AbstractWith the Advent of Information Technology in the medical world, various radiological modalities produce a variety of digital medical files most often datasets and images. The surge of digital radiological modalities in modern hospitals and research institutes around the world, has led to the creation of a vast amount of medical digital assets, like signals and images. These files as any digital asset should be protected from unwanted modification of their contents, especially as they contain vital medical information. Thus their protection and authentication seems of great importance and this need will rise along with the future standardization of exchange of data between hospitals or between patients and doctors. Modern telecommunication infrastructure supports the possibility of delivering quality health care without the physical presence of medical experts. The integrity of biomedical data being transmitted through communication channels must be established before their utilization. A well known authentication method is studied. Watermarking, a technique first introduced for multimedia files, provides a method for authentication and protection and has been recently applied to medical images. With this in mind, this work proposes techniques for hiding sensitive patient metadata within the actual medical signal, which are stored into a patients medical record. In specific, the focus is on Electroencephalogram (EEG) Signal and MRI images and how to embed numerical metadata within the data. A prerequisite of this embedding is, not to destroy the data usability and indeed show that the usefulness of the data is not affected, because of the imperceptible distortion induced through the embedding of the metadata within the actual data. When dealing with medical data this means that watermarking algorithms should not change the diagnosis of a physician. For example, when dealing with EEG signals, common tasks are the detection of seizure or other brain related illnesses. Therefore, the diagnosis on the watermarked signal should not deviate from the diagnosis on the original signal. The privacy of the embedded data is assured because the private metadata will not directly embed on the signal, but instead embed a surrogate random sequence, that is generated by a cryptographically safe pseudorandom number generator (PRNG) using the metadata as the input and a secret key as the seed. Hence, avoid leaking or revealing any information about the patients sensitive information to the public. Even though the privacy of sensitive data attributes can be addressed through encryption, such an approach is inherently a blocking factor in data dissemination. This technique is then adapted here for interleaving patient information and message authentication code with medical images in a reversible manner, that is using lossless compression. The resulting scheme enables on a side the exact recovery of the original data that can be unambiguously authenticated, and on the other side, the patient information to be saved or transmitted in a confidential way. To ensure greater security the patient information is encrypted before being embedded into data.

Chapter 1IntroductionDigital steganography and watermarking are the two kinds of data hiding technology to provide hidden communication and authentication. The word steganography is derived from the Greek words stegos meaning cover and grafia meaning writing defining it as covered writing. In contrast to Cryptography, where the enemy is allowed to detect, intercept and modify messages without being able to violate certain security premises, the goal of steganography is to hide a secret message inside harmless medium in such a way that it is not possible even to detect that there is a secret message. The medium for data hiding is also called as cover, host and carrier. To human eyes, data usually contains known forms, like images, videos, sounds and text. Most internet data naturally includes unwarranted headers too. These are media exploited using steganograpy techniques. Images are the most powerful medium for data hiding because of the limitation of Human visual System (HVS). Basic idea of watermarking is to embed covert information into a digital signal, like digital audio, image, or video, to trace ownership or protect privacy. Data hiding can be used in a large amount of data formats in the digital world of today. The most popular data formats used are .bmp, .doc, .gif, .jpeg, .mp3, .txt and .wav mainly because of their popularity on the Internet.1.1 Requirement of information hiding system An information-hiding system is characterized using four different aspects: capacity, security, perceptibility and robustness shown in Fig. 1. Capacity refers to the amount of information that can be hidden in the cover medium. Security refers the inability of the hacker to extract hidden information. Perceptibility means the inability to detect the hidden information. Robustness is the amount of modification the stego-medium can withstand before an adversary can destroy the hidden information.

CapacitySecurityPerceptibilityRobustness

Figure 1.1: Aspect of information-hiding systemDuring the past five years, the development of infrastructures "Internet" has allowed an important evolution towards the transmission of data via the networks. This technology is devoted mainly to deliver and transfer the contents from the numerical documents which are not well protected and do not have file and history [1]. The primary applications of watermarking are to protect copyrights and integrity verification [2]. The main reason for protecting copyrights is to prevent image piracy when the transmitter sends it on the internet [3]. For integrity verification, it is important to ensure that the medical image originated from a specific source and that it has not been changed, manipulated or falsified. Medical data are stored for the following three purposes: [4]. Diagnosis Database Long-term storageSince an obtained medical data must be kept perfectly without any loss of information before the data is diagnosed by a doctor, the data should be compressed by lossless algorithm or should be stored without using compression. Note that a data compressed by lossless algorithm can be restored completely at the cost of low compression rate, while lossy algorithm loses information of the data in some degree in order to achieve high compression rate. When the data is diagnosed by a doctor at distant site, it cannot be exposed to public by using secured channel to transmit it [5, 6]. However, since any person with privilege can access to data which are contained in database and can modify them maliciously, the integrity of the data must be protected by using watermarking, which is called integrity watermark. Meanwhile, web-based database system contains valuable medical data resources for not only research purpose but also commercial purpose [8]. Therefore the copyright and intellectual property of the database should be also protected by a watermark, which is called copyright watermark [9]. Moreover, for long-term storage, the protection of the integrity and copyright of data is also critical issue [10]. First, when a person (archiver) stored a data in the long-term storage system long ago and a different person (viewer) refers to the data, the viewer can confirm the integrity of the data only through a watermark embedded in the data. Second, when a patient does not want his/her medical images open to the public, the copyright of the data is thought to belong to the patient. Therefore the patient can protect the copyright of the data by using watermarking. It is usual that a medical data is diagnosed before storing the data in the long-term storage, so the significant part of the data is already determined. The significant part is called ROI (Region Of Interest), which must be preserved without lack of any information even for the long-term storage. Distant learning is one of applications using database of medical data, which may refer to the image of newly discovered medical case, and they may be images with ROI part for long-term storage. Therefore it is desirable that the copyright and integrity of the medical image with ROI part are protected by digital watermarking. However it is impossible to embed signature information into the ROI part since the ROI must be kept without any distortion.1.2 Classification of Watermarking1.2.1 According to signal processing methods [7]i) Spatial domain Direct usage of data to embed and extract Watermark. This technique is susceptible to signal processing operations like signal compression, cropping, filtering.ii) Transform domain Conversion of data to another format to embed and extract. e.g. Conversion to polar co-ordinate systems of 3D models, makes it robust against scaling. The mark embedding in frequency domain like DCT and DWT.1.2.2 According to application point of view [8]i) Robust Watermarking Robust watermarks are designed to resist against attacks. Is mainly aimed at Copyright protection.ii) Fragile Watermarking Is mainly aimed at content authentication. It can be altered or destroyed when the digital content is modified.

Chapter 2Literature ReviewReversible watermarking or data hiding in medical images and signals is gaining great interest lately. Several reversible schemes have been proposed up to now. They can be divided into three major categories based on Feng and colleagues [11]: schemes that apply data compression, like in [12] schemes that use difference expansion [13] schemes that use histogram bin exchanging [14]In the first case, in order to recover the original image, this whole image, or part of it, is embedded in the original. Side information is also needed for recovery and has to be embedded as well in the host. In order to increase the embedding capacity, compression is introduced [12], [15], [16], [17]. However this type of reversible watermarking lack robustness, as any loss of the compressed data may destroy the embedded data [11].In the second category, Difference Expansion (DE) is applied to embed information. Small values are generated in order to represent the features of the image, taken from an integer transformation, which can be an integer wavelet transformation or another similar function [13]. These schemes are also fragile under attack. Even though, they are pixel-wise, loss of one pixel will not destroy the next pixel and thus the image, however it will destroy the completeness of the location map causing mismatch to all the later pixels. In the third kind of schemes, histogram bin shifting has been proposed in order to tackle the robustness issue. In that scheme the embedded target is replaced by the histogram of the block [14], [18]. In Vleeschoover et al.s scheme the original image is segmented into several blocks of pixels and then follows the embedding process [18]. In the medical field, the reversible watermarking or data hiding technique is gaining great interest due to the strict importance of medical image security and protection. Some newly developed approaches exist for the Magnetic Resonance Imaging (MRI) images. The MRI modality offers images with extreme clarity of representation of the patients internal organs and soft tissues and its significance is high in the medical world. Usually with the separation of medical images into regions of interest and regions of non-interest, the usual trend is that the authentication payload is inserted into the RONIs [19]. In this latter work, a mixed reversible scheme was proposed for head MRI images. Two levels of protection schemes were introduced. In the first level, a robust RONI watermarking scheme was applied. Once the ROI is located, a unique identification number (C1) and a digital signature (S1) derived from it are generated and inserted inside the RONI in a robust manner. In the second level, a protection of the image generated in the first level is introduced. A digital signature of the entire image (S2) is computed and inserted with a unique identification number (C2) according to a reversible scheme. This level necessitates the removal of the reversible watermark before the integrity verification. The lossy fragile watermarking is performed with the Least Significant Bit (LSB) scheme. The lossy robust watermarking inserts one bit of the message by modifying in one block B, the relationship between the value of one selected pixel and the mean value of B. In another image tampered proofing approach, for brain MRI images, belonging to the schemes that apply compression, the host image was divided into blocks of equal size. Then, the recovery and the verification data were created from each block using vector quantization. These latter data were then embedded in the two least significant bit of every block [20]. Moreover, as and in multimedia watermarking the methods can be also divided in those concerning analysis in the frequency or spatial domain. In the first category, the image is transformed into the frequency domain and then some frequency components are being replaced by the watermark. For example Shih and Wu describe a method for robust MRI watermarking that use the Discrete Cosine Transform (DCT) and the Discrete Wavelet transform (DWT). In this work, the ROI was compressed by lossless compression while the rest by lossy compression. Information like a digital signature and textual data were embedded inside the RONI in the frequency domain [21]. A considerable amount of research on reversible data hiding has been done over the past few years. Four important techniques are discussed here.

2.1 Integer Transform TechniqueIn this scheme, an integer transform is used to embed 1-bit watermark into one pixel pair in a way that the sum of the pixel pair remains unchanged. Based on the invariability of sum values and the equality between the parities of sum values and difference values, the extraction of watermarks and the recovery of pixel pairs can be easily achieved. Shaowei Weng et al.[22][23][24] proposed an integer transform in which the forward transform is defined asx = x + d/2 + b y = y d/2 b (1)Where b is used to denote one bit watermark, and d is the difference between the pixels x and y.Actually, x + y equals x + y. x + y and d have the same parity. x and y are the watermarkedimage pixels corresponding to x and y.On the decoding side, the sum of x and y is calculated first. Therefore x + y are determined. The difference value of x and y is calculated and denoted as d. The actual difference d can be calculated as d = (d + LSB(d))/2 b ..(2)The value of d and the watermark bit b can be uniquely deduced because the parity of d is known and b is a binary number. For example, if x = 7, y = 5 and b = 0, then x = 8, y=4 after embedding. On the receiver side, (d + LSB(d))/2 is calculated as 2. The parity of d can be guaranteed to be the same as d. The parity of d is odd parity. The parity of d is odd if and only if b = 0. As a result, watermark bit b is correctly extracted and the value of d is obtained. Once d and x + y are obtained then the original pixel values x and y are calculated asx = (x + y + d)/2 y = (x + y d)/2 (3)

2.2 Difference Expansion TechniqueYongjian Hu [25] uses the predicted image pixel error instead of the pixel-pair difference for Difference Expansion (DE) embedding. A predictor below can exploit the neighboring information to predict an image pixel.max(a,c), if b = xi1 xi 1, if di > P and xi < xi1 (8)where yi is the watermarked value of pixel i. If di = P, modify xi according to the message bit b as followyi = xi + b, if di = P and xi >= xi1 xi b, if di = P and xi < xi1 .(9)At the receiving end, the recipient extracts message bits from the watermarked image by scanning the image in the same order as during the embedding. The message bit b can be extracted byb = 0, if |yi xi1| = P 1, if |yi xi1| = P + 1 (10)where xi1 denotes the restored value of yi1. The original pixel value can be restored byyi + 1, if | yi xi1| > P and yi < xi1xi = yi 1, if | yi xi1| > P and yi > xi1 yi , otherwise (11)Thus, an exact copy of the original host image is obtained. These steps complete the data hiding and extraction process in which only one peak point is used. Large hiding capacities can be obtained by repeating the data hiding process. However, recipients may not be able to retrieve both the embedded message and the original host image without knowledge of the peak points of every hiding pass. A binary tree structure used to deal with communication of multiple peak points. Modification of a pixel may not be allowed if the pixel is saturated (0 or 255). To prevent overflow and underflow, histogram shifting technique is used that narrows the histogram from both sides.

2.4 Interpolation Technique In this technique [29], the difference between interpolation value and corresponding pixel value is used to embed bit 1 of 0 by expanding it additively or leaving it unchanged. It is different from most differential expansion approaches in two important aspects:1) It uses interpolation-error, instead of interpixel difference or prediction- error, to embed data.2) It expands difference, which is interpolation-error here, by addition instead of bit-shifting.First, interpolation values of pixels are calculated using interpolation technique, which works by guessing a pixel value from its surrounding pixels. Then interpolation-errors are obtained by e = x x . (12)where x are the interpolation values of pixels x. The secret bit b is embedded by additively expanding the interpolation error values. The additive interpolation-error expansion is formulated as e + sign(e) x b, e = LM orRM e = e + sign(e) x 1, e (LN,LM)U(RM,RN) e, otherwise (13)where LM and RM denote the corresponding values of the two highest points of interpolation-errors histogram and LN and RN denote the corresponding values of the two lowest points of interpolation-errors histogram. The watermarked pixels x becomes x = x + e .......(14)During the extracting process, the interpolation value x is computed with the same interpolation algorithm and the corresponding interpolation-errors are obtained. Once the interpolation errors, LM, RM, LN and RN are known, the embedded secret data can be extracted. Then the inverse function of additive interpolation-error expansion is applied to recover the original interpolation-errors. Finally, we can restore the original pixels x by adding interpolation value x and the interpolation error e.After secret messages are embedded, some overhead information is needed to extract the covert information and restore the original image. The overhead information are the information to identify those pixels containing embedded bit (LM,LN,RM and RN) and the information to solve the overflow/underflow problem [30].

Chapter 3Problem Identification/MotivationDigital watermarking is the act of hiding information in multimedia data (images, audio or video), for the purposes of content protection or authentication. In digital watermarking, the secret information (usually in the form of a bit-stream), the watermark, is embedded into a multimedia data (cover data), in such a way, that distortion of the cover data due to watermarking is almost negligible perceptually. In addition, in reversible watermarking, the cover data restored after the watermark extraction is identical to the original cover data, bit-by bit. Reversible watermarking finds widespread use in military and medical applications, where distortionfree recovery of the original data after watermark extraction is of utmost importance. A number of reversible watermarking algorithms have already been proposed by various authors, though overwhelming majority of those algorithms has been for grayscale images.The major motivation behind this work is to develop a platform, capable of analyzing and evaluating reversible watermarking algorithms, theoretically as well as through simulations. Currently the focus is on "reversible watermarking of medical data (images/signals)". The steps through which the aim is to achieve this goal:

3.1 Goals(1) An extensive literature survey.(2) Investigating the performance of existing reversible watermarking schemes under specific circumstances, such as an extremely noisy environment, where the cover data is highly vulnerable to get tampered.(3) Investigating the applicability of existing reversible watermarking algorithms to areas other than grayscale imagery, such as color imagery, halftone imagery as well as other forms of multimedia, viz. video and audio.(4) Developing a mechanism to localize tampering in reversibly watermarked medical data, to reduce false rejection rate when authentication fails at the receiver side.(5) Reducing runtime requirements of existing reversible watermarking algorithms, which often require very large complex operations to execute. The aim is to reduce the time complexity by efficient software implementations of those algorithms.(6) Finally, developing a theoretical evaluation platform for analyzing and evaluating the reversible watermarking algorithms.The objectives of the work covering all six steps discussed above have been discussed in this report. The work done till now in the direction of achieving the objectives, includes an extensive literature survey covering five classes of reversible watermarking algorithms based on four different principles of operation , which are (i) integer transform (ii) data compression (iii) histogram bin shifting (iv) prediction of pixel values A simulation based study of the performances of all these algorithms in extremely noisy military environment has been presented in this report. In this study we investigated the effect of high data error rates on different reversible watermarking algorithms, i.e., the amount of distortion of a grayscale image and signal, when it is reversibly watermarked, then sent over such a highly noisy channel, and finally recovered at the receiver side.

Chapter 4Problem Statement/Objective4.1 problem statementMost of the research on digital watermarking has been focused on either direct watermarking of the signal/image or bit stream embedding where the signal/image is represented in a compressed format. Just as in image and video watermarking, the use of perceptual models is an important component in generating an effective and acceptable watermarking scheme for signal/image [31]. Many of the requirements for audio watermarking are similar to signal/image watermarking, such as imperceptibility, robustness to signal alterations such as compression, filtering, and cropping etc [32]. The approach described in [33] consists of generating a PN-sequence for the watermark and processing it with a filter that approximates the frequency masking properties of the human vision system (HVS), followed by a time-domain weighting for temporal masking. Correlation properties of PN-sequences are desirable for detection and applying an auditory model guarantees imperceptibilitya critical feature for high quality signal where copyright protection may be most critical. An overview paper on how perceptual models have been exploited for signal compression can be found in [34][35]. Watermark embedding consists of adding a perceptually weighted PN-sequence to the signal/image file while watermark detection consists of a correlation detector to determine whether the watermark is or is not present in the received signal/image.

4.1.1 The Human Visual System To hide the watermark in the signal, it is useful to exploit the weaknesses of the human visual system [36, 37, 38, and 39]. However, the study of human perception is not confined only to the understanding of the optical device that represents the eye: Between the visions of the image that is printed on the retina and its interpretation in the human brain. The HVS model used in this work has been suggested in [40, 41]. This model is also used in many insertion algorithm and detection of the watermark.(A) Texture sensitivity The degradations caused by the signature will be very low in the homogeneous areas of the signal/image [42, 43], but may be more intense in highly textured areas for which the eye will not be able to differentiate between the signal/image from the image and the signal from the signature [44, 45]. The stronger the texture is less visible is the watermark. The texture sensitivity can be estimated by applying the LWT transformation on the signal/image. The result is in the form of integer so no need to perform quantization.(B) Frequency sensitivityThe insertion in the low frequency band of LWT blocks correspond to homogeneous zones in the signal/image features a marking robust but distorts apparently the watermarked image [46, 47, and 48]. Contrariwise, an insertion in the high frequency band corresponding to the changes of the signal/image, will characterize an imperceptible marking but fragile. Hence the needs to choose a low frequency band that provide a maximum resistance to attacks. 4.2 ObjectiveObjectives of my research work can be divided into five broad sections: Analysis of performance of reversible watermarking algorithms in extreme environments. Reversible watermarking techniques find application in military and medical signal/image, where integrity of the cover image/signal is of utmost importance. However, in practice, many military and medical data transmissions occur over communication channels whose noiselevels are so high that the receiving system is unable to correct all errors in the received data. In such a case, we are bound to get nonzero distortion in the recovered cover image/signal as well as the extracted watermark, in spite of using reversible watermarking techniques. We aim to investigate the effect of high data error rates on different reversible watermarking algorithms, which would help users to choose the most suitable reversible watermarking scheme, depending on whether the distortion of the retrieved cover image or the distortion of the retrieved watermark is the primary concern. Application of reversible watermarking to areas other than grayscale image security. The aim is to investigate the applicability of existing reversible watermarking algorithms to color images as well as other forms of multimedia, viz. video and audio. Another goal is to develop efficient reversible watermarking scheme for halftone images, which are widely used in books, magazines, printer outputs, fax documents [49]-[51] etc. and routinely transmitted over computer networks in large numbers, in printing and publishing industries [52]. Such algorithms are few in the existing literature, so this problem needs to be addressed. Tamper localization in reversible watermarking algorithms. The goal of reversible watermarking is mainly twofold: (1) to authenticate an image at the receiver side; (2) to recover the cover image without the loss of even one bit. Till date a number of reversible watermarking algorithms have been developed which help to achieve both these goals. In those algorithms, the cover image is accepted at the receiver side once it is authenticated; but it is entirely rejected if even a single bit is proved to be corrupted. This leads to false rejection of a large number of pixels. We aim to develop a scheme to reduce this false rejection rate, by exact localization of the tampered pixels. In other words, we aim to minimize the area of the cover image, surrounding the tampered pixel(s), which gets rejected if authentication fails at the receiver side. Efficient software implementation. The existing reversible watermarking algorithms suffer from large runtime requirements when implemented using the standard image processing software such as MATLAB. This is because of the various complex operations needed for achieving the reversible property of those algorithms. One such necessary operation which is absent in other classes of watermarking is the mechanism of managing additional cover image retrieval information. Taking into account this major issue, the efficient implementation of reversible image watermarking needs to be addressed. We aim to develop efficient software (e.g. through multi-threaded programming) as well as hardware implementations of reversible watermarking algorithms. Development of a theoretical evaluation platform. Having solved all the above mentioned issues we aim to develop a theoretical platform for analyzing and evaluating reversible watermarking algorithms. This would help a user to select an algorithm, or a particular class of algorithms, suited for his need.

Chapter 5Proposed Methodology

5.1 MethodologyIn order to embed metadata within the medical image/signal, utilize notions from data watermarking and channel coding. The sensitive metadata (social security number (SSN), birth date, and so on) will be embedded as a hidden watermark within the medical measurements of the patient. In order to provide additional protection and data resilience the watermark is first encrypted than embedded. The embedding will introduce virtually no distortion. Notice, that the watermarking technique should be chosen according to the requirement whether a robust watermarking is required or a fragile watermarking is required. An overview of this architecture is provided in Figure 5.1.1

AuthenticationOriginal EEG Final EEG

+ FragileTemper Detection=+

Meta DataData Retrieval

Key

Figure 5.1.1: Overview of Approach

Once the metadata are effectively fused within the medical signal, there are three supported modes of operation:i) Tamper Detection by examining the presence of the fragile watermark.ii) Data Authentication through correlation with the originally embedded metadata. For example, if the SSN of a patient is embedded in an EEG signal, then using the SSN and a secret,one can verify that the data indeed belong to the patient with a specific Social Security Number.iii) Data Retrieval The rightful owner of the data can provide the secret key to someone else, who is now at a position to retrieve the embedded metadata from the medical signal.5.2 Approach UsedIn the proposed paper a multilevel of security is provided. The technique is based on a message being encoded and hidden in a data in wavelet domain in such a way as to make the existence of the message unknown to an observer shown in figure 5.1.1. The encryption of watermark, while being able to maintain a high level of security for the patient identity. Only those individuals with a key will be able to know the identity of the patient [48]. To increase the security the data is embedded into transformed domain. After transformation the signal/image is divided into two bands, first is low frequency band and second is high frequency band. The low frequency band contains important information of the signal/image and high frequency band contains relatively less information of signal [46]. Most of the attacks are done onto high frequency band as the attackers do not want to harm the high information areas as shown in figure 5.2.1. For this purpose lifting based integer wavelet transform is used. The algorithm organizes wavelet coefficients to generate wavelet blocks, and applies a novel method to classify these wavelet blocks based on human visual system (HVS) [45]. The watermark is first encrypted and then converted in to form of zero and one. The watermark is pseudo randomly distributed over the transformed signal/image. The extraction is done same as encryption in reverse manner. The receiver picks the watermark sample bits and decrypts them shown in figure 5.2.2.

Figure 5.2.1: Watermark Embedding Procedure

Figure 5.2.2: Watermark Extraction Procedure

5.2.1 Watermark ConstructionLet us describe now how the private metadata are embedded into the hidden watermark. The social security number (SSN) of the patient is used as watermark. The SSN of patient is available in decimal number. This number is encrypted using 8 bit PN sequence. The number is modulo 2 added with the generated PN sequence. The encrypted watermark sequence is converted from decimal to binary. The basic procedure is shown in the figure 5.2.2.1.

SSN999-99Text to Image 999-99Character to Binary1000.10PN Sequence

Figure 5.2.1.1: Encrypted watermark in binary format

5.2.2 Lifting Based Integer Wavelet TransformThe wavelet transform is a valuable tool for Multi resolution analysis that has been widely used in image processing applications [38]. The wavelet transform has a number of advantages over other transforms as it provides a multi resolution description, it allows superior modeling of the HVS, the high-resolution sub bands allow easy detection of features such as edges or textured areas in transform domain. In the transform coding of images, the image is projected onto a set of basis functions, and the resultant transform coefficients are encoded [39]. Efficient coding requires that the transform compact the energy into a small number of coefficients. The LWT transform the signal into two band, low frequency and high frequency band shown in figure 5.2.2.1. In this work one level decomposition of signal is done. Second level decomposition can also be used but the embedding capacity will be decrease.

1st level decomposition 2nd level decompositionFigure 5.2.2.1: 2 level Wavelet Decomposition

a) 1-D Discrete Wavelet TransformA general 1-D discrete wavelet transform can be written as [3]: (3)

(x) = 1 0