WAVELET BASED SPIHT COMPRESSION FOR DICOM IMAGES

Transcription

1 Degree Project WAVELET BASED SPIHT COMPRESSION FOR DICOM IMAGES Supervisor: Sven Nordebo School of Computer Sciences, Physics and Mathematics Submitted for the degree of Master in Electrical Engineering Specialized in Signal Processing and Wave Propagation Vimal Rathinasamy, Iyyappan Dhasarathan, Tang Cui Subject: Master Thesis Level: Second Course code: 5ED06E

2

3 ACKNOWLEDGEMENTS It is a great privilege for us to thank our beloved professor and advisor Sven Nordebo first for the motivation and encouragement he has given us, which led us finish our thesis in a successful manner. We were very fortunate to work under him as the kind of support he provided us was exceptional, without which we could not have finished our thesis in appropriate time. So, we are highly grateful to our beloved Professor and Advisor Sven Nordebo for his support. It s our duty to thank the Swedish Educational Department for giving us an opportunity to be here and make our dream come true. The standard of Education has been fabulous in Sweden and we are very fortunate to be educated under this high standard. We cordially thank Swedish Government for this wonderful opportunity. We thank our dear parents without whom our dream of doing this Master s degree programme would not have been possible. So, we pay our heartfelt gratitude to our beloved parents. We thank all our department faculty for educating us in various courses which led us achieve a great success in this programme. We have to say that it s only because of the knowledge gained through all those Professors in several subjects in this programme, we are able to finish our thesis ultimately. So, we are grateful to all our professors of our department and we always look for their wishes in our career. Finally, we would like to thank all our friends who supported us in some way for successfully finishing this thesis. Thanks everyone!

4 Wavelet Based SPIHT Compression for DICOM Images Vimal Rathinasamy, T131 Iyyappan Dhasarathan, Tang Cui,

5 ABSTRACT Generally, image viewers do not include the scalability for the image compression and ecient encoding and decoding for easy transmission. They also never consider the specic requirements of the heterogeneous networks constituted by the Global Packet Radio Service(GPRS), Universal Mobile Telecommunication System(UMTS), Wireless Local Area Network(WLAN) and Digital Video Broadcasting(DVB-H). This work contains the medical application with viewer for the Digital Imaging and Communications in Medicine(DICOM) images as its core content. This application discusses the scalable wavelet-based compression, retrival and the decompression of the DICOM images. This proposed application is compatible with the mobile phones activated in the heterogeneous netwoks. This paper also explains about the performance issues when this application is used in prototype heterogenous networks. 2

6 Contents 1 INTRODUCTION Background Why Do We Need Compression? What are the Principles behind Compression? Dierent Classes of Compression Techniques Image Compression Process Objective of the Thesis Thesis Organization WAVELET TRANSFORM OVERVIEW Wavelet Transform Scaling Shifting Scale and Frequency Discrete Wavelet Transform One-Stage Filtering Multiple-Level Decomposition Wavelet Reconstruction Reconstructing Approximations and Details D Wavelet Transform D Transform Hierarchy Line based Wavelet Integer lifting SPIHT (SET PARTITIONING IN HIERARCHICAL TREES) Introduction Approach Set Partitioning Algorithm Image Quality Progressive Image Transmission Optimized Embedded Coding Lossless Compression Rate or Distortion Specication Encoding/Decoding Speed Hierarchical Tree SPIHT coding algorithm Algorithm Performance Evaluation

7 3.12 Error Metrics SIMULATION RESULTS, CONCLUSION AND FUTURE WORK Simulation Conclusion and Future Work

8 Chapter 1 INTRODUCTION 1.1 Background The size of a graphics le can be minimized in bytes without degrading the quality of the image to an unacceptable level using Image compression. So that more images can be stored in the given memory space. This also minimizes the sending and receiving time of the images, say for an example: through Internet. Several methods are there for compressing the images. For Internet, the most popular graphic image formats employed for compression are JPEG format and GIF format. JPEG method is used particularly for the photographs whereas the GIF method is used when the images include line arts and simple geometric shapes. Fractals and wavelets are the other methods of Image compression. These methods are not widely used for internet images. However, they oer very good compression ratios than GIF and JPEG for some images. PNG format is another method that may replace GIF format. Compressing raw binary data is signicantly dierent from compressing images. If general-purpose compression programs are used then the result would be less than optimal. This is because the statistical properties of the images can be exploited well only by the encoders specically designed for them. Sometimes, some of the ner details in the image can be sacriced for the sake of little more bandwidth or storage space. In other words, lossy compression can be used in such areas. Generally, a text le can be compressed without the introduction of errors up to a certain extent. This is called lossless compression. But after that extent errors are unavoidable. In text and program les it is so important that we use lossless compression because a single error in text or program le will change the meaning of the text or cause the program not to run. A small loss in image compression is always not noticeable. There is no concern till the critical point. Beyond that it's not possible! The compression factor can be high if there is loss tolerance or else it must be less. So, graphic images can be with high compression ratio than that of the text les or program les. 5

9 1.2 Why Do We Need Compression? Ecient compression is one of the major aspects of image storage. For example an image of 1024 x 1024 x 24 would require the storage memory of 3MB and needs 7 minutes for transmitting and utilizing in a high speed ISDN (64 Kbit/s). But if the image is compressed at the ratio of 10:1 the memory required for storage would be just 300 KB and the transmission time drops under 6 seconds. In the time required for sending an uncompressed le through Appletalk network, we can transfer compressed seven 1 MB les to a oppy [3]. In any kind of environment, the large les are always a biggest setback in systems. This shows how desperately we need compression for managing transmittable dimensions. Apart from compression methods, we can also increase the bandwidth but this will not provide ecient outputs. The gures in Table.1 show the qualitative transition from simple text to fullmotion video data and the disk space, transmission bandwidth, and transmission time needed to store and transmit such uncompressed data. Multimedia Data Size/Duration Bits/pixel Uncompressed Size Transmission Time A page of text 11 X 8.5 Varying one 4-8KB sec Telephone Quality 10 sec 8bps 80 KB 22.2sec Grayscale Image 512 X 512 8bpp 262KB 1 min 13 sec Color Image 512 X bpp 786 KB 3 min 39 sec Medical Image 2048 X bpp 5.16MB 23 min 54 sec SHD Image 2048 X bpp 12.58MB 58 min 15 sec Full-motion Video 640X480,1min(30frames/sec) 24bpp 1.66GB 5 days 8 hrs Table 1: Multimedia data types and uncompressed storage space, transmission bandwidth, and transmission time required. The prex kilo- denotes a factor of 1000 rather than 1024[15]. The gures shown in the table clearly states the need of the sucient storage space, wide transmission bandwidth, long transmission time for audio, video, image data. This kind of problems can be easily solved with the help of compression. So, the original data is compressed before trasmission and storage and decompressed at the receiving end. For example, if the compression ratio is 32:1, the required space, bandwidth and transmission time can be reduced by a factor of 32 with an acceptable quality[15]. 1.3 What are the Principles behind Compression? It's a general fact that most images have their neighboring pixels correlated to eachother. This correlation contains less information. So, our aim is to remove this less correlated representation of the image [5]. Image compression addresses the problem lying behind the reduction of the amount of data that is required for representing a digital image. Removal of redundant data is the key basis of image compression. In mathematical point, 6

10 2-D pixel array is transformed into statistically uncorrelated dataset. This is done before transmitting or storing the image. Later, the original image can be reproduced or an approximation is set in the decompression process. The key concepts of the compression are irrelevancy and redundancy reduction. Removing duplication from the original image is carried by redundancy reduction whereas the irrelevancy reduction omits the part of the signal which can not be noticed by the signal receivers like Human Visual System [5]. The three kinds of redundancy are as follows: a) Spatial Redundancy or correlation between neighboring pixel values. b) Spectral Redundancy or correlation between dierent color planes or spectral bands. c)temporal Redundancy or correlation between adjacent frames in a sequence of images (in video applications). 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. 1.4 Dierent Classes of Compression Techniques Two ways of classifying compression techniques are mentioned here. (a) Lossless vs. Lossy compression: In lossless compression scheme, the reconstructed image is numerically equivalent to the original image. However, only a modest compression could be achieved by this scheme. In lossy compression scheme, the reconstructed image will be with degradation when compared with the original image. This is because, this method avoids all redundancies in the image. However, this method achieves very high compression ratio. Under normal viewing conditions, no losses are visible. So it is visibly lossless! Due to the quantization of data there is information loss in lossy coding. Sorting the data into dierent bits and representing each bit with a value is called as the quantization process. The value used for representing each bit is known as the reconstruction value and each item in the bit has the same reconstruction value. This is why there is information loss until there is own bit for each item in quantization process. (b) Predictive vs. Transform coding: In predictive coding the future values are predicted by the information that has already been sent or available and then coded. It is very easy to implement and will easily adapt to the local image characteristics since this is done in the spatial or image domain. The good example for this predictive coding is Dierential Pulse Code Modulation (DPCM)[16]. Transform coding rst transforms the image from its spatial domain to a dierent representation using a well-known transform and then codes the transformed values. In the expense of greater computation this method provides higher data compression than predictive coding. 7

11 1.5 Image Compression Process A typical lossy image compression system is shown in gure 1.1. It consists of three closely connected components namely (a) Source Encoder (b) Quantizer, and (c) Entropy Encoder. Compression is accomplished by applying a linear transform to decorrelate the image data and then quantizing the resulting transform coecients and nally entropy coding the quantized values. Source Encoder (or Linear Transformer): A signicant number of linear transforms have been developed through the years, which include Discrete Fourier Transform (DFT), Discrete Cosine Transform (DCT), Discrete Wavelet Transform (DWT) and many more, each with its own positives and negatives. Quantizer: The Quantizer reduces the precision of the transformed coecient values to reduc e the number of bits required to store the transformed coef- cients. Here the main source of compression is the encoder and since it is many-to-one mapping this is a lossy process. Quantization can be performed on each individual coecient, which is known as Scalar Quantization (SQ). Quantization can also be performed on a group of coecients together, and this is known as Vector Quantization (VQ). Both uniform and non-uniform quantizers can be used depending on the problem at hand[17]. Entropy Encoder: The quantized values are further losslessly compressed by the encoder to give an overall better compression results. The encoder then accuretely determines the probabilities of the quantized values using a model and then using this it produces an appropriate code such that the resulting output code stream will be smaller than the input code stream. The most commonly used entropy encoders are the Human encoder and the arithmetic encoder, although for applications requiring fast execution, simple run-length encoding (RLE) has proven very eective[17]. It is important to note that a properly designed quantizer and entropy encoder are absolutely necessary along with optimum signal transformation to get the best possible compression. 8

12 1.6 Objective of the Thesis Our aim is to design an application for compressing the DICOM images using discrete wavelet transform and compress it with SPIHT, for the easy mobile access of the DICOM images over heterogeneous networks like UMTS, GPRS and so on. 1.7 Thesis Organization Chapter 2 discusses the details of wavelets. Chapter 3 discusses the details of SPIHT algorithm. Chapter 4 discuses the results and nally concludes the thesis and identify the direction for the future work References. 9

13 Chapter 2 WAVELET TRANSFORM OVERVIEW 2.1 Wavelet Transform Wavelets are mathematical functions dened over a nite interval and having an average value of zero that transform data into dierent frequency components, representing each component with a resolution matched to its scale [4]. The basic concept behind the wavelet transform is to represent any arbitrary function as a superposition of a set of such wavelets or basis functions. These basis functions are called the baby wavelets and these baby wavelets are obtained from a single prototype wavelet known as the mother wavelet by dilations or contractions (scaling) and translations (shifts)[4]. While analyzing physical situations where the signal has discontinuities and sharp spikes, they are ecient and advantageous over traditional Fourier methods. Image compression, turbulence, human vision, radar and earthquake prediction are new wavelet applications developed in recent years. In wavelet transform the basis functions are wavelets. Wavelets tend to be irregular and symmetric. All wavelet functions, w(2kt - m), are derived from a single mother wavelet, w(t). This wavelet is a small wave or pulse like the one shown in gure 2.1. Normally it starts at time t = 0 and ends at t = T. The shifted wavelet w(t - m) starts at t = m and ends at t = m + T. The scaled wavelets w(2kt) start at t = 0 and end at t = T/2k. Their graphs are w(t) compressed by the factor 10

14 of 2k as shown in gure 2.2. For example, when k = 1, the wavelet is shown in gure 2.2(a). If k = 2 and 3, they are shown in (b) and (c), respectively. The wavelets are called orthogonal when their inner products are zero. The smaller the scaling factor is, the wider the wavelet is. Wide wavelets are comparable to low-frequency sinusoids and narrow wavelets are comparable to highfrequency sinusoids. 2.2 Scaling Wavelet analysis produces a time-scale view of a signal. Scaling a wavelet means stretching (or) compressing it. The scale factor is used to express the compression of wavelets and often denoted by the letter a. If the scale factor is smaller, the more compressed is the wavelet. The scale is inversely related to the frequency of the signal in wavelet analysis. 2.3 Shifting Shifting a wavelet means delaying (or) hastening its onset. Delaying a function f(t) by k is mathematically represented as; f(t-k) [23] and is shown in gure

15 2.4 Scale and Frequency The higher the scales, the more the wavelets are stretched. Therefore, the portion of the signal with which it is being compared is longer and then the signal features being measured by the wavelet coecients are coarser. The relation between the scale and the frequency is shown in gure 2.4. Thus, there is a correspondence between wavelet scales and frequency as revealed by wavelet analysis: Low scale 1) Compressed wavelet 2) Rapidly changing details 3) High frequency. High scale 1) Stretched wavelet 2) Slowly changing, coarse features 3) Low frequency. 2.5 Discrete Wavelet Transform Calculating wavelet coecients at each possible scale is a work which generates an awful lot of data. If the scales and positions are chosen based on powers of two, the so-called dyadic scales and positions, then calculating wavelet coecients are ecient and just as accurate. This is obtained from discrete wavelet transform (DWT). 12

16 2.5.1 One-Stage Filtering The low-frequency content is the most important part for many signals. It is the identity of the signal. The high-frequency content imparts details to the signal. In wavelet analysis, the approximations and details are obtained after ltering. The approximations are the high-scale and low frequency components of the signal whereas the details are the low-scale and high frequency components of the signal. Two signals are emerged when the original signal passes through two complementary lters. It may produce doubling of samples and so to prevent this problem, downsampling is used. DWT coecients re produced by the process on the right which includes downsampling Multiple-Level Decomposition The iteration of the decomposition process by decomposing successive approximations in turn such that one signal is broken down into many lower resolution components is known as the wavelet decomposition tree and is shown in gure Wavelet Reconstruction The reconstruction of the image is achieved by the inverse discrete wavelet transform (IDWT). The values are rst upsampled and then passed to the lters. Filtering and downsampling are the basis of the wavelet analysis, whereas upsampling and ltering are the basis of wavelet reconstruction process. Lengthening a signal component by inserting zeros between samples is known as the upsampling process. 13

17 2.5.4 Reconstructing Approximations and Details It is possible that the original signal can be reconstructed from the coecients of the approximations and details and this process produces a reconstructed approximation which has the same length as the original signal and which is a real approximation of it. These reconstructed details and approximations are known to be true constituents of the original signal. The details and approximations are produced by downsampling process and are only half the length of the original signal. So, they cannot be directly combined to reproduce the signal[24]. Therefore, it is required to reconstruct the approximations and the details before combining them. The reconstructed signal is schematically represented as in g D Wavelet Transform A signal is passed through a low pass and high pass lter, h and g, respectively, then down sampled by a factor of two, constituting one level of transform. Repeating the ltering and decimation process on the lowpass branch outputs make multiple levels or "scales" of the wavelet transform only. The process is 14

18 typically carried out for a nite number of levels K, and the resulting coecients are called wavelet coecients. The one-dimensional forward wavelet transform is dened by a pair of lters s and t that are convolved with the data at either the even or odd locations. The lters s and t used for the forward transform are called analysis lters. and n L l i = (s j X 2i+j ) (2.1) j= n L n H h i = (t j X 2i+1+j ). (2.2) j= n H Although l and h are two separate output streams, together they have the same total number of coecients as the original data. The output stream l, which is commonly referred to as the low-pass data may then have the identical process applied again repeatedly. The other output stream, h (or high-pass data), generally remains untouched. The inverse process expands the two separate lowand high-pass data streams by inserting zeros between every other sample, convolves the resulting data streams with two new synthesis lters s and t, and adds them together to regenerate the original double size data stream[25]. y i = n H j= n H (t jl i+j) + n l where l 2i = l i, l 2i+1 = 0, h 2i+1 = h i, h 2i = 0. j= n H (s jh i+j), (2.3) To meet the denition of a wavelet transform, the analysis and synthesis lters s, t, s and t must be chosen so that the inverse transform perfectly reconstructs the original data. Since the wavelet transform maintains the same number of coecients as the original data, the transform itself does not provide any compression[25]. However, the structure provided by the transform and the expected values of the coecients give a form that is much more amenable to compression than the original data. Since the lters s, t, s and t are chosen to be perfectly invertible, the wavelet transform itself is lossless. Later application of the quantization step will cause some data loss and can be used to control the degree of compression. A 1-D subband decomposition process is carried out by the forward wavelet-based transform. Here a 1-D set of samples is converted into two bands called the low-pass subband (Li) and high-pass subband (Hi). The low-pass subband corresponds to a down sampled low-resolution version of the original image and the high-pass subband corresponds to the residual information of the original image. This residual information is needed for the perfect reconstruction of the original image from the low-pass subband D Transform Hierarchy The 1-D wavelet transform can be extended to a two-dimensional (2-D) wavelet transform using separable wavelet lters using which the 2-D transform can be 15

19 obtained by applying a 1-D transform to all the rows of the input signal, and then repeating it on all of the columns[26]. The original image of a one-level (K=1), 2-D wavelet transform, with corresponding notation is shown in gure 2.7. The example is repeated for a threelevel (K =3) wavelet expansion in gure 2.8. In all of the discussion K represents the highest level of the decomposition of the wavelet transform. The 2-D subband decomposition is extended method of the 1-D version of decomposition. Here the 1-D method is done twice; rst in one direction (horizontal) and then in the orthogonal (vertical) direction. For instance, the low-pass subbands (Li) resulting from the horizontal direction is further decomposed in the vertical direction which leads to LLi and LHi subbands. The high pass subband (Hi) is further decomposed into HLi and HHi in similar manner. After this process it is possible that the Lli subband can be further decomposed into four subbands with same method. This produces multiple transform levels or multiple decomposition levels. As shown in gure 2.8. the rst level of transform or decomposition results in the four subbands; LL1, HL1, LH1 and HH1. In this LL1 is further transformed into four subbands; LL2, HL2, LH2 and HH2 in the second level. And the subband LL2 is used for the third level of transform or decomposition. In any band, LLi is the low-resolution subband and the high-pass subbands LHi, HLi, HHi are the horizontal, vertical and diagonal subband because they represent the horizontal, vertical, and diagonal residual information of the original image. To obtain a two-dimensional wavelet transform, the one-dimensional transform 16

20 is applied rst along the rows and then along the columns to produce four subbands: low-resolution, horizontal, vertical, and diagonal. (The vertical subband is created by applying a horizontal high-pass, which yields vertical edges.) At each level, the wavelet transform can be reapplied to the low-resolution subband to further decorrelate the image. Figure 2.9 illustrates the image decomposition, dening level and subband conventions used in the AWIC algorithm. The nal conguration contains a small low-resolution subband. In addition to the various transform levels, the phrase level 0 is used to refer to the original image data. When the user requests zero levels of transform, the original image data (level 0) is treated as a low-pass band and processing follows its natural ow. On each source image, wavelet transform is performed and then based on fusion rules fusion decision map is generated. Then the fused wavelet co-ecient map can be obtained from the wavelet co-ecients of the original image according to the fusion decision map. Then by inverse wavelet transform the fused image can be obtained. From the above gure it is clear that fusion rules play a vital role in fusion process. 2.6 Line based Wavelet The lifting sheme has some unique properties that is not found in other transforms. 17

21 The gure 2.10 shows some of those few properties. The inverse transform can be applied such that the signs of the scaling factors are changed, replacing "split" by "merge", move from right to left (reverse the data ow). For the lifting scheme, this invertibility is true. Lifting can be done such that we never need the samples other than the previous lifting step. So, we can update the old stream with the new one at the summation point. When we iterate the lter bank using the in-place lifted lters it will result in interlaced co-ecients. This is not immediately clear from the image. We can split the inputs in the even and odd samples and in-place lifting steps are performed. After one complete step the high-pass ltered samples and wavelet coecients sit in the odd numbered places whereas the low-pass ltered samples sit in the even numbered places. Now, we will perform the next transform but in low-pass ltered samples. So, again odd and even samples are obtained in that area. The odd numbered samples are then transformed into wavelet coecients while the even numbered samples are processed and interlaced with the wavelet coecients. The third important property is not mentioned but from the gure it is clear that the lifting is not casual. But we can make it casual by delaying the signal although it will never be real time. In some cases we can design the casual lifting transform. The last important property is the computational complexity. But it is actually proven that for long lters the lifting cuts the computational complexity in a half than that of the standard iterated FIR Filter bank algorithm. This wavelet transform already has the complexity of N. In other words, it is much better than FFT with its complexity with N log(n) and lifting speeds things up to another factor of 2. This transform is the fast wavelet transform and thus it can be referred as fast lifting wavelet transform of FLWT Integer lifting It should be made sure that the wavelet coecients are integers and this is the last stage of the wavelet transform. Generally, in classical transforms that also includes the non-lifted wavelet transforms, the wavelet coecients are assumed as the oating point numbers. This is because the lte coecients used in transform lters are the oating point numbers. In the lifting sceme the integer data maintenance can be easy though the dynamic range of data might increase. This is the easy invertibilityy property of lifting. By rewriting the basic lifting step, we have from [13], x new (z) x(z) + s(z)y(z), (2.4) because y(z) signal is not changed in the lifting step and the lter operation result can be rounded as follows, x new (z) x(z) + s(z)y(z), (2.5) x(z) x new (z) s(z)y(z). (2.6) Thus it is so clear that the lifting operation is reversible irrespective of which kind of rounding method we use. 18

22 However care must be taken that we did not consider the scaling step. Scaling is the part of lifting transform and it does not yield integer results. The important point is that the transform coecients have to be scaled. This is a good solution for this problem. This is particularly good in denoising applications. If scaling is ignored, the scaling factor must be as close to 1 as possible. This is made possible by using non-uniqueness in lifting factorization. One more solution can be to factor the scaling into lifting steps. As told earlier, integer lifting transform will not guarantee to preserve the dyanamic range of input signal. There are some schemes that can keep the dynamic range since the dynamic range doubles usually. The two complement representation of integers in a computer and a wrap-around overows cause in the representation are employed by the lifting transform with the so-called precision preservation property. The main problem with this transform is that the high coecients are represented by the small values and thus it is quite dicult to take decisions on coecient values. 19

23 Chapter 3 SPIHT (SET PARTITIONING IN HIERARCHICAL TREES) 3.1 Introduction In [1], a wavelet-based still image coding algorithm known as set partitioning in hierarchical trees (SPIHT) is developed that generates a continuously scalable bit stream. This means that a single encoded bit stream can be used to produce images at various bit-rates and quality, without any drop in compression. The decoder simply stops decoding when a target rate or reconstruction quality has been reached. In the SPIHT algorithm, the image is rst decomposed into a number of sub bands using hierarchical wavelet decomposition. The sub bands obtained for a two-level decomposition are shown in gure 3.1. The sub band coecients are then grouped into sets known as spatial-orientation trees, which eciently exploit the correlation between the frequency bands. The coecients in each spatial orientation tree are then progressively coded bit-plane by bitplane, starting with the coecients with highest magnitude and at the lowest pyramid levels. Arithmetic coding can also be used to give further compression. In general, increasing the number of levels gives better compression although the improvement becomes negligible beyond 5 levels. In practice the number of possible levels can be limited by the image dimensions since the wavelet decomposition can only be applied to images with even dimensions. The use of arithmetic coding only results in a slight improvement for a 5 level decomposition. The embedded zerotree wavelet (EZW) coding was rst introduced by J.M Shapiro and has since become a much studied topic in image coding. The EZW coding technique is a fairly simple and ecient technique for compressing the information in an image. Our focus in this project is to analyze the Set Partition in Hierarchical Tree algorithm in the EZW technique and to obtain observations by implementing the structure and testing it. In order to compress a binary le, some prior information must be known about 20

24 the properties and structure of the le in order to exploit the abnormalities and assume the consistencies. The information that we know about the image le that is produced from wavelet transformation is that it can be represented in a binary tree format with the root of the tree having a much larger probably of containing a greater pixel magnitude level than that of the branches of the root. The algorithm that takes advantage of this information is the Set Partition in Hierarchical Tree (SPHT) algorithm. This project has been preformed previously but has produced an inecient explanation of the implementation of the algorithm and some of its diculties. In this project, we hope to be able to identify the problems and give more insight to the development and implementation of this algorithm than in previous projects Approach Matlab oers a set of wavelet tools to be able to produce an image with the needed properties. The concept of wavelet transformation was not our focus in this project but in order to understand how the SPHT algorithm works, the properties of wavelet transformation would need to be identied. Matlab was able to create adequate testing pictures for this project. To adequately comprehend the advantages of the SPHT algorithm, a top level understanding will be needed to identify its characteristics and dierences from other algorithms Set Partitioning Algorithm The SPHT algorithm is unique in that it does not directly transmit the contents of the sets, the pixel values, or the pixel coordinates. What it does transmit is the decisions made in each step of the progression of the trees that dene the structure of the image. Because only decisions are being transmitted, the pixel value is dened by what points the decisions are made and their outcomes, while the coordinates of the pixels are dened by which tree and what part of that tree the decision is being made on. The advantage to this is that the decoder can have an identical algorithm to be able to identify with each of the decisions and create identical sets along with the encoder. 21

25 The part of the SPIHT that designates the pixel values is the comparison of each pixel value to 2 n c i,j < 2 n+1 with each pass of the algorithm having a decreasing value of n. In this way, the decoding algorithm will not need to passed the pixel values of the sets but can get that bit value from a single value of n per bit depth level. This is also the way in which the magnitude of the compression can be controlled. By having an adequate number for n, there will be many loops of information being passed but the error will be small, and likewise if n is small, the more variation in pixel value will be tolerated for a given nal pixel value. A pixel value that is 2 n c i,j is said to be signicant for that pass. By sorting through the pixel values, certain coordinates can be tagged at "signicant" or "insignicant" and then set into partitions of sets. The trouble with traversing through all pixel values multiple times to decide on the contents of each set is an idea that is inecient and would take a large amount of time. Therefore the SPIHT algorithm is able to make judgments by simulating a tree sort and by being able to only traverse into the tree as much as needed on each pass. This works exceptionally well because the wavelet transform produces an image with properties that this algorithm can take advantage of. This "tree" can be dened as having the root at the very upper left most pixel values and extending down into the image with each node having four (2 x 2 pixel group) ospring nodes(see gure 3.1). The SPIHT method is not an extension from the traditional methods of image compression, and it represents an important advance in the eld. The SPIHT (set partitioning in hierarchical trees) is an ecient image coding method using the wavelet transform. Recently, image-coding using the wavelet transform has attracted great attention. Among the many coding algorithms, the embedded zero tree wavelet coding by Shapiro and its improved version, the set partitioning in hierarchical trees (SPIHT) by Said and Pearlman have been very successful. Compared with JPEG which is the current standard for still image compression, the EZW and the SPIHT methods are more ecient and are able to reduce the blocking artifact[14]. The method provides the following which requires special attention: 1) Good image quality and high PSNR especially for the color images 2) It is optimized for progressive image transmission 3) Produces a fully embedded coded le 4) Simple quantization algorithm 5) Can be used for lossless compression 6) Can code to exact bit rate or distortion 7) Fast coding/decoding (nearly symmetric) 8) Has wide applications, completely adaptive 9) Ecient combination with error protection These properties[19] are discussed in the following. Generally, dierent compression methods were developed that has at least one of the following properties but SPIHT really is outstanding since it has all those qualities simultaneously. 22

26 3.2 Image Quality SPHIT has the very good ability when tested to nd the minimum rate in reproducing the image that is indistinguishable with the original. SPHIT is even more perfect to encode the color images as it allocates the bits automatically for the local optimality for the color components. Basically other algorithms encode the color components seperately following the global statistics of the individual components. But SPIHT is dierent from this. The compression that is visually lossless can be obtained in some color images with the compression ratios from : Progressive Image Transmission Systems like WWW servers which has progressive image transmissions, have the quality of dispaying images which are inecient. This is because these widely used schemes use very primitive progressive image transmission method. On the other hand we have SPHIT which is the state-of-the-art method that is designed for optimal pregressive transmission which still beats most nonprogressive transmission methods. SPHIT makes it possible by producing fully embedded coded le such that the quality of the displayed image le, at any moment is the best available for the number of bits received upto the moment. Thus SPHIT can be used in applications where the user can go through the image so quickly and decide if it should really be downloaded or is it enough to be saved or need renement. 3.4 Optimized Embedded Coding Embedded coding scheme is dened as: when the two les produced by the encoder have the size M and N bits such that if M>N then the le with the size N is identical to the rst N bits of the M-size le. In practice, if you need to compress an image for thee remote users who require the same image with dierent reproduction quality and you nd that those qualities can be obtained from the images that are compressed to at least 8 Kb, 30 Kb and 800 Kb. If you use non-embedded encoder like JPEG inorder to 23

27 avoid transmission cost or time then you must prepare one le for each user. In the other hand if you use embedded encoder like SPHIT then you can compress the image le to single 80 Kb le, then send the rst 8 Kb of the le to the rst user, then rst 30 Kb to the second user and the full le to the next user. Surprisingly in SPHIT, all the three users would get an image quality for the same le size, superior to the sophisticated non-embedded encoders available today. This is possible in SPHIT because it optimizes the embedded coding process and codes the most important information rst. 3.5 Lossless Compression SPIHT codes the individual bits of transform image wavelet coecients following a bit-plane sequence and thus it is possible to reproduce full image perfectly by each bit since it codes all bits of transform. However, only if the numbers are stored as innite-precision numbers, the wavelet transform can yield perfect reconstruction. In practice, rounding recovery approach is followed to reconstruct the image perfectly but this method is not encouraged due to its ineciency. The codec that uses the transformation to produce ecient progressive transmission till the lossless recovery is among the SPHIT. The surprising results produced can show that for lossless compression this codec used is as ecient as the most eective lossless encoders which denitely does not include lossless JPEG. The property in SPIHT that produces progressive transmission with practically no reduction in compression eciency applies to the lossless compression too. 3.6 Rate or Distortion Specication Most of the image compression methods developed today has no precise rate control. For some methods, the user sets the target rate and the program tries to give the result ina rate that is not too far away from what you have specied. In certain methods, the user gives the "quality factor" and waits for the results to look if the size suits your needs. The embedded property of SPIHT has the exact bitrate control which does not aect at all the image quality or performance since no bits are wasted in padding or whatsoever. It also provides the Mean Square-Error (MSE) distortion control. Though this is not the best measure for image quality it provides far superior quality specication than other criteria. 3.7 Encoding/Decoding Speed The more the compression simplicity the high the encoding/decoding speed. The SPIHT algorithm is so symmetric such that the encoding time is almost same as the decoding time unlike complex compression algorithms where the encode time will be signicantly more than the decode time. 24

28 3.8 Hierarchical Tree In this subsection, we will describe the proposed algorithm to code the wavelet coecients. In general, a wavelet decomposed image typically has non-uniform distribution of energy within and across subbands. This motivates us to partition each subband into dierent regions depending on their signicance and then assign these regions with dierent quantization levels. The proposed coding algorithm is based on the set partitioning in hierarchical trees (SPIHT) algorithm, which is an elegant bit-plane encoding method that generates M embedded bit sequence through M stages of successive quantization. Let s 0, s 1,..s M 1 denote the encoder's output bit sequence of each stage. These bit sequences are ordered in such a way that s 0 consists of the most signicant bit, s 1 consists of the next most signicant bit, and so on. The SPIHT algorithm forms a hierarchical quadtree data structure for the wavelet transformed coecients. A spatial orientation tree (SOT) is shown in gure 3.3 that the set of root node and corresponding descendents are known as a spatial orientation tree (SOT). In the tree, each node has either no leaves or four ospring, that are from 2 x 2 adjacent pixels. In the highest decomposition level, the pixels on the LL subimage are the tree roots and they are also grouped in 2 x 2 adjacent pixels. But, the upper-left pixel in 2 x 2 adjacent pixels has no descendant as shown in gure.3.3. Other three pixels have four children each. The real implementation of SPIHT is illustrated below. To make it simple, the following sets of coordinates are dened. (1) O(i, j): set of coordinates of all ospring of node (i, j); (2) D(i, j): set of coordinates of all descendants of the node (i, j); (3) H: set of coordinates of all spatial orientation tree roots (nodes in the highest pyramid level); (4) L(i, j)=d(i, j)-o(i, j). Thus, except at the highest and lowest levels, we have O(i, j)=(2i, 2j), (2i, 2j+1), (2i+1, 2j), (2i+1, 2j+1). Dene the following function. { 1, max(i,j)eτ { c S n (τ) = i,j } 2 n, 0, otherwise 25

29 S n (τ) denotes the signicance of a set of coordinates τ, where the preset significant threshold used in the nth stage is denoted by T(n). The SPIHT coding algorithm is described as follows. First, T(0) is assumed to be 2 M 1. Here M is chosen such that the largest coecient magnitude c max, satises 2 M 1 c max < 2 M. In the coecient magnitude, the encoding is progressive to successfully use a sequence of thresholds T(n)=2 (M 1) n, n=0,1,2,...m-1. These thresholds are of the powe of '2', so that the encoding can be taken as the bit plane encoding of the wavelet coecients. All the coecients with the magnitudes between T(n) and 2T(n), at stage n are signicant and their positions and the sign bits are encoded. This is called sorting pass process. Then each coecient with the magnitude at least 2T(n) is rened by encoding the 'n'th most signicant bit. This is known as renement pass. The signicant coecient encoding position and the signicant coecient scanning can be achieved by the three lists: (LSP) the list of signicant pixels, (LIP) the list of insignicant pixels, and (LIS) the list of insignicant set. Any entry into the LSP and LIP is an individual pixel which is represented or indicated by the coordinates (i,j). Each entry into the LIS is regarded as the set: either D(i,j) or L(i,j), such that LIS is indicated as type A if it is D(i,j) and type B if it is L(i,j) [19]. 3.9 SPIHT coding algorithm Step 1: (Initialization) Output, log 2 (max i,j { C i,j }) set the LSP as an empty list, add the coordinates (i, j) H to the LIP, add the coordinates (i, j) H with descendants to the list LIS, as type A entries, Step 2: (Sorting Pass) 2.1) for each entry (i, j) in the LIP do: output Sn(i, j), if Sn(i, j)=1 move (i, j) to the LSP, 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, 1) O(i, j) do: output Sn(k, l), if Sn(k, l) = 1 then add (k, l) to the LSP, output the sign of c k,l, if Sn(k, l)=0 then add (k, l) to the end of the LIP, 26

30 *if L(i, j) 0 then move (i, j) to the end of the LIS, as an entry of type B, go to Step 2.2.2). otherwise remove entry (i, j) from the LIS, 2.2.2) if the entry is of type B output Sn(L(i, j)), if Sn(L(i, j)) = 1 then *add each (k, 1) O(i, j) to the end of the LIS as an entry of type A, *remove (i,j) from the LIS, Step 3: (Renement 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 signicant bit of c i,j, Step 4: (Quantization-Step Update) Decrement n by 1 and go to Step 2 [21] Algorithm O(i,j): set of coordinates of all ospring of node (i,j); children only D (i,j): set of coordinates of all descendants of node (i,j); children, grandchildren, great-grand, etc. H (i,j): set of all tree roots (nodes in the highest pyramid level); parents L (i,j): D (i,j) - O(i,j) (all descendents except the ospring); grandchildren, great-grand, etc. 27

31 28

32 29

33 30

34 3.11 Performance Evaluation Compression eciency is measured by the compression ratio and is estimated by the ratio of the original image size over the compressed data size. The complexity of an image compression algorithm is calculated by the number of data operations required to perform both encoding and decoding processes. Practically, it is sometimes expressed by the number of operations. For a lossy compression scheme, a distortion measurement is a criterion for determining how much information has been lost when the reconstructed image is produced from the compressed data. The most often used measurement is the mean square error (MSE). In the MSE measurement the total squared dierence between the original signal and the reconstructed one is averaged over the entire signal. Mathematically, MSE = ( 1 N 1 N ) (x i x i ) 2, (3.1) where x i is the reconstructed value of x i. N is the number of pixels. The mean square error is commonly used because of its convenience. A measurement of MSE in decibels on a logarithmic scale is the Peak Signal-to-Noise Ratio (PSNR), which is a popular standard objective measure of the lossy codec. We use the PSNR as the objective measurement for compression algorithms throughout this thesis. It is dened as follows, i=0 P SNR = 10 log 10 MAX 2 1 w h w i=1 h j=1 (o(i, j) c(i, j))2, (3.2) where w and h are the width and height of the image respectively, o is the original image data, and c is the compressed image data. MAX is the maximum value that a pixel can have, 255. Compressing raw binary data is signicantly dierent from comressing images. If general-purpose compression programs are used then the result would be less than optimal. This is because the statistical proper ties of the images can be exploited well only by the encoders specically designed for them. Sometimes, some of the ner details in the image can be sacriced for the sake of little more bandwidth or storage space. In other words, lossy compression can be used in such areas. Lossless compression is about compressed data, when decompressed is the exact replica of the original data. This is the case when documents and programs are compressed. Because they need to be exactly reproduced when decompressed unlike music and images that need not be exactly reproduced on decompression. Most of the time an approximation to the original image is enough as long as the losses between the original image and the compressed image are tolerable. 31

35 3.12 Error Metrics There are two error metrics that are used to compare dierent image compression methods. They are Mean Square-Error (MSE) and Peak Signal-to-Noise Ratio (PSNR). The cumulative squared error between the original and the compressed image is shown by MSE and the peak error is shown by the PSNR[18]. Mathematically, they are written as follows, MSE = 1 MN N y=1 x=1 M [I(x, y) I (x, y)] 2. (3.3) 225 P SNR = 20 log 10 ( ). (3.4) sqrt(mse) where I(x, y) denotes the original image and I (x, y) denotes the approximation to the original image which is also called as the decompressed image. M, N are the image dimensions. The lower value of MSE says that the errors are less. Due to inverse relation of MSE and PSNR, the errors will be less when the PSNR is high. Logically, signal is the image and noise is the errors produced in the reconstructed image. So, if signal to noise ratio is peak and MSE is less for an image when comparison then one can make a conrmation that this is the better one! The term PSNR is an engineering term for the ratio of the maximum possible power of the signal to the power of corrupting noise that aects the delity of its representation. Since many signals have a wide dynamic range, PSNR is expressed in logarithmic decibel scale. PSNR is the measure of the quality of reconstruction of the compressed image. It can be easily dened through the MSE which for two m X n monochrome images I and K where one of them is assumed as the noisy approximation to the other and is dened as, The PSNR is dened as: MSE = 1 mn m 1 i=0 n 1 I(i, j) K(i, j) 2. (3.5) j=0 P SNR = 10 log 10 ( MAX2 I MSE ) = 20 log 10( MAX I MSE ). (3.6) The maximum possible pixel value is dened by MAX i. When the pixels are 8 bits per sample then it is 255. (i.e) if the pixels are represented by PCM with B bits per sample then MAX I is 2 B 1. The color images have three RGB values per pixel, for which the PSNR is dened as the same except the Mean Square-Error (MSE) is the sum over all squared value dierences divided by image size and by three. Typically, in lossy image and video compression, the PSNR values are from 30 to 50 db, where higher is always better for PSNR! 32

36 Chapter 4 SIMULATION RESULTS, CONCLUSION AND FUTURE WORK 4.1 Simulation Here comes the simulation part to see how the outputs look like. After writing the codes for Discrete Wavelet Transform followed by SPIHT coding using Matlab, the original image is passed through it for decomposing it by seperating high frequency and low frequency signals. Then the resulting image is SPHIT encoded and sent through the decompressing section which includes SPIHT decoding and Inverse Discrete Wavelet Transform to get the reconstructed image or the approximation to the original image. The steps of execution of the program code is explained below with the simulation results. When the code is executed, a GUI(Graphical User Interface) window opens as shown in the gure

37 34

38 When we browse for a DICOM image or any other image, the chosen image will get displayed in the GUI, as shown in the gure 4.2. So, if we encode it using the 'Encode' tab, the chosen DICOM image rst gets its trasformation by Discrete Wavelet Transform and as a result four subbands are produced which are the seperation between low and high-pass signals. Then, the transformed image is encoded using SPIHT algorithm to produce the bitstream for the image signal. This process is shown in the gure

39 After the encoding process, the bitstream is sent to the receiving end, such that the user in the receiving end needs to decode it before the he/she can see the reconstructed image. The window, how it looks like after decoding the image is shown in the gure

40 37

41 The reconstructed image or decompressed image is shown in the gure 4.6, which is almost the replica of the original image. Also, it had been discussed already that the more the compression ratio, the more the losses in the reconstructed image, although it is called as the Lossless compression! 4.2 Conclusion and Future Work In this thesis we have developed a technique for line based wavelet transforms. We pointed out that this s transform can be assigned to the encoder or the decoder and that it can hold compressed data. We provided an analysis for the case where both encoder and decoder are symmetric in terms of memory needs and complexity. We described highly scalable SPIHT coding algorithm that can work with very low memory in combination with the line-based transform, and showed that its performance can be competitive with state of the art image coders, at a fraction of their memory utilization. To the best of our knowledge, our work is rst to propose a detailed implementation of a low memory wavelet image coder by making it attractive both in terms of speed and memory needs. Further improvements of our system especially in terms of speed can be achieved by introducing a lattice factorization of the wavelet kernel or by using the lifting steps. This will reduce the computational complexity and complement the memory reductions mentioned in this work. 38