A DWT Based Approach for Image Steganography

ABSTRACT: In this paper we propose a new steganography technique which embeds the secret messages in frequency domain. According to different users demands on the embedding capacity and image quality, the proposed algorithm is divided into two modes and 5 cases. Unlike the space domain approaches, secret messages are embedded in the high frequency coefficients resulted from Discrete Wavelet Transform. Coefficients in the low frequency sub-band are preserved unaltered to improve the image quality. Some basic mathematical operations are performed on the secret messages before embedding. These operations and a well-designed mapping Table keep the messages away from stealing, destroying from unintended users on the internet and hence provide satisfactory security.

INTRODUCTIONIn a highly digitalized world we live today, computers help transforming analog data into digital forms before storing and/or processing. In the mean while, the internet develops very fast and hence becomes an important medium for digital data transmission. However, being a fully open medium, the internet brought us not only convenience but also some hazards and risks. If the data to be transmitted are confidential, it is convenient as well for some malicious users to illegally copy, destroy, or change them on the internet. As a result, information security becomes an essential issue . Various schemes for data hiding are developed recently . According to the purposes of data hiding, these schemes are classified into two categories: watermarking and steganography. Watermarking is a protecting technique which protects (claims) the authors property right for images by some hidden watermarks. On the other hand, steganography techniques apply some cover images to protect the confidential data from unintended internet users. According to the domain where watermarks or confidential data are embedded, both categories can be further classified as the time domain methods and the frequency domain methods. Watermarking designs are usually consistent with the following features .(1) Imperceptibility: Human eyes cannot distinguish the difference between the watermarked image and the original version. In other words, the watermarked images still preserve high imagequality. (2) Security: The watermarked image cannot be copied, modified, or deleted by anyanimus observer. (3) Robustness: The watermark still can be extracted out within certain acceptable quality even the image has endured some signal processing or noises before extraction. (4) Statistically undetectable: It is extremely hard (or impossible) to detect the watermark by statistical and/or mathematical analysis. (5) Blind detection: The extracting procedures have not to access the original image.For spatial domain watermarking methods the processing is applied on the image pixel values directly. In other words, the watermark is embedded in image by modifying the pixel values. The advantage of this type of watermarking is easy and computationally fast. The disadvantage is its low ability to bear some signal processing or noises. For frequency domain methods, the first step is to transform the image data into frequency domain coefficients by some mathematical tools (e.g. FFT, DCT, or DWT). Then, according to the different data characteristics generated by these transforms, embed the watermark into the coefficients in frequency domain. After the watermarked coefficients are transformed back to spatial domain, the entire embedding procedure is completed. The advantage of this type of watermarking is the high ability to face some signal processing or noises. However, methods of this type are computationally complex and hence slower. The second category of data hiding is called Steganography. The methods are designed to embed some confidential data into some cover-media (such as texts, voices, images,and videos). After the confidential data are embedded, they are then transmitted together with the cover-media. The major objective is to prevent some unintended observer from stealing or destroying those confidential data. There are two things to be considered when designing a steganography system: (1) Invisibility: Human eyes cannot distinguish the difference between the original image and thestego-image (the image with confidential data embedded in). (2) Capacity: The more data an image can carry the better it is. However, large embedded data usually degrade the image quality significantly. How one can increase the capacity without ruining the invisibility is the key problem. The design of a steganography system also can be categorized into spatial domain methods and frequency domain ones The advantages and disadvantages are the same as those we mentioned about watermarking methods earlier.

DIGITAL IMAGE PROCESSINGBACKGROUND: Digital image processing is an area characterized by the need for extensive experimental work to establish the viability of proposed solutions to a given problem. An important characteristic underlying the design of image processing systems is the significant level of testing & experimentation that normally is required before arriving at an acceptable solution. This characteristic implies that the ability to formulate approaches &quickly prototype candidate solutions generally plays a major role in reducing the cost & time required to arrive at a viable system implementation. What is DIP? An image may be defined as a two-dimensional function f(x, y), where x & y are spatial coordinates, & the amplitude of f at any pair of coordinates (x, y) is called the intensity or gray level of the image at that point. When x, y & the amplitude values of f are all finite discrete quantities, we call the image a digital image. The field of DIP refers to processing digital image by means of digital computer. Digital image is composed of a finite number of elements, each of which has a particular location & value. The elements are called pixels. Vision is the most advanced of our sensor, so it is not surprising that image play the single most important role in human perception. However, unlike humans, who are limited to the visual band of the EM spectrum imaging machines cover almost the entire EM spectrum, ranging from gamma to radio waves. They can operate also on images generated by sources that humans are not accustomed to associating with image. There is no general agreement among authors regarding where image processing stops & other related areas such as image analysis& computer vision start. Sometimes a distinction is made by defining image processing as a discipline in which both the input & output at a process are images. This is limiting & somewhat artificial boundary. The area of image analysis (image understanding) is in between image processing & computer vision. There are no clear-cut boundaries in the continuum from image processing at one end to complete vision at the other. However, one useful paradigm is to consider three types of computerized processes in this continuum: low-, mid-, & high-level processes. Low-level process involves primitive operations such as image processing to reduce noise, contrast enhancement & image sharpening. A low- level process is characterized by the fact that both its inputs & outputs are images. Mid-level process on images involves tasks such as segmentation, description of that object to reduce them to a form suitable for computer processing & classification of individual objects. A mid-level process is characterized by the fact that its inputs generally are images but its outputs are attributes extracted from those images. Finally higher- level processing involves Making sense of an ensemble of recognized objects, as in image analysis & at the far end of the continuum performing the cognitive functions normally associated with human vision. Digital image processing, as already defined is used successfully in a broad range of areas of exceptional social & economic value.What is an image? An image is represented as a two dimensional function f(x, y) where x and y are spatial co-ordinates and the amplitude of f at any pair of coordinates (x, y) is called the intensity of the image at that point. Gray scale image: A grayscale image is a function I (xylem) of the two spatial coordinates of the image plane.I(x, y) is the intensity of the image at the point (x, y) on the image plane.I (xylem) takes non-negative values assume the image is bounded by a rectangle [0, a] [0, b]I: [0, a] [0, b] [0, info)

Color image: It can be represented by three functions, R (xylem) for red, G (xylem) for green and B (xylem) for blue. An image may be continuous with respect to the x and y coordinates and also in amplitude. Converting such an image to digital form requires that the coordinates as well as the amplitude to be digitized. Digitizing the coordinates values is called sampling. Digitizing the amplitude values is called quantization.Coordinate convention: The result of sampling and quantization is a matrix of real numbers. We use two principal ways to represent digital images. Assume that an image f(x, y) is sampled so that the resulting image has M rows and N columns. We say that the image is of size M X N. The values of the coordinates (xylem) are discrete quantities. For notational clarity and convenience, we use integer values for these discrete coordinates. In many image processing books, the image origin is defined to be at (xylem)=(0,0).The next coordinate values along the first row of the image are (xylem)=(0,1).It is important to keep in mind that the notation (0,1) is used to signify the second sample along the first row. It does not mean that these are the actual values of physical coordinates when the image was sampled. Following figure shows the coordinate convention. Note that x ranges from 0 to M-1 and y from 0 to N-1 in integer increments. The coordinate convention used in the toolbox to denote arrays is different from the preceding paragraph in two minor ways. First, instead of using (xylem) the toolbox uses the notation (race) to indicate rows and columns. Note, however, that the order of coordinates is the same as the order discussed in the previous paragraph, in the sense that the first element of a coordinate topples, (alb), refers to a row and the second to a column. The other difference is that the origin of the coordinate system is at (r, c) = (1, 1); thus, r ranges from 1 to M and c from 1 to N in integer increments. IPT documentation refers to the coordinates. Less frequently the toolbox also employs another coordinate convention called spatial coordinates which uses x to refer to columns and y to refers to rows. This is the opposite of our use of variables x and y.

Image as Matrices: The preceding discussion leads to the following representation for a digitized image function: f (0,0) f(0,1) .. f(0,N-1) f(1,0) f(1,1) f(1,N-1) f(xylem)= . . . . . . f(M-1,0) f(M-1,1) f(M-1,N-1) The right side of this equation is a digital image by definition. Each element of this array is called an image element, picture element, pixel or pel. The terms image and pixel are used throughout the rest of our discussions to denote a digital image and its elements. A digital image can be represented naturally as a MATLAB matrix: f(1,1) f(1,2) . f(1,N) f(2,1) f(2,2) .. f(2,N) . . . f = f(M,1) f(M,2) .f(M,N) Where f(1,1) = f(0,0) (note the use of a monoscope font to denote MATLAB quantities). Clearly the two representations are identical, except for the shift in origin. The notation f(p ,q) denotes the element located in row p and the column q. For example f(6,2) is the element in the sixth row and second column of the matrix f. Typically we use the letters M and N respectively to denote the number of rows and columns in a matrix. A 1xN matrix is called a row vector whereas an Mx1 matrix is called a column vector. A 1x1 matrix is a scalar. Matrices in MATLAB are stored in variables with names such as A, a, RGB, real array and so on. Variables must begin with a letter and contain only letters, numerals and underscores. As noted in the previous paragraph, all MATLAB quantities are written using mono-scope characters. We use conventional Roman, italic notation such as f(x ,y), for mathematical expressionsReading Images: Images are read into the MATLAB environment using function imread whose syntax is imread(filename) Format name Description recognized extension TIFF Tagged Image File Format .tif, .ti JPEG Joint Photograph Experts Group .jpg, .jpeg GIF Graphics Interchange Format .gif BMP Windows Bitmap .bmp PNG Portable Network Graphics .png XWD X Window Dump .xwd Here filename is a spring containing the complete of the image file(including any applicable extension).For example the command line >> f = imread (8. jpg);reads the JPEG (above table) image chestxray into image array f. Note the use of single quotes () to delimit the string filename. The semicolon at the end of a command line is used by MATLAB for suppressing output. If a semicolon is not included. MATLAB displays the results of the operation(s) specified in that line. The prompt symbol(>>) designates the beginning of a command line, as it appears in the MATLAB command window. When as in the preceding command line no path is included in filename, imread reads the file from the current directory and if that fails it tries to find the file in the MATLAB search path. The simplest way to read an image from a specified directory is to include a full or relative path to that directory in filename. For example, >> f = imread ( D:\myimages\chestxray.jpg); reads the image from a folder called my images on the D: drive, whereas >> f = imread( . \ myimages\chestxray .jpg); reads the image from the my images subdirectory of the current of the current working directory. The current directory window on the MATLAB desktop toolbar displays MATLABs current working directory and provides a simple, manual way to change it. Above table lists some of the most of the popular image/graphics formats supported by imread and imwrite.Data Classes: Although we work with integers coordinates the values of pixels themselves are not restricted to be integers in MATLAB. Table above list various data classes supported by MATLAB and IPT are representing pixels values. The first eight entries in the table are refers to as numeric data classes. The ninth entry is the char class and, as shown, the last entry is referred to as logical data class. All numeric computations in MATLAB are done in double quantities, so this is also a frequent data class encounter in image processing applications. Class unit 8 also is encountered frequently, especially when reading data from storages devices, as 8 bit images are most common representations found in practice. These two data classes, classes logical, and, to a lesser degree, class unit 16 constitute the primary data classes on which we focus. Many ipt functions however support all the data classes listed in table. Data class double requires 8 bytes to represent a number uint8 and int 8 require one byte each, uint16 and int16 requires 2bytes and unit 32.Image Types:The toolbox supports four types of images: 1 .Intensity images 2. Binary images 3. Indexed images 4. R G B images Most monochrome image processing operations are carried out using binary or intensity images, so our initial focus is on these two image types. Indexed and RGB colour images.

Intensity Images: An intensity image is a data matrix whose values have been scaled to represent intentions. When the elements of an intensity image are of class unit8, or class unit 16, they have integer values in the range [0,255] and [0, 65535], respectively. If the image is of class double, the values are floating _point numbers. Values of scaled, double intensity images are in the range [0, 1] by convention.Binary Images: Binary images have a very specific meaning in MATLAB.A binary image is a logical array 0s and1s.Thus, an array of 0s and 1s whose values are of data class, say unit8, is not considered as a binary image in MATLAB .A numeric array is converted to binary using function logical. Thus, if A is a numeric array consisting of 0s and 1s, we create an array B using the statement. B=logical (A) If A contains elements other than 0s and 1s.Use of the logical function converts all nonzero quantities to logical 1s and all entries with value 0 to logical 0s.Using relational and logical operators also creates logical arrays.To test if an array is logical we use the I logical function: islogical(c) If c is a logical array, this function returns a 1.Otherwise returns a 0. Logical array can be converted to numeric arrays using the data class conversion functions.Indexed Images:An indexed image has two components:A data matrix integer, x.A color map matrix, map. Matrix map is an m*3 arrays of class double containing floating_ point values in the range [0, 1].The length m of the map are equal to the number of colors it defines. Each row of map specifies the red, green and blue components of a single color. An indexed images uses direct mapping of pixel intensity values color map values. The color of each pixel is determined by using the corresponding value the integer matrix x as a pointer in to map. If x is of class double ,then all of its components with values less than or equal to 1 point to the first row in map, all components with value 2 point to the second row and so on. If x is of class units or unit 16, then all components value 0 point to the first row in map, all components with value 1 point to the second and so on. RGB Image: An RGB color image is an M*N*3 array of color pixels where each color pixel is triplet corresponding to the red, green and blue components of an RGB image, at a specific spatial location. An RGB image may be viewed as stack of three gray scale images that when fed in to the red, green and blue inputs of a color monitor Produce a color image on the screen. Convention the three images forming an RGB color image are referred to as the red, green and blue components images. The data class of the components images determines their range of values. If an RGB image is of class double the range of values is [0, 1]. Similarly the range of values is [0,255] or [0, 65535].For RGB images of class units or unit 16 respectively. The number of bits use to represents the pixel values of the component images determines the bit depth of an RGB image. For example, if each component image is an 8bit image, the corresponding RGB image is said to be 24 bits deep. Generally, the number of bits in all component images is the same. In this case the number of possible color in an RGB image is (2^b) ^3, where b is a number of bits in each component image. For the 8bit case the number is 16,777,216 colors.LITERATURE REVIEW:Since the rise of the Internet one of the most important factors of information technology and communication has been the security of information. Cryptography was created as a technique for securing the secrecy of communication and many different methods have been developed to encrypt and decrypt data in order to keep the message secret. Unfortunately it is sometimes not enough to keep the contents of a message secret, it may also be necessary to keep the existence of the message secret. The technique used to implement this, is called steganography. Steganography is the art and science of invisible communication. This is accomplished through hiding information in other information, thus hiding the existence of the communicated information. The word steganography is derived from the Greek words stegos meaning cover and grafia meaning writing defining it as covered writing. In image steganography the information is hidden exclusively in images. The idea and practice of hiding information has a long history. In Histories the Greek historian Herodotus writes of a nobleman, Histaeus, who needed to communicate with his son-in-law in Greece. He shaved the head of one of his most trusted slaves and tattooed the message onto the slaves scalp. When the slaves hair grew back the slave was dispatched with the hidden message. In the Second World War the Microdot technique was developed by the Germans. Information, especially photographs, was reduced in size until it was the size of a typed period. Extremely difficult to detect, a normal cover message was sent over an insecure channel with one of the periods on the paper containing hidden information. Today steganography is mostly used on computers with digital data being the carriers and networks being the high speed delivery channels.Steganography differs from cryptography in the sense that where cryptography focuses on keeping the contents of a message secret, steganography focuses on keeping the existence of a message secret. Steganography and cryptography are both ways to protect information from unwanted parties but neither technology alone is perfect and can be compromised. Once the presence of hidden information is revealed or even suspected, the purpose of steganography is partly defeated. The strength of steganography can thus be amplified by combining it with cryptography.

Two other technologies that are closely related to steganography are watermarking and fingerprinting. These technologies are mainly concerned with the protection of intellectual property, thus the algorithms have different requirements than steganography. These requirements of a good steganographic algorithm will be discussed below. In watermarking all of the instances of an object are marked in the same way. The kind of information hidden in objects when using watermarking is usually a signature to signify origin or ownership for the purpose of copyright protection. With fingerprinting on the other hand, different, unique marks are embedded in distinct copies of the carrier object that are supplied to different customers. This enables the intellectual property owner to identify customers who break their licensing agreement by supplying the property to third parties.

In watermarking and fingerprinting the fact that information is hidden inside the files may be public knowledge sometimes it may even be visible while in steganography the imperceptibility of the information is crucial. A successful attack on a steganographic system consists of an adversary observing that there is information hidden inside a file, while a successful attack on a watermarking or fingerprinting system would not be to detect the mark, but to remove it.Research in steganography has mainly been driven by a lack of strength in cryptographic systems. Many governments have created laws to either limit the strength of a cryptographic system or to prohibit it altogether, forcing people to study other methods of secure information transfer. Businesses have also started to realize the potential of steganography in communicating trade secrets or new product information. Avoiding communication through well-known channels greatly reduces the risk of information being leaked in transit. Hiding information in a photograph of the company picnic is less suspicious than communicating an encrypted file.

This project intends to offer a state of the art overview of the different algorithms used for image steganography to illustrate the security potential of steganography for business and personal use. After the overview it briefly reflects on the suitability of various image steganography techniques for various applications. This reflection is based on a set of criteria that we have identified for image steganography.

Steganography conceptsAlthough steganography is an ancient subject, the modern formulation of it is often given in terms of the prisoners problem proposed by Simmons, where two inmates wish to communicate in secret to hatch an escape plan. All of their communication passes through a warden who will throw them in solitary confinement should she suspect any covert communication.

The warden, who is free to examine all communication exchanged between the inmates, can either be passive or active. A passive warden simply examines the communication to try and determine if it potentially contains secret information. If she suspects a communication to contain hidden information, a passive warden takes note of the detected covert communication, reports this to some outside party and lets the message through without blocking it. An active warden, on the other hand, will try to alter the communication with the suspected hidden information deliberately, in order to remove the information.

Different kinds of steganographyAlmost all digital file formats can be used for steganography, but the formats that are more suitable are those with a high degree of redundancy. Redundancy can be defined as the bits of an object that provide accuracy far greater than necessary for the objects use and display. The redundant bits of an object are those bits thatcan be altered without the alteration being detected easily. Image and audio files especially comply with this requirement, while research has also uncovered other file formats that can be used for information hiding. Figure 1 shows the four main categories of file formats that can be used for steganography.

Figure 1: Categories of steganography

Hiding information in text is historically the most important method of steganography. An obvious method was to hide a secret message in every nth letter of every word of a text message. It is only since the beginning of the Internet and all the different digital file formats that is has decreased in importance. Text steganography using digital files is not used very often since text files have a very small amount of redundant data. Given the proliferation of digital images, especially on the Internet, and given the large amount of redundant bits present in the digital representation of an image, images are the most popular cover objects for steganography.

This project will focus on hiding information in images in the next sections.To hide information in audio files similar techniques are used as for image files. One different technique unique to audio steganography is masking, which exploits the properties of the human ear to hide information unnoticeably. A faint, but audible, sound becomes inaudible in the presence of another louder audible sound.This property creates a channel in which to hide information. Although nearly equal to images in steganographic potential, the larger size of meaningful audio files makes them less popular to use than images.

The term protocol steganography refers to the technique of embedding information within messages and network control protocols used in network transmission. In the layers of the OSI network model there exist covert channels where steganography can be used An example of where information can be hidden is in the header of a TCP/IP packet in some fields that are either optional or are never used.

Image steganographyAs stated earlier, images are the most popular cover objects used for steganography. In the domain of digital images many different image file formats exist, most of them for specific applications. For these different image file formats, different steganographic algorithms exist.

Image definitionTo a computer, an image is a collection of numbers that constitute different light intensities in different areas of the image. This numeric representation forms a grid and the individual points are referred to as pixels. Most images on the Internet consists of a rectangular map of the images pixels (represented as bits) where each pixel is located and its colour. These pixels are displayed horizontally row by row. The number of bits in a colour scheme, called the bit depth, refers to the number of bits used for each pixel.

The smallest bit depth in current colour schemes is 8, meaning that there are 8 bits used to describe the colour of each pixel. Monochrome and greyscale images use 8 bits for each pixel and are able to display 256 different colours or shades of grey. Digital colour images are typically stored in 24-bit files and use the RGBcolour model, also known as true colour. All colour variations for the pixels of a 24-bit image are derived from three primary colours: red, green and blue, and each primary colour is represented by 8 bits. Thus in one given pixel, there can be 256 different quantities of red, green and blue, adding up to more than 16-millioncombinations, resulting in more than 16-million colour. Not surprisingly the larger amount of colours that can be displayed, the larger the file size..Image steganography techniques:Image steganography techniques can be divided into two groups: those in the Image Domain and those in the Transform Domain.

Image also known as spatial domain techniques embed messages in the intensity of the pixels directly, while for transform also known as frequency domain, images are first transformed and then the message is embedded in the image.

Image domain techniques encompass bit-wise methods that apply bit insertion and noise manipulation and are sometimes characterised as simple systems. The image formats that are most suitable for image domain steganography are lossless and the techniques are typically dependent on the image format.

Steganography in the transform domain involves the manipulation of algorithms and image transforms. These methods hide messages in more significant areas of the cover image, making it more robust. Many transform domain methods are independent of the image format and the embedded message may survive conversion between lossy and lossless compression.

Image DomainLeast Significant BitLeast significant bit (LSB) insertion is a common, simple approach to embedding information in a cover image. The least significant bit (in other words, the 8th bit) of some or all of the bytes inside an image is changed to a bit of the secret message. When using a 24-bit image, a bit of each of the red, green and blue colour components can be used, since they are each represented by a byte. In other words, one can store 3 bits in each pixel. An 800 600 pixel image, can thus store a total amount of 1,440,000 bits or 180,000 bytes of embedded data.

For example a grid for 3 pixels of a 24-bit image can be as follows:(00101101 00011100 11011100)(10100110 11000100 00001100)(11010010 10101101 01100011)When the number 200, which binary representation is 11001000, is embedded into the least significant bits of this part of the image, the resulting grid is as follows:(00101101 00011101 11011100)(10100110 11000101 00001100)(11010010 10101100 01100011)Although the number was embedded into the first 8 bytes of the grid, only the 3 underlined bits needed to be changed according to the embedded message. On average, only half of the bits in an image will need to be modified to hide a secret message using the maximum cover size. Since there are 256 possible intensities ofeach primary colour, changing the LSB of a pixel results in small changes in the intensity of the colours. These changes cannot be perceived by the human eye - thus the message is successfully hidden. With a well-chosen image, one can even hide the message in the least as well as second to least significant bit.

In the above example, consecutive bytes of the image data from the first byte to the end of the message are used to embed the information. This approach is very easy to detect. A slightly more secure system is for the sender and receiver to share a secret key that specifies only certain pixels to be changed. Should anadversary suspect that LSB steganography has been used, he has no way of knowing which pixels to target without the secret key.

In its simplest form, LSB makes use of BMP images, since they use lossless compression. Unfortunately to be able to hide a secret message inside a BMP file, one would require a very large cover image. Nowadays, BMP images of 800 600 pixels are not often used on the Internet and might arouse suspicion. For this reason, LSB steganography has also been developed for use with other image file formats.

LSB and Palette Based ImagesPalette based images, for example GIF images, are another popular image file format commonly used on the Internet. By definition a GIF image cannot have a bit depth greater than 8, thus the maximum number of colours that a GIF can store is 256. GIF images are indexed images where the colours used in the image are stored in a palette, sometimes referred to as a colour lookup table. Each pixel is represented as a single byte and the pixel data is an index to the colour palette. The colours of the palette are typically ordered from the most used colour to the least used colours to reduce lookup time.

GIF images can also be used for LSB steganography, although extra care should be taken. The problem with the palette approach used with GIF images is that should one change the least significant bit of a pixel, it can result in a completely different colour since the index to the colour palette is changed. If adjacent palette entries are similar, there might be little or no noticeable change, but should the adjacent palette entries be very dissimilar, the change would be evident. One possible solution is to sort the palette so that the colour differences between consecutive colours are minimized. Another solution is to add new colours which are visually similar to the existing colours in the palette. This requires the original image to have less unique colours than the maximum number of colours (this value depends on the bit depth used). Using this approach, one should thus carefully choose the right cover image. Unfortunately any tampering with the palette of an indexed image leaves a very clear signature, making it easier to detect.A final solution to the problem is to use greyscale images. In an 8-bit greyscale GIF image, there are 256 different shades of grey. The changes between the colours are very gradual, making it harder to detect.WAVELETSA wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have specific properties that make them useful for signal processing. Wavelets can be combined, using a "shift, multiply and sum" technique called convolution, with portions of an unknown signal to extract information from the unknown signal.For example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly a 32nd note. If this wavelet were to be convolved at periodic intervals with a signal created from the recording of a song, then the results of these convolutions would be useful for determining when the Middle C note was being played in the song. Mathematically, the wavelet will resonate if the unknown signal contains information of similar frequency - just as a tuning fork physically resonates with sound waves of its specific tuning frequency. This concept of resonance is at the core of many practical applications of wavelet theory.As a mathematical tool, wavelets can be used to extract information from many different kinds of data, including - but certainly not limited to - audio signals and images. Sets of wavelets are generally needed to analyze data fully. A set of "complementary" wavelets will deconstruct data without gaps or overlap so that the deconstruction process is mathematically reversible. Thus, sets of complementary wavelets are useful in wavelet based compression/decompression algorithms where it is desirable to recover the original information with minimal loss.

1.1. Wavelet transformsA wavelet is a mathematical function used to divide a given function or continuous-time signal into different scale components. Usually one can assign a frequency range to each scale component. Each scale component can then be studied with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets. The wavelets are scaled and translated copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillating waveform (known as the "mother wavelet"). Wavelet transforms have advantages over traditional Fourier transforms for representing functions that have discontinuities and sharp peaks, and for accurately deconstructing and reconstructing finite, non-periodic and/or non-stationary signals.Wavelet transforms are classified into discrete wavelet transforms (DWTs) and continuous wavelet transforms (CWTs). Note that both DWT and CWT are continuous-time (analog) transforms. They can be used to represent continuous-time (analog) signals. CWTs operate over every possible scale and translation whereas DWTs use a specific subset of scale and translation values or representation grid.Wavelet Transform:a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform.The integral wavelet transform is the integral transform defined as

The wavelet coefficients are then given by

Here, a = 2 j is called the binary dilation or dyadic dilation, and b = k2 j is the binary or dyadic position.

There are a large number of wavelet transforms each suitable for different applications. For a full list see list of wavelet-related transforms but the common ones are listed below: Continuous wavelet transform (CWT) Discrete wavelet transform (DWT) Fast wavelet transform (FWT) Stationary wavelet transform (SWT) Lifting Scheme Complex Wavelet Transform(CWT) 1.1.1. Continuous Wavelet Transform (CWTA continuous wavelet transform (CWT) is used to divide a continuous-time function into wavelets. Unlike Fourier transform, the continuous wavelet transform possesses the ability to construct a time-frequency representation of a signal that offers very good time and frequency localization. In mathematics, the continuous wavelet transform of a continuous, square-integrable function x(t) at a scale a > 0 and translational value is expressed by the following integral

where (t) is a continuous function in both the time domain and the frequency domain called the mother wavelet and represents operation of complex conjugate. The main purpose of the mother wavelet is to provide a source function to generate the daughter wavelets which are simply the translated and scaled versions of the mother wavelet. To recover the original signal x (t), inverse continuous wavelet transform can be exploited.

is the dual function of (t). And the dual function should satisfy

Sometimes, , where

Is called the admissibility constant and is the Fourier transform of . For a successful inverse transform, the admissibility constant has to satisfy the admissibility condition:.It is possible to show that the admissibility condition implies that, so that a wavelet must integrate to zero.Mother wavelet:In general, it is preferable to choose a mother wavelet that is continuously differentiable with compactly supported scaling function and high vanishing moments. A wavelet associated with a multi resolution analysis is defined by the following two functions: the wavelet function (t), and the scaling function.

The scaling function is compactly supported if and only if the scaling filter h has a finite support, and their supports are the same. For instance, if the support of the scaling function is [N1, N2], then the wavelet is [(N1-N2+1)/2,(N2-N1+1)/2]. On the other hand, the kth moments can be expressed by the following equation

If m0 = m1 = m2 = ..... = mp 1 = 0, we say (t) has p vanishing moments. The number of vanishing moments of a wavelet analysis represents the order of a wavelet transform. According to the Strang-Fix conditions, the error for an orthogonal wavelet approximation at scale a = 2 i globally decays as aL, where L is the order of the transform. In other words, a wavelet transform with higher order will result in better signal approximations.Mathematically, the process of Fourier analysis is represented by the Fourier transform: Which is the sum over all time of the signal f (t) multiplied by a complex exponential. (Recall that a complex exponential can be broken down into real and imaginary sinusoidal components.) The results of the transform are the Fourier coefficients , which when multiplied by a sinusoid of frequency yield the constituent sinusoidal components of the original signal.

Graphically, the process looks like

Similarly, the continuous wavelet transform (CWT) is defined as the sum over all time of the signal multiplied by scaled, shifted versions of the wavelet function

The results of the CWT are many wavelet coefficients C, which are a function of scale and position. Multiplying each coefficient by the appropriately scaled and shifted wavelet yields the constituent wavelets of the original signal:

Continuous waveletsReal-valued Beta wavelet Hermitian wavelet Hermitian hat wavelet Mexican hat wavelet Shannon waveletComplex-valued Complex Mexican hat wavelet Morlet wavelet Shannon wavelet Modified Morlet wavelet1.1.2. Discrete Wavelet Transform (DWT)A discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information (location in time).Definition:One level of the transformThe DWT of a signal x is calculated by passing it through a series of filters. First the samples are passed through a low pass filter with impulse response g resulting in a convolution of the two:

The signal is also decomposed simultaneously using a high-pass filter h. the outputs giving the detail coefficients (from the high-pass filter) and approximation coefficients (from the low-pass). It is important that the two filters are related to each other and they are known as a quadrature mirror filter.However, since half the frequencies of the signal have now been removed, half the samples can be discarded according to Nyquists rule. The filter outputs are then sub sampled by 2 (Mallat's and the common notation is the opposite, g- high pass and h- low pass):

This decomposition has halved the time resolution since only half of each filter output characterizes the signal. However, each output has half the frequency band of the input so the frequency resolution has been doubled.Examples:Haar wavelets:The first DWT was invented by the Hungarian mathematician Alfred Haar. For an input represented by a list of 2n numbers, the Haar wavelet transform may be considered to simply pair up input values, storing the difference and passing the sum.

This process is repeated recursively, pairing up the sums to provide the next scale: finally resulting in 2n 1 differences and one final sum.Daubechies waveletsThe most commonly used set of discrete wavelet transforms was formulated by the Belgian mathematician Ingrid Daubechies in 1988. This formulation is based on the use of recurrence relations to generate progressively finer discrete samplings of an implicit mother wavelet function; each resolution is twice that of the previous scale. In her seminal paper, Daubechies derives a family of wavelets, the first of which is the Haar wavelet. Interest in this field has exploded since then, and many variations of Daubechies' original wavelets were developed.Calculating wavelet coefficients at every possible scale is a fair amount of work, and it generates an awful lot of data. What if we choose only a subset of scales and positions at which to make our calculations? It turns out, rather remarkably, that if we choose scales and positions based on powers of two -- so-called dyadic scales and positions -- then our analysis will be much more efficient and just as accurate. We obtain such an analysis from the discrete wavelet transform (DWT). An efficient way to implement this scheme using filters was developed in 1988 by Mallat. The Mallat algorithm is in fact a classical scheme known in the signal processing community as a two-channel sub band coder. This very practical filtering algorithm yields a fast wavelet transform -- a box into which a signal passes, and out of which wavelet coefficients quickly emerge.Given a signal s of length N, the DWT consists of log2N stages at most. Starting from s, the first step produces two sets of coefficients: approximation coefficients cA1, and detail coefficients cD1.

These vectors are obtained by convolving s with the low-pass filter Lo_ D for approximation, and with the high-pass filter Hi_ D for detail, followed by dyadic decimation. More precisely, the first step is

The length of each filter is equal to 2N. If n = length (s), the signals F and G are of length n + 2N - 1, and then the coefficients cA1 and cD1 are of length The next step splits the approximation coefficients cA1 in two parts using the same scheme, replacing s by cA1 and producing cA2 and cD2, and so on. Applications of discrete wavelet transform:Generally, an approximation to DWT is used for data compression if signal is already sampled, and the CWT for signal analysis. Thus, DWT approximation is commonly used in engineering and computer science, and the CWT in scientific research.

Wavelet transforms are now being adopted for a vast number of applications, often replacing the conventional Fourier Transform. Many areas of physics have seen this paradigm shift, including molecular dynamics, astrophysics, density-matrix localization, seismology, optics, turbulence and quantum mechanics. This change has also occurred in image processing, blood-pressure, heart-rate and ECG analyses, DNA analysis, protein analysis, climatology, general signal processing, speech recognition, computer graphics and multi fractal analysis. In computer vision and image processing, the notion of scale-space representation and Gaussian derivative operators is regarded as a canonical multi-scale representation.One use of wavelet approximation is in data compression. Like some other transforms, wavelet transforms can be used to transform data, then encode the transformed data, resulting in effective compression. For example, JPEG 2000 is an image compression standard that uses bi-orthogonal wavelets. This means that although the frame is over complete, it is a tight frame, and the same frame functions (except for conjugation in the case of complex wavelets) are used for both analysis and synthesis, i.e., in both the forward and inverse transform. For details see wavelet compression.A related use is for smoothing/de-noising data based on wavelet coefficient thresholding, also called wavelet shrinkage. By adaptively thresholding the wavelet coefficients that correspond to undesired frequency components smoothing and/or de-noising operations can be performed.Wavelet transforms are also starting to be used for communication applications. Wavelet OFDM is the basic modulation scheme used in HD-PLC (a power line communications technology developed by Panasonic), and in one of the optional modes included in the IEEE 1901 standard. Wavelet OFDM can achieve deeper notches than a traditional FFT OFDM system does not require a guard interval (which usually represents significant overhead in FFT OFDM systems.

1.1.3. Fast Wavelet Transform (FWT)The Fast Wavelet Transform is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets. The transform can be easily extended to multidimensional signals, such as images, where the time domain is replaced with the space domain.It has as theoretical foundation the device of a finitely generated, orthogonal multi resolution analysis (MRA). In the terms given there, one selects a sampling scale J with sampling rate of 2J per unit interval, and projects the given signal f onto the space VJ; in theory by computing the scalar products

Where is the scaling function of the chosen wavelet transform; in practice by any suitable sampling procedure under the condition that the signal is highly oversampled, so

Is the orthogonal projection or at least some good approximation of the original signal in.

The MRA is characterized by its scaling sequenceor, as Z-transform, And its wavelet sequence or (Some coefficients might be zero). Those allow to compute the wavelet coefficients, at least some range k=M,-----, J-1, without having to approximate the integrals in the corresponding scalar products. Instead, one can directly, with the help of convolution and decimation operators, compute those coefficients from the first approximation s (J)Forward DWT:One computes recursively, starting with the coefficient sequence s(J) and counting down from k=J-1 down to some M