to the decoder are refined to an additional bit of precision. More specifically, during a subordinate pass, the width of the effective quantizer step size, which defines an uncertainty interval for the true magnitude of the coefficient, is cut in half. For each magnitude on the subordinate list, this refinement can be encoded using a binary alphabet with a 1 symbol indicating that the true value falls in the upper half of the old uncertainty interval and a 0 symbol indicating the lower half. The string of symbols from this binary alphabet that is generated during a subordinate pass is then entropy coded. Note that prior to this refinement, the width of the uncertainty region is exactly equal to the current threshold. After the completion of a subordinate pass the magnitudes on the subordicate list are sorted in decreasing magnitude, to the extent that the decoder has the information to perform the same sort. The process continues to alternate between dominant passes and subordinate passes where the threshold is halved before each dominant pass [10]. In the decoding operation, each decoded symbol, both during a dominant and a subordinate pass, refines and reduces the width of the uncertainty interval in which the true value of the coefficient (or coefficients, in the case of a zerotree root) may occur. The reconstruction value used can be anywhere in that uncertainty interval. The encoding stops when some target stopping condition is met, such as when the bit budget is exhausted. The encoding can cease at any time and the resulting bit stream contains all lower rate encodings [10]. 3.2 Set Partitioning in Hierarchical Trees The SPIHT coder is a highly refined version of the EZW algorithm and is a powerful image compression algorithm that produces an embedded bit stream from which the best reconstructed images in the mean square error sense can be extracted at various bit rates. SPIHT stands for Set Partitioning in Hierarchical Trees. The term Hierarchical Trees refers to the quadtrees that we defined in our discussion of EZW. Set Partitioning refers to the way these quadtrees divide up, partition, the wavelet transform values at a given threshold [11]. By a careful analysis of this partitioning of transform values, Said and Pearlman were able to greatly improve the EZW algorithm, significantly increasing its compressive power. SPIHT is a waveletbased image compression coder that offers a variety of good characteristics [12]. These characteristics include: Good image quality with a high PSNR, Fast coding and decoding, A fully progressive bitstream, Can be used for lossless compression, Ability to code for exact bit rate or PSNR. 183 The main goal of this transformation is to first transfer most important information, those that will result in minimum distortion [13]. To measure distortion takes the minimum square error (MSE). Coefficients of greater importance is transmitted first because they contain the most information. In addition, the transfer coefficients should be transmitted most significant bits first. One important fact in the design of the sorting algorithm is that there is no need to sort all coefficients. Actually, an algorithm which simply selects the coefficients such that 2n ci,j < 2n+1, with n decremented in each pass. Given n, if ci,j 2n, then the coefficient is said to be significant; otherwise it is called insignificant. The sorting algorithm divides the sets of pixels into partitioning subsets Tm and performs the magnitude test (, ), [14]. If the decoder receives a no as that answer, that is the subset is insignificant, then it knows that all coefficients in Tm are insignificant. If the answer is yes, that is the subset is significant, then a certain rule shared by the decoder and encoder is used to partition Tm into new subsets and the significance test is then applied to the new subsets. This set division process continues until the magnitude test is done to all single coordinate significant subsets in order to identify each significant coefficient [14]. To reduce the number of magnitude comparisons, a set partitioning rule that uses an expected ordering in the hierarchy defined by the sub band pyramid, is used [15]. The objective is to create new partitions such that subsets expected to be insignificant contain a large number of elements, and subsets expected to be significant contain only one element. The relationship between magnitude comparisons and message bits is given by the significance function ( )= 1, 0, (, ), 2 (5) Fig. 5. shows how the spatial orientation tree is defined in a pyramid constructed with recursive fourband splitting. Each node of the tree corresponds to a pixel, and is identified by the pixel coordinate. Its direct descendants (offspring) correspond to the pixels of the same spatial orientation in the next finer level of the pyramid. The tree is defined in such a way that each node has either no offspring or four offsprings, which always form a group of 2X2 adjacent pixels. The pixels in the highest level of the pyramid are the tree roots and are also grouped in 2X2 adjacent pixels. However, their offspring branching is different, and in each group one of them (indicated by the star in Fig. 5. has no descendants. Parts of the spatial orientation trees are used as the partitioning subsets in the sorting [14]. Fig. 5. Parent offspring dependencies in spatial orientation tree
6 184 To make it possible to distinguish sets of pixels, we introduced the following phrases: A (i,j): set of coordinates of all children of node (i,j) D (i,j): set of coordinates of all descendants of node (i,j) H: set of coordinates of all nodes at the highest level (root). It also true: L(i, j) = D(i, j) O(i, j). These relations between coordinates of pixels are valid at each level, except at the highest and lowest level: O(i, j) = {(2i, 2j), (2i, 2j+1), (2i+1, 2j), (2i+1, 2j+1)}. Rules for division sets are: 1. Initial division is formed a set {(i, j)} and D (i, j) for all (i, j) H. 2. If D (i, j) is significant, then it is divided into L (i, j) and the four sets with one element where (k, l) O (i, j). 3. If L (i, j) is significant, then it is divided into four sets of D (k, l) where (k, l) O (i, j). Wavelets coefficients which are not significant at the nth bitplane level may be significant at (n1)th bitplane or lower. This information is arranged, according to its significance, in three separate lists [16]: List of insignificant sets (LIS), List of insignificant pixels (LIP) and List of significant pixels (LSP). otherwise remove entry (i,j) from the LIS (2.2.2) If the entry is of type B then Output Sn{L(i, j)} If Sn{L(i, j)} = 1, then Add each (k,l) O(i, j) to the end of the LIS as entry of type A Remove (i,j) from the LIS In each list, the members are defined by coordinates (i, j). LIP and LSP lists containing the coordinates of pixels. LIS is a set D (i, j) or a set L (i, j). To be able to distinguish, say that the set D (i, j) is type A, and a set L (i, j) is type B. During the sorting in algorithm, pixels in the LIP list are tested and the significant coefficients are written to the LSP. In a similar way, sets are tested. If one has become significant, is deleted from the list and divided into subsets. The SPIHT algorithm can be summarized as follows: (1) Initialization: = log,,. Set the LSP as empty list and add the coordinates (i,j) H to the LIP and only those with descendents 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 Sn(i, j) (2.1.2) If Sn(i, j) = 1, then move (i,j) to the LSP and output the sign of ci,j (2.2) For each entry (i,j) in the LIS do: (2.2.1) If the entry is of type A then Output Sn{D(i, j)} If Sn{D(i, j)} = 1, then for each (k,l) O(i, j), do: Output Sn(k, l) If Sn(k, l) = 1, then add (k, l) to the LSP and output the sign of ci,j o If Sn(k, l) = 0, then add (k, l) to the end of LIP If L(i, j) 0, then move (i,j) to the end of the LIS as entry of type B, and go to step 2.2.2; o o (3) Refinement Pass: For each entry (i,j) in the LSP except those included in the last sorting pass (i.e. with the same n), output the nth most significant bit of ci,j ; (4) Quantization step update: decrement n by 1 and go to step 2.Some of the best results highest PSNR values for given compression ratios for a wide variety of images have been obtained with SPIHT. Hence, it has become the benchmark stateoftheart algorithm for image compression. 3.3 Wavelet Difference Reduction One of the defects of SPIHT is that it only implicitly locates the position of significant coefficients. This makes it difficult to perform operations which depend on the position of significant transform values, such as region selection on compressed data. Region selection, also known as region of interest (ROI), means a portion of a compressed image that requires increased resolution [17]. This can occur, for, example, with a portion of a low resolution medical image that has been sent at a low bpp rate in order to arrive quickly. Such compressed data operations are possible with the WDR algorithm of Tian and Wells. The term difference reduction refers to the way in which WDR encodes the locations of significant wavelet transform values, which will be describe below. Although WDR will not typically produce higher PSNR values than SPIHT, WDR can produce perceptually superior images, especially at high compression ratios [17]. After choosing threshold with which it will compare the coefficients of wavelet transformation, follows the significance pass phase of the algorithm. The output of the significance pass phase of the algorithm consists of the signs of significant value with the bit patterns that concisely describe the precise location of significant coefficients. Refinement pass phase algorithm performs quantization bits received in the significance pass stage [18]. The best way to understand this method is to consider a simple example. Suppose that the significant values are w(2) = +34.2, w(3) = 33.5, w(7) = +48.2, w(12) = , and w(34) = The indices for these significant values are 2, 3, 7, 12, and 34. Rather than working with these values, WDR works with their successive differences: 2, 1, 4, 5, 22. In this latter list, the first number is the starting index and each successive number is the number of steps needed to reach the next index. The binary expansions of these successive differences are (10)2, (1)2, (100)2, (101)2 and (10110)2. Since the most significant bit for each of these expansions is always 1, this bit can be dropped and the signs of the significant transform values can be used instead as separators in the symbol stream. The resulting symbol stream for this example is then When the most significant bit is removed, the original sequence of bits is replaced by the reduced number of bits.
7 Notice, in particular, that the reduced binary expansion of 1 is empty. The reduced binary expansion of 2 is just the 0 bit, the reduced binary expansion of 3 is just the 1 bit, and so on. The WDR algorithm simply consists of replacing the significance pass in the procedure with the following step: (1) Initialization: Choose initial threshold, T = T0, such that all transform values satisfy w(m) < T0 and at least one transform value satisfies ( ) /2. Set the initial scan order to be the baseline scan order. (2) Update threshold: Let Tk = Tk1/2. (3) Significance pass: Perform the following procedure on the insignificant indices in the baseline scan order: Initialize stepcounter C = 0 Let Cold =0 Do Get next insignificant index m Increment stepcounter C by 1 If ( ) then Output sign w(m) and set wq(m)=tk Move m to end of sequence of significant indices Let n = CCout Set Cold = C If n > 1 then Output reduced binary expansion of n Else if ( ) < then Let wq(m) retain its initial value of 0. Loop until end of insignificant indices. Output endmarker. (4) Refinement pass: Scan through significant values found with higher threshold values Tj, for j < k (if k = 1 skip this step). For each significant value w(m), do the following: then If ( ) [ ( ), ( ) + Output bit 0 Else if ( ) [ ( ) +, ), then ( )+2, Output bit 1 Replace value of wq(m) by wq(m) + Tk. (5) Loop: Repeat steps 2 through 4. The output for the endmarker is a plus sign, followed by the reduced binary expansion of n = C+1Cold, and a final plus sign. The procedure continues until the list of coefficients obtained by wavelet transformation is exhausted. 185 features of WDR: low complexity, region of interest, embeddedness, and progressive PSNR [19]. ASWDR adapts the scanning order so as to predict locations of new significant values. If a prediction is correct, then the output specifying that location will just be the sign of the new significant value, the reduced binary expansion of the number of steps will be empty. Therefore a good prediction scheme will significantly reduce the coding output of WDR. The prediction method used by ASWDR is the following: If w(m) is significant for threshold T, then the values of the children of m are predicted to be significant for halfthreshold T/2. As the pseudocode presented below will show, the only difference between ASWDR and WDR is in the predictive scheme employed by ASWDR to create new scanning orders. Consequently, if ASWDR typically encodes more values than WDR does, this must be due to the success of the predictive scheme. Notice that the significance pass portion of this procedure is the same as the WDR significance pass described above and that the refinement pass is the same, too. The one new feature is the insertion of a step for creating a new scanning order. The ASWDR algorithm can be summarized as follows: ((1) Initialization: Choose initial threshold, T = T0, such that all transform values satisfy w(m) < T0 and at least one transform value satisfies ( ) /2. Set the initial scan order to be the baseline scan order. (2) Update threshold: Let Tk = Tk1/2. (3) Significance pass: Perform the following procedure on the insignificant indices in the baseline scan order: Initialize stepcounter C = 0 Let Cold =0 Do Get next insignificant index m Increment stepcounter C by 1 If ( ) then Output sign w(m) and set wq(m)=tk Move m to end of sequence of significant indices Let n = CCout Set Cold = C If n > 1 then Output reduced binary expansion of n Else if ( ) < then Let wq(m) retain its initial value of 0. Loop until end of insignificant indices. Output endmarker. It is not hard to see that WDR is of no greater computational complexity than SPIHT. For one thing, WDR does not need to search through quadtrees as SPIHT does. The calculations of the reduced binary expansions adds some complexity to WDR, but they can be done rapidly with bitshift operations. (4) Refinement pass: Scan through significant values found with higher threshold values Tj, for j < k (if k = 1 skip this step). For each significant value w(m), do the following: If ( ) [ ( ), ( ) + ), then Output bit 0 Else if ( ) [ ( ) +, ( ) + 2, then Output bit 1 Replace value of wq(m) by wq(m) + Tk. 3.4 Adaptively Scanned Wavelet Difference Reduction ASWDR adapts the scanning procedure used by WDR in order to predict locations of significant transform values at half thresholds. These methods retain all of the important (5) Create new scan order: For each level j in the wavelet transform (except for j = 1), scan through the significant values using the old scan order. The initial part of the new scan order at level j 1 consists of the indices for
8 186 insignificant values corresponding to the child indices of these level j significant values. Then, scan again through the insignificant values at level j using the old scan order. Append to the initial part of the new scan order at level j  1 the indices for insignificant values corresponding to the child indices of these level j significant values. Note: No change is made to the scan order at level L, where L is the number of levels in the wavelet transform. compression codecs it is used as an approximation to human perception of reconstruction quality, therefore in some cases one reconstruction may appear to be closer to the original than another, even though it has a lower PSNR, a higher PSNR would normally indicate that the reconstruction is of higher quality. One has to be extremely careful with the range of validity of this metric; it is only conclusively valid when it is used to compare results from the same codec, or codec type, and same content. It is most easily defined via the mean squared error (MSE) which for two m n monochrome images I and K where one of the images is considered a noisy approximation of the other is defined as: = (7) [ (, ) (, )] (6) Loop: Repeat steps 2 through 5. The creation of the new scanning order only adds a small degree of complexity to the original WDR algorithm. Moreover, ASWDR retains all of the attractive features of WDR: simplicity, progressive transmission capability, and ROI capability. The PSNR is defined as: 4. ENTROPY CODING The output symbol stream is an input to an entropy encoder to complete the last stage of the compression without adding distortion. The lossless entropy encoding process replaces the symbol stream produced in the quantization stage with a sequence of binary codewords which is called a bit stream. The probability of the corresponding symbol is proportional to the length of a codeword. The smallest possible number of bits that is required to represent a symbol sequence can be defined as the entropy of the symbol source: = log (6) Here the pi is the probability of the ith symbol. In the optimal case, the sum of the probability would be equaled to 1 and the ith symbol would be log. We can define the entropy as the expected length of binary code over all possible symbols [2]. = 10 log = 20 log ( ) (8) Here, MAXI is the maximum possible pixel value of the image. Generally, when samples are represented using linear PCM with B bits per sample, MAXI is 2B 1. For color images with three RGB values per pixel, the definition of PSNR is the same except the MSE is the sum over all squared value differences divided by image size and by three. When the two images are identical, the MSE will be zero. For this value the PSNR is undefined. The Table 1 shows the result of compression ratio and PSNR value using each of the four algorithm for Vegetables truecolor image of size 512 x 512. Table 1. CR & PSNR results for 512 x 512 Vegetables 5. IMAGE COMPRESSION RESULTS AND DISCUSSION A measure of achieved compression is given by the compression ratio (CR) and the BitPerPixel (BPP) ratio. CR and BPP represent equivalent information. CR indicates that the compressed image is stored using CR % of the initial storage size while BPP is the number of bits used to store one pixel of the image. For a grayscale image the initial BPP is 8. For a truecolor image the initial BPP is 24, because 8 bits are used to encode each of the three colors (RGB color space) [20]. The challenge of compression methods is to find the best compromise between a low compression ratio and a good perceptual result. Also, two of the error metrics used to compare the various image compression techniques are the Mean Square Error (MSE) and the Peak Signal to Noise Ratio (PSNR). The phrase Peak Signal to Noise ratio, often abbreviated PSNR, is an engineering term for the ratio between the maximum possible power of a signal and the power of corrupting noise that affects the fidelity of its representation. Because many signals have a very wide dynamic range, PSNR is usually expressed in terms of the logarithmic decibel scale. The PSNR is most commonly used as a measure of quality of reconstruction of lossy compression codecs. The signal in this case is the original data, and the noise is the error introduced by compression. When comparing The Table 2 shows the result of compression ratio and PSNR value using each of the four algorithm for Statue truecolor image of size 256 x 256. Table 2. CR & PSNR results for 256 x 256 Statue
9 The Table 3 shows the result of compression ratio and PSNR value using each of the four algorithm for Cameraman intensity image of size 256 x Table 6. CR & PSNR results for 1024 x 1024 Mammogram Table 3. CR & PSNR results for 256 x 256 Cameraman The TAable 4 shows the result of compression ratio and PSNR value using each of the four algorithm for Lena indexed image of size 512 x 512. Table 4. CR & PSNR results for 512 x 512 Lena The Table 5 shows the result of compression ratio and PSNR value using each of the four algorithm for MRI indexed image of size 128 x 128. Table 5. CR & PSNR results for 128 x 128 MRI We show the results of compression of six test images, so in TABLES I  VI we give PSNR and CR values for several test images. These data show that SPIHT produces higher PSNR values than the three other algorithms in case all six images. SPIHT is wellknown for its superior performance when PSNR is used as the error measure. We can see that EZW, WDR and ASWDR give almost the same PSNR values for truecolor images Vegetables and Statue and for intensity image Cameraman. For indexed images Lena, MRI and Mammogram, ASWDR gives much higher PSNR values compared with WDR, almost as good as those for SPIHT, while the PSNR values that give EZW are mainly similar to ASWDR values. High PSNR values, however, are not the sole criteria for the performance of lossy compression algorithms. We shall discuss other criteria below. It can be seen that the compression ratio increase, when the BPP is increased. CR values vary from image to image very much, so we can not say which algorithm provides lower CR values for a specific type of image. For example, both images, MRI and Mammogram, are indexed images but in case MRI image all six algorithms provides very high CR values, while in case Mammogram image all six algorithms provides very low CR values. The difference is really large, MRI image is stored using % of the initial storage size when we used ASWDR and BPP=1, and Mammogram image is stored using 2.09 % of the initial storage size when we also used ASWDR and BPP=1. The Table 7 shows the result of compression duration value using each of the four algorithm for all six images. Duration of compression is reflected in seconds. Table 7. Duration of compression expressed in seconds for all six images The Table 6 shows the result of compression ratio and PSNR value using each of the four algorithm for mammogram indexed image of size 1024 x In general, the duration of compression depends on the size of the image that we compress and on performance of the computer on which we perform compression. Table 7 shows that SPIHT algorithm has the least duration of compression for all images, compared with the other three algorithms. In other words, the algorithm is the fastest. This is due to reduced the number of magnitude comparisons, a set partitioning rule that uses an expected
10 188 ordering in the hierarchy defined by the sub band pyramid, is used. The objective is to create new partitions such that subsets expected to be insignificant contain a large number of elements, and subsets expected to be significant contain only one element. EZW and WDR provides similar duration of compression, while ASWDR s duration of compression is a significantly increased. This must be due to the success of the predictive scheme. Fig shows original images and compares images compressed with EZW, SPIHT, WDR and ASWDR. Compression results in terms of quality compressed images are similar in the case EZW, WDR and ASWDR algorithm. SPIHT provides poorer quality of compressed images, and the reason is its characteristic poor preservation of edges within the image. (d) (a) (a) (b) (e) Fig. 8. Results for Cameraman (a) Original image (b) EZW coding (c) SPIHT coding (d) WDR coding (e) ASWDR coding (c) (b) (d) (c) (e) Fig. 9. Results for Lena (a) Original image (b) EZW coding (c) SPIHT coding (d) WDR coding (e) ASWDR coding (d) (e) Fig. 6. Results for Vegetables (a) Original image (b) EZW coding (c) SPIHT coding (d) WDR coding (e) ASWDR coding (a) (a) (b) (d) (c) (b) (c) (d) (e) Fig. 10. Results for MRI (a) Original image (b) EZW coding (c) SPIHT coding (d) WDR coding (e) ASWDR coding (e) Fig. 7. Results for Statue (a) Original image (b) EZW coding (c) SPIHT coding (d) WDR coding (e) ASWDR coding (a) (a) (b) (c) (b) (d) (c) (e) Fig. 11. Results for Mammogram (a) Original image (b) EZW coding (c) SPIHT coding (d) WDR coding (e) ASWDR coding
11 The EZW, WDR and ASWDR algorithms preserve more of the fine details in the image. SPIHT erases many fine details. Although EZW, WDR and ASWDR will not produce higher PSNR values than SPIHT, as observed from Tables 1 7, they can produce perceptually superior images, especially at high compression rates. High compression ratio images like these are used in reconnaissance and in medical applications, where fast transmission and ROI (region selection) are employed, as well as multiresolution detection. The WDR and ASWDR algorithms do allow for ROI while SPIHT does not. Furthermore, their superior performance in displaying edge details at low bit rates facilitates multiresolution detection. We tested some image s features such as standard deviation and mean pixel intensity value. We have obtained results which show that the standard deviation values and mean pixel intensity values, obtained before and after image compression, are almost the same for all four algorithms. Standard deviation and mean pixel intensity value we calculated as the following. Let (i; j) be the spatial location in the mammogram at row i and column j; uij be the pixel brightness at (i; j). For each MbyN image, the standard deviation is the square root of the variance and is given by the following equation: = ( ) (9) where µ is the mean of the input matrix u and is given by the following equation: = (10) The Table 8 shows the standard deviation values and mean pixel intensity values, obtained before and after image compression, using each of the four algorithm for images Cameraman, Lena, MRI and Mammogram. Table 8. Standard deviation and mean pixel intensity values, obtained before and after image compression As seen from Table 8, standard deviation values and mean pixel intensity values of images compressed using EZW, WDR and ASWDR are identical. Using SPIHT, we got a bit worse results, ie., the difference between standard deviation values and mean pixel intensity values of original images and these values of compressed images is slightly greater. However, all four algorithms provide excellent results in terms of keeping important information of image. 6. CONCLUSION Discrete wavelet transformation analyzes the image at different frequency bands, different resolutions, decomposing the image in a coarse approximations and detailed 189 informations. Using twodimensional wavelet analysis images can be effectively compressed without sacrificing image quality. The EZW algorithm was one of the first algorithms to show the full power of waveletbased image compression. The SPIHT coder is a highly refined version of the EZW algorithm, he is faster and significantly increases the PSNR values. Some of the SPIHT s gaps are bad preserve edges within the image and poorer quality of compressed images. These and many other defects of SPIHT overcome by using WDR algorithm, which is simpler and provides better quality of compressed images, especially when it comes to the high degree of compression. Though WDR produces better perceptual results than SPIHT, there is still room for improvement. One such algorithm is the ASWDR algorithm. The adjective adaptively scanned refers to the fact that this algorithm modifies the scanning order used by WDR in order to achieve better performance. We compare PSNR values for WZW, SPIHT, WDR and for ASWDR on several images at various compression ratios. In every case, SPIHT has higher PSNR values. We saw that EZW, WDR and ASWDR give almost the same PSNR values for truecolor and intensity images. For indexed images, ASWDR gives much higher PSNR values compared with WDR, almost as good as those for SPIHT, while the PSNR values that give EZW are mainly similar to ASWDR values. We also showed that CR values vary from image to image very much, so we can not say which algorithm provides lower CR values for a specific type of image. Another one important features of image compression algorithms is the speed of compression. SPIHT algorithm has the least duration of compression for all images, compared with the other three algorithms. EZW and WDR provides similar duration of compression, while ASWDR s duration of compression is a significantly increased. Initial results that we obtained show that it is possible to compress medical images such as mamograms and MRI images using any of these four algorithms without loss of important information. Each technique can be well suited with different images based upon the user requirements. Our proposal is to use ASWDR algorithm for indexed images compression, especially in case of medical images. For other type of images, when a good quality of compressed images has a crucial importance, we suggest any of three algorithms, EZW, WDR or ASWDR. SPIHT is a good choice when the quality of compressed images does not has significance and necessary are good values for the parameters that characterize the compression, such as great PNSR, low CR and short duration of compression. The various wavelet based image coding schemes are discussed in this paper. Each of these schemes finds use in different applications owing to their unique characteristics. Though there a number of coding schemes available, the need for improved performance and wide commercial usage, demand newer and better techniques to be developed. 7. REFERENCES [1] O. Khalifa, Wavelet coding design for image data compression, The International Arab Journal of Information Technology, Vol.2, No.2, pp , [2] J.J. Ding, Introduction to Medical Image Compression Using Wavelet Transform, National Taiwan University Graduate Institute of Communication Engineering, 2007.
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