A PROJECT REPORT ON COMPARISON & IMPROVEMENT OF IMAGE FUSION USING WAVELETS ABSTRACT Title: Analysis of pixel level Multi sensor medical image fusion Aim: The goal of image fusion is to create new images that are more suitable for the purposes of human visual perception, object detection and target recognition. Description: The objective of the image fusion is to combine the source images of the same scene to form one composite image that contains a more accurate description of the scene than any individual source images. Image fusion methods can be broadly classified into two - spatial domain fusion and transform domain fusion. The fusion methods such as averaging, Brovey method, principal component analysis (PCA) and IHS based methods fall under spatial domain approaches. In this project, we propose a new multiresolution data fusion scheme based on the principal component analysis (PCA) transform and the pixel-level weights wavelet transform. In order to get a more ideal fusion result, a linear local mapping which based on the PCA is used to create a new "origin" image of the image fusion. Daubechies wavelet is chosen as the wavelet basis. Wavelet based fusion techniques have been reasonably effective in combining perceptually important image features. Shift invariance of the wavelet transform is important in ensuring robust sub band fusion. Applications of image fusion also includes Image Classification , Aerial and Satellite imaging, Medical imaging , Robot vision , Concealed weapon detection, Multi- focus image fusion, Digital camera application, Concealed weapon detection,Battle field monitoring. Experimental results confirm that the proposed algorithm is the best image sharpening method and can best maintain the spectral information of the original image. Also, the proposed technique performs better than the other ones, more robust and effective, from both subjective visual effects and objective statistical analysis results. The performance of the image fusion is evaluated by normalized least square error, entropy, overall cross entropy, standard deviation and mutual information. Reference: Multi-sensor image data fusion based on pixel-level weights of wavelet and the PCA transform (IEEE-2007). INTRODUCTION TO DIGITAL IMAGE PROCESSING Image: A digital image is a computer file that contains graphical information instead of text or a program. Pixels are the basic building blocks of all digital images. Pixels are small adjoining squares in a matrix across the length and width of your digital image. They are so small that you don‘t see the actual pixels when the image is on your computer monitor. Pixels are monochromatic. Each pixel is a single solid color that is blended from some combination of the 3 primary colors of Red, Green, and Blue. So, every pixel has a RED component, a GREEN component and BLUE component. The physical dimensions of a digital image are measured in pixels and commonly called pixel or image resolution. Pixels are scalable to different physical sizes on your computer monitor or on a photo print. However, all of the pixels in any particular digital image are the same size. Pixels as represented in a printed photo become round slightly overlapping dots. Pixel Values: As shown in this bitonal image, each pixel is assigned a tonal value, in this example 0 for black and 1 for white. PIXEL DIMENSIONS are the horizontal and vertical measurements of an image expressed in pixels. The pixel dimensions may be determined by multiplying both the width and the height by the dpi. A digital camera will also have pixel dimensions, expressed as the number of pixels horizontally and vertically that define its resolution (e.g., 2,048 by 3,072). Calculate the dpi achieved by dividing a document's dimension into the corresponding pixel dimension against which it is aligned. Example: Fig: An 8" x 10" document that is scanned at 300 dpi has the pixel dimensions of 2,400 pixels (8" x 300 dpi) by 3,000 pixels (10" x 300 dpi). Images in MATLAB: The basic data structure in MATLAB is the array, an ordered set of real or complex elements. This object is naturally suited to the representation of images, real- valued ordered sets of color or intensity data. MATLAB stores most images as two-dimensional arrays (i.e., matrices), in which each element of the matrix corresponds to a single pixel in the displayed image. (Pixel is derived from picture element and usually denotes a single dot on a computer display.) For example, an image composed of 200 rows and 300 columns of different colored dots would be stored in MATLAB as a 200-by-300 matrix. Some images, such as color images, require a three-dimensional array, where the first plane in the third dimension represents the red pixel intensities, the second plane represents the green pixel intensities, and the third plane represents the blue pixel intensities. This convention makes working with images in MATLAB similar to working with any other type of matrix data, and makes the full power of MATLAB available for image processing applications. IMAGE REPRESENTATION An image is stored as a matrix using standard Matlab matrix conventions. There are four basic types of images supported by Matlab: 1. Binary images 2. Intensity images 3. RGB images 4. Indexed images Binary Images: In a binary image, each pixel assumes one of only two discrete values: 1 or 0. A binary image is stored as a logical array. By convention, this documentation uses the variable name BW to refer to binary images. The following figure shows a binary image with a close-up view of some of the pixel values. Fig: Pixel Values in a Binary Image Grayscale Images: A grayscale image (also called gray-scale, gray scale, or gray-level) is a data matrix whose values represent intensities within some range. MATLAB stores a grayscale image as an individual matrix, with each element of the matrix corresponding to one image pixel. By convention, this documentation uses the variable name I to refer to grayscale images. The matrix can be of class uint8, uint16, int16, single, or double. While grayscale images are rarely saved with a color map, MATLAB uses a color map to display them. For a matrix of class single or double, using the default grayscale color map, the intensity 0 represents black and the intensity 1 represents white. For a matrix of type uint8, uint16, or int16, the intensity intmin (class (I)) represents black and the intensity intmax (class (I)) represents white. The figure below depicts a grayscale image of class double. Fig: Pixel Values in a Grayscale Image Define Gray Levels Color Images: A color image is an image in which each pixel is specified by three values — one each for the red, blue, and green components of the pixel's color. MATLAB store color images as an m-by-n-by-3 data array that defines red, green, and blue color components for each individual pixel. Color images do not use a color map. The color of each pixel is determined by the combination of the red, green, and blue intensities stored in each color plane at the pixel's location. Graphics file formats store color images as 24-bit images, where the red, green, and blue components are 8 bits each. This yields a potential of 16 million colors. The precision with which a real-life image can be replicated has led to the commonly used term color image. A color array can be of class uint8, uint16, single, or double. In a color array of class single or double, each color component is a value between 0 and 1. A pixel whose color components are (0, 0, 0) is displayed as black, and a pixel whose color components are (1, 1, 1) is displayed as white. The three color components for each pixel are stored along the third dimension of the data array. For example, the red, green, and blue color components of the pixel (10,5) are stored in RGB(10,5,1), RGB(10,5,2), and RGB(10,5,3), respectively. The following figure depicts a color image of class double. Fig: Color Planes of a True color Image Indexed Images: An indexed image consists of an array and a colormap matrix. The pixel values in the array are direct indices into a colormap. By convention, this documentation uses the variable name X to refer to the array and map to refer to the colormap. The colormap matrix is an m-by-3 array of class double containing floating-point values in the range [0, 1]. Each row of map specifies the red, green, and blue components of a single color. An indexed image uses direct mapping of pixel values to colormap values. The color of each image pixel is determined by using the corresponding value of X as an index into map. A colormap is often stored with an indexed image and is automatically loaded with the image when you use the imread function. After you read the image and the colormap into the MATLAB workspace as separate variables, you must keep track of the association between the image and colormap. However, you are not limited to using the default colormap--you can use any colormap that you choose. The relationship between the values in the image matrix and the colormap depends on the class of the image matrix. If the image matrix is of class single or double, it normally contains integer values 1 through p, where p is the length of the colormap. The value 1 points to the first row in the colormap, the value 2 points to the second row, and so on. If the image matrix is of class logical, uint8 or uint16, the value 0 points to the first row in the colormap, the value 1 points to the second row, and so on. The following figure illustrates the structure of an indexed image. In the figure, the image matrix is of class double, so the value 5 points to the fifth row of the colormap. Fig: Pixel Values Index to Colormap Entries in Indexed Images Digital Image File Types: The 5 most common digital image file types are as follows: 1. JPEG is a compressed file format that supports 24 bit color (millions of colors). This is the best format for photographs to be shown on the web or as email attachments. This is because the color informational bits in the computer file are compressed (reduced) and download times are minimized. 2. GIF is an uncompressed file format that supports only 256 distinct colors. Best used with web clip art and logo type images. GIF is not suitable for photographs because of its limited color support. 3. TIFF is an uncompressed file format with 24 or 48 bit color support. Uncompressed means that all of the color information from your scanner or digital camera for each individual pixel is preserved when you save as TIFF. TIFF is the best format for saving digital images that you will want to print. Tiff supports embedded file information, including exact color space, output profile information and EXIF data. There is a lossless compression for TIFF called LZW. LZW is much like 'zipping' the image file because there is no quality loss. An LZW TIFF decompresses (opens) with all of the original pixel information unaltered. 4. BMP is a Windows (only) operating system uncompressed file format that supports 24 bit color. BMP does not support embedded information like EXIF, calibrated color space and output profiles. Avoid using BMP for photographs because it produces approximately the same file sizes as TIFF without any of the advantages of TIFF. 5. Camera RAW is a lossless compressed file format that is proprietary for each digital camera manufacturer and model. A camera RAW file contains the 'raw' data from the camera's imaging sensor. Some image editing programs have their own version of RAW too. However, camera RAW is the most common type of RAW file. The advantage of camera RAW is that it contains the full range of color information from the sensor. This means the RAW file contains 12 to 14 bits of color information for each pixel. If you shoot JPEG, you only get 8 bits of color for each pixel. These extra color bits make shooting camera RAW much like shooting negative film. You have a little more latitude in setting your exposure and a slightly wider dynamic range. Image Coordinate Systems: Pixel Coordinates Generally, the most convenient method for expressing locations in an image is to use pixel coordinates. In this coordinate system, the image is treated as a grid of discrete elements, ordered from top to bottom and left to right, as illustrated by the following figure. Fig: The Pixel Coordinate System For pixel coordinates, the first component r (the row) increases downward, while the second component c (the column) increases to the right. Pixel coordinates are integer values and range between 1 and the length of the row or column. There is a one-to-one correspondence between pixel coordinates and the coordinates MATLAB uses for matrix subscripting. This correspondence makes the relationship between an image's data matrix and the way the image is displayed easy to understand. For example, the data for the pixel in the fifth row, second column is stored in the matrix element (5, 2). You use normal MATLAB matrix subscripting to access values of individual pixels. For example, the MATLAB code I (2, 15) Returns the value of the pixel at row 2, column 15 of the image I. Spatial Coordinates: In the pixel coordinate system, a pixel is treated as a discrete unit, uniquely identified by a single coordinate pair, such as (5, 2). From this perspective, a location such as (5.3, 2.2) is not meaningful. At times, however, it is useful to think of a pixel as a square patch. From this perspective, a location such as (5.3, 2.2) is meaningful, and is distinct from (5, 2). In this spatial coordinate system, locations in an image are positions on a plane, and they are described in terms of x and y (not r and c as in the pixel coordinate system). The following figure illustrates the spatial coordinate system used for images. Notice that y increases downward. Fig: The Spatial Coordinate System This spatial coordinate system corresponds closely to the pixel coordinate system in many ways. For example, the spatial coordinates of the center point of any pixel are identical to the pixel coordinates for that pixel. There are some important differences, however. In pixel coordinates, the upper left corner of an image is (1,1), while in spatial coordinates, this location by default is (0.5,0.5). This difference is due to the pixel coordinate system's being discrete, while the spatial coordinate system is continuous. Also, the upper left corner is always (1,1) in pixel coordinates, but you can specify a non default origin for the spatial coordinate system. Another potentially confusing difference is largely a matter of convention: the order of the horizontal and vertical components is reversed in the notation for these two systems. As mentioned earlier, pixel coordinates are expressed as (r, c), while spatial coordinates are expressed as (x, y). In the reference pages, when the syntax for a function uses r and c, it refers to the pixel coordinate system. When the syntax uses x and y, it refers to the spatial coordinate system. Digital image processing: Digital image processing is the use of computer algorithms to perform image processing on digital images. As a subfield of digital signal processing, digital image processing has many advantages over analog image processing; it allows a much wider range of algorithms to be applied to the input data, and can avoid problems such as the build-up of noise and signal distortion during processing. Image digitization: An image captured by a sensor is expressed as a continuous function f(x,y) of two co-ordinates in the plane. Image digitization means that the function f(x,y) is sampled into a matrix with M rows and N columns. The image quantization assigns to each continuous sample an integer value. The continuous range of the image function f(x,y) is split into K intervals. The finer the sampling (i.e., the larger M and N) and quantization (the larger K) the better the approximation of the continuous image function f(x,y). Image Pre-processing: Pre-processing is a common name for operations with images at the lowest level of abstraction -- both input and output are intensity images. These iconic images are of the same kind as the original data captured by the sensor, with an intensity image usually represented by a matrix of image function values (brightness). The aim of pre-processing is an improvement of the image data that suppresses unwanted distortions or enhances some image features important for further processing. Four categories of image pre- processing methods according to the size of the pixel neighborhood that is used for the calculation of new pixel brightness: o Pixel brightness transformations. o Geometric transformations. o Pre-processing methods that use a local neighborhood of the processed pixel. o Image restoration that requires knowledge about the entire image. Image Segmentation: Image segmentation is one of the most important steps leading to the analysis of processed image data. Its main goal is to divide an image into parts that have a strong correlation with objects or areas of the real world contained in the image.Two kinds of segmentation 1. Complete segmentation: This results in set of disjoint regions uniquely corresponding with objects in the input image. Cooperation with higher processing levels which use specific knowledge of the problem domain is necessary. 2. Partial segmentation: in which regions do not correspond directly with image objects. Image is divided into separate regions that are homogeneous with respect to a chosen property such as brightness, color, reflectivity, texture, etc. In a complex scene, a set of possibly overlapping homogeneous regions may result. The partially segmented image must then be subjected to further processing, and the final image segmentation may be found with the help of higher level information. Segmentation methods can be divided into three groups according to the dominant features they employ 1. First is global knowledge about an image or its part; the knowledge is usually represented by a histogram of image features. 2. Edge-based segmentations form the second group; and 3. Region-based segmentations Image enhancement The aim of image enhancement is to improve the interpretability or perception of information in images for human viewers, or to provide `better' input for other automated image processing techniques. Image enhancement techniques can be divided into two broad categories: 1. Spatial domain methods, which operate directly on pixels, and 2. Frequency domain methods, which operate on the Fourier transform of an image. Unfortunately, there is no general theory for determining what `good‘ image enhancement is when it comes to human perception. If it looks good, it is good! However, when image enhancement techniques are used as pre-processing tools for other image processing techniques, then quantitative measures can determine which techniques are most appropriate. IMAGE FUSION Introduction: In computer vision, Multisensor Image Fusion is the process of combining relevant information from two or more images into a single image. The resulting image will be more informative than any of the input images. In remote sensing applications, the increasing availability of space borne sensors gives a motivation for different image fusion algorithms. Several situations in image processing require high spatial and high spectral resolution in a single image. Most of the available equipment is not capable of providing such data convincingly. The image fusion techniques allow the integration of different information sources. The fused image can have complementary spatial and spectral resolution characteristics. But, the standard image fusion techniques can distort the spectral information of the multispectral data, while merging. In satellite imaging, two types of images are available. The panchromatic image acquired by satellites is transmitted with the maximum resolution available and the multispectral data are transmitted with coarser resolution. This will be usually, two or four times lower. At the receiver station, the panchromatic image is merged with the multispectral data to convey more information. Many methods exist to perform image fusion. The very basic one is the high pass filtering technique. Later techniques are based on DWT, uniform rational filter bank, and Laplacian pyramid. Multisensor data fusion has become a discipline to which more and more general formal solutions to a number of application cases are demanded. Several situations in image processing simultaneously require high spatial and high spectral information in a single image. This is important in remote sensing. However, the instruments are not capable of providing such information either by design or because of observational constraints. One possible solution for this is data fusion Standard Image Fusion Methods: Image fusion methods can be broadly classified into two - spatial domain fusion and transform domain fusion. The fusion methods such as averaging, Brovey method, principal component analysis (PCA) and IHS based methods fall under spatial domain approaches. Another important spatial domain fusion method is the high pass filtering based technique. Here the high frequency details are injected into upsampled version of MS images. The disadvantage of spatial domain approaches is that they produce spatial distortion in the fused image. Spectral distortion becomes a negative factor while we go for further processing, such as classification problem, of the fused image. The spatial distortion can be very well handled by transform domain approaches on image fusion. The multiresolution analysis has become a very useful tool for analyzing remote sensing images. The discrete wavelet transform has become a very useful tool for fusion. Some other fusion methods are also there, such as Laplacian pyramid based, Curvelet transform based etc. These methods show a better performance in spatial and spectral quality of the fused image compared to other spatial methods of fusion. Applications: 1. Image Classification 2. Aerial and Satellite imaging 3. Medical imaging 4. Robot vision 5. Concealed weapon detection 6. Multi-focus image fusion 7. Digital camera application 8. Concealed weapon detection 9. Battle field monitoring Satellite Image Fusion: Several methods are there for merging satellite images. In satellite imagery we can have two types of images Panchromatic images - An image collected in the broad visual wavelength range but rendered in black and white. Multispectral images - Images optically acquired in more than one spectral or wavelength interval. Each individual image is usually of the same physical area and scale but of a different spectral band. The SPOT PAN satellite provides high resolution (10m pixel) panchromatic data while the LANDSAT TM satellite provides low resolution (30m pixel) multispectral images. Image fusion attempts to merge these images and produce a single high resolution multispectral image. The standard merging methods of image fusion are based on Red-Green-Blue (RGB) to Intensity-Hue-Saturation (IHS) transformation. The usual steps involved in satellite image fusion are as follows: 1. Register the low resolution multispectral images to the same size as the panchromatic image 2. Transform the R,G and B bands of the multispectral image into IHS components 3. Modify the panchromatic image with respect to the multispectral image. This is usually performed by Histogram Matching of the panchromatic image with Intensity component of the multispectral images as reference 4. Replace the intensity component by the panchromatic image and perform inverse transformation to obtain a high resolution multispectral image. Medical Image Fusion: Image fusion has recently become a common term used within medical diagnostics and treatment. The term is used when patient images in different data formats are fused. These forms can include magnetic resonance image (MRI), computed tomography (CT), and positron emission tomography (PET). In radiology and radiation oncology, these images serve different purposes. For example, CT images are used more often to ascertain differences in tissue density while MRI images are typically used to diagnose brain tumors. For accurate diagnoses, radiologists must integrate information from multiple image formats. Fused, anatomically-consistent images are especially beneficial in diagnosing and treating cancer. Companies such as Keosys, MIMvista, IKOE, and BrainLAB have recently created image fusion software to use in conjunction with radiation treatment planning systems. With the advent of these new technologies, radiation oncologists can take full advantage of intensity modulated radiation therapy (IMRT). Being able to overlay diagnostic images onto radiation planning images results in more accurate IMRT target tumor volumes. FUSION ALGORITHMS: The details of wavelets and PCA algorithm and their use in image fusion along with simple average fusion algorithm are described in this section. Principal Component Analysis: The PCA involves a mathematical procedure that transforms a number of correlated variables into a number of uncorrelated variables called principal components. It computes a compact and optimal description of the data set. The first principal component accounts for as much of the variance in the data as possible and each succeeding component accounts for as much of the remaining variance as possible. First principal component is taken to be along the direction with the maximum variance. The second principal component is constrained to lie in the subspace perpendicular of the first. Within this subspace, this component points the direction of maximum variance. The third principal component is taken in the maximum variance direction in the subspace perpendicular to the first two and so on. The PCA is also called as Karhunen-Loève transform or the Hotelling transform. The PCA does not have a fixed set of basis vectors like FFT, DCT and wavelet etc. and its basis vectors depend on the data set. Let X be a d-dimensional random vector and assume it to have zero empirical mean. Orthonormal projection matrix V would be such that Y=VTX with the following constraints. The covariance of Y, i.e., cov(Y) is a diagonal and inverse of V is equivalent to its transpose ( V-1 = VT). Using matrix algebra …………… (1) Multiplying both sides of above eqn by V, one gets One could write V as V= [V1, V2… , Vd] and cov(Y) as …………… (2) Substituting Eqn (1) into the Eqn (2) gives [λ1V1, λ2V2, …. , λdVd] = [cov(X) V1, cov(X) V2, … , cov(X) Vd] ………. (3) This could be rewritten as λiVi = cov(X) V1 ……… (4) where i=1,2,...,d and Vi is an eigenvector of cov(X ) . PCA Algorithm: Let the source images (images to be fused) be arranged in two-column vectors. The steps followed to project this data into 2-D subspaces are: 1. Organize the data into column vectors. The resulting matrix Z is of dimension 2 x n. 2. Compute the empirical mean along each column. The empirical mean vector M has a dimension of 1 x 2. 3. Subtract the empirical mean vector M from each column of the data matrix Z. The resulting matrix X is of dimension 2 x n. 4. Find the covariance matrix C of X i.e. =XXT mean of expectation = cov(X) 5. Compute the eigenvectors V and eigenvalue D of C and sort them by decreasing eigenvalue. Both V and D are of dimension 2 x 2. 6. Consider the first column of V which corresponds to larger eigenvalue to compute P1 and P2 as And Image Fusion by PCA: The information flow diagram of PCA-based image fusion algorithm is shown in figure below. The input images (images to be fused) I1 (x, y) and I2 (x, y) are arranged in two column vectors and their empirical means are subtracted. The resulting vector has a dimension of n x 2, where n is length of the each image vector. Compute the eigenvector and eigenvalues for this resulting vector are computed and the eigenvectors corresponding to the larger eigenvalue obtained. The normalized components P1 and P2 (i.e., P1 + P2 = 1) using equation (3) are computed from the obtained eigenvector. The fused image is: Figure: Information flow diagram in image fusion scheme employing PCA. Image Fusion by Simple Average: This technique is a basic and straightforward technique and fusion could be achieved by simple averaging corresponding pixels in each input image as: Image Fusion by Wavelet Transforms: Fourier analysis: Signal analysts already have at their disposal an impressive arsenal of tools. Perhaps the most well-known of these is Fourier analysis, which breaks down a signal into constituent sinusoids of different frequencies. Another way to think of Fourier analysis is as a mathematical technique for transforming our view of the signal from time-based to frequency-based. Figure 2 For many signals, Fourier analysis is extremely useful because the signal‘s frequency content is of great importance. So why do we need other techniques, like wavelet analysis? Fourier analysis has a serious drawback. In transforming to the frequency domain, time information is lost. When looking at a Fourier transform of a signal, it is impossible to tell when a particular event took place. If the signal properties do not change much over time — that is, if it is what is called a stationary signal—this drawback isn‘t very important. However, most interesting signals contain numerous non stationary or transitory characteristics: drift, trends, abrupt changes, and beginnings and ends of events. These characteristics are often the most important part of the signal, and Fourier analysis is not suited to detecting them. Short-Time Fourier Analysis In an effort to correct this deficiency, Dennis Gabor (1946) adapted the Fourier transform to analyze only a small section of the signal at a time—a technique called windowing the signal.Gabor‘s adaptation, called the Short-Time FourierTransform (STFT), maps a signal into a two-dimensional function of time and frequency. Figure 3 The STFT represents a sort of compromise between the time- and frequency-based views of a signal. It provides some information about both when and at what frequencies a signal event occurs. However, you can only obtain this information with limited precision, and that precision is determined by the size of the window. While the STFT compromise between time and frequency information can be useful, the drawback is that once you choose a particular size for the time window, that window is the same for all frequencies. Many signals require a more flexible approach—one where we can vary the window size to determine more accurately either time or frequency. Wavelet Analysis Wavelet analysis represents the next logical step: a windowing technique with variable-sized regions. Wavelet analysis allows the use of long time intervals where we want more precise low-frequency information, and shorter regions where we want high- frequency information. Figure 4 Here‘s what this looks like in contrast with the time-based, frequency-based, and STFT views of a signal: Figure 5 You may have noticed that wavelet analysis does not use a time-frequency region, but rather a time-scale region. For more information about the concept of scale and the link between scale and frequency, see ―How to Connect Scale to Frequency?‖ What Can Wavelet Analysis Do? One major advantage afforded by wavelets is the ability to perform local analysis, that is, to analyze a localized area of a larger signal. Consider a sinusoidal signal with a small discontinuity — one so tiny as to be barely visible. Such a signal easily could be generated in the real world, perhaps by a power fluctuation or a noisy switch. Figure 6 A plot of the Fourier coefficients (as provided by the fft command) of this signal shows nothing particularly interesting: a flat spectrum with two peaks representing a single frequency. However, a plot of wavelet coefficients clearly shows the exact location in time of the discontinuity. Figure 7 Wavelet analysis is capable of revealing aspects of data that other signal analysis techniques miss, aspects like trends, breakdown points, discontinuities in higher derivatives, and self-similarity. Furthermore, because it affords a different view of data than those presented by traditional techniques, wavelet analysis can often compress or de-noise a signal without appreciable degradation. Indeed, in their brief history within the signal processing field, wavelets have already proven themselves to be an indispensable addition to the analyst‘s collection of tools and continue to enjoy a burgeoning popularity today. What Is Wavelet Analysis? Now that we know some situations when wavelet analysis is useful, it is worthwhile asking ―What is wavelet analysis?‖ and even more fundamentally, ―What is a wavelet?‖ A wavelet is a waveform of effectively limited duration that has an average value of zero. Compare wavelets with sine waves, which are the basis of Fourier analysis. Sinusoids do not have limited duration — they extend from minus to plus infinity. And where sinusoids are smooth and predictable, wavelets tend to be irregular and asymmetric. Figure 8 Fourier analysis consists of breaking up a signal into sine waves of various frequencies. Similarly, wavelet analysis is the breaking up of a signal into shifted and scaled versions of the original (or mother) wavelet. Just looking at pictures of wavelets and sine waves, you can see intuitively that signals with sharp changes might be better analyzed with an irregular wavelet than with a smooth sinusoid, just as some foods are better handled with a fork than a spoon. It also makes sense that local features can be described better with wavelets that have local extent. The Continuous Wavelet Transform: 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 F(w), which when multiplied by a sinusoid of frequency w yields the constituent sinusoidal components of the original signal. Graphically, the process looks like: Figure 9 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 result of the CWT is a series 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: Figure 10 Scaling We‘ve already alluded to the fact that wavelet analysis produces a time-scale view of a signal and now we‘re talking about scaling and shifting wavelets. What exactly do we mean by scale in this context? Scaling a wavelet simply means stretching (or compressing) it. To go beyond colloquial descriptions such as ―stretching,‖ we introduce the scale factor, often denoted by the letter a. If we‘re talking about sinusoids, for example the effect of the scale factor is very easy to see: Figure 11 The scale factor works exactly the same with wavelets. The smaller the scale factor, the more ―compressed‖ the wavelet. Figure 12 It is clear from the diagrams that for a sinusoid sin (wt) the scale factor ‗a‘ is related (inversely) to the radian frequency ‗w‘. Similarly, with wavelet analysis the scale is related to the frequency of the signal. Shifting Shifting a wavelet simply means delaying (or hastening) its onset. Mathematically, delaying a function (t) by k is represented by (t-k) Figure 13 Five Easy Steps to a Continuous Wavelet Transform: The continuous wavelet transform is the sum over all time of the signal multiplied by scaled, shifted versions of the wavelet. This process produces wavelet coefficients that are a function of scale and position. It‘s really a very simple process. In fact, here are the five steps of an easy recipe for creating a CWT: 1. Take a wavelet and compare it to a section at the start of the original signal. 2. Calculate a number C that represents how closely correlated the wavelet is with this section of the signal. The higher C is, the more the similarity. More precisely, if the signal energy and the wavelet energy are equal to one, C may be interpreted as a correlation coefficient. Note that the results will depend on the shape of the wavelet you choose. Figure 14 3. Shift the wavelet to the right and repeat steps 1 and 2 until you‘ve covered the whole signal. Figure 15 4. Scale (stretch) the wavelet and repeat steps 1 through 3. Figure 16 5. Repeat steps 1 through 4 for all scales. When you‘re done, you‘ll have the coefficients produced at different scales by different sections of the signal. The coefficients constitute the results of a regression of the original signal performed on the wavelets. How to make sense of all these coefficients? You could make a plot on which the x- axis represents position along the signal (time), the y-axis represents scale, and the color at each x-y point represents the magnitude of the wavelet coefficient C. These are the coefficient plots generated by the graphical tools. Figure 17 These coefficient plots resemble a bumpy surface viewed from above. If you could look at the same surface from the side, you might see something like this: Figure 18 The continuous wavelet transform coefficient plots are precisely the time-scale view of the signal we referred to earlier. It is a different view of signal data than the time- frequency Fourier view, but it is not unrelated. Scale and Frequency: Notice that the scales in the coefficients plot (shown as y-axis labels) run from 1 to 31. Recall that the higher scales correspond to the most ―stretched‖ wavelets. The more stretched the wavelet, the longer the portion of the signal with which it is being compared, and thus the coarser the signal features being measured by the wavelet coefficients. Figure 19 Thus, there is a correspondence between wavelet scales and frequency as revealed by wavelet analysis: • Low scale a=> Compressed wavelet => Rapidly changing details => High frequency ‗w‘. • High scale a=>Stretched wavelet=>Slowly changing, coarse features=>Low frequency ‗w‘. The Scale of Nature: It‘s important to understand the fact that wavelet analysis does not produce a time- frequency view of a signal is not a weakness, but a strength of the technique. Not only is time-scale a different way to view data, it is a very natural way to view data deriving from a great number of natural phenomena. Consider a lunar landscape, whose ragged surface (simulated below) is a result of centuries of bombardment by meteorites whose sizes range from gigantic boulders to dust specks. If we think of this surface in cross-section as a one-dimensional signal, then it is reasonable to think of the signal as having components of different scales—large features carved by the impacts of large meteorites, and finer features abraded by small meteorites. Figure 20 Here is a case where thinking in terms of scale makes much more sense than thinking in terms of frequency. Inspection of the CWT coefficients plot for this signal reveals patterns among scales and shows the signal‘s possibly fractal nature. Figure 21 Even though this signal is artificial, many natural phenomena — from the intricate branching of blood vessels and trees, to the jagged surfaces of mountains and fractured metals — lend themselves to an analysis of scale. The Discrete Wavelet Transform: 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. Let‘s examine this in more depth. One-Stage Filtering: Approximations and Details: For many signals, the low-frequency content is the most important part. It is what gives the signal its identity. The high-frequency content on the other hand imparts flavor or nuance. Consider the human voice. If you remove the high-frequency components, the voice sounds different but you can still tell what‘s being said. However, if you remove enough of the low-frequency components, you hear gibberish. In wavelet analysis, we often speak of approximations and details. The approximations are the high-scale, low- frequency components of the signal. The details are the low-scale, high-frequency components. The filtering process at its most basic level looks like this: Figure 23 The original signal S passes through two complementary filters and emerges as two signals. Unfortunately, if we actually perform this operation on a real digital signal, we wind up with twice as much data as we started with. Suppose, for instance that the original signal S consists of 1000 samples of data. Then the resulting signals will each have 1000 samples, for a total of 2000. These signals A and D are interesting, but we get 2000 values instead of the 1000 we had. There exists a more subtle way to perform the decomposition using wavelets. By looking carefully at the computation, we may keep only one point out of two in each of the two 2000-length samples to get the complete information. This is the notion of own sampling. We produce two sequences called cA and cD. Figure 24 The process on the right which includes down sampling produces DWT Coefficients. To gain a better appreciation of this process let‘s perform a one-stage discrete wavelet transform of a signal. Our signal will be a pure sinusoid with high- frequency noise added to it. Here is our schematic diagram with real signals inserted into it: Figure 25 The MATLAB code needed to generate s, cD, and cA is: s = sin(20*linspace(0,pi,1000)) + 0.5*rand(1,1000); [cA,cD] = dwt(s,'db2'); where db2 is the name of the wavelet we want to use for the analysis. Notice that the detail coefficients cD is small and consist mainly of a high-frequency noise, while the approximation coefficients cA contains much less noise than does the original signal. [length(cA) length(cD)] ans = 501 501 You may observe that the actual lengths of the detail and approximation coefficient vectors are slightly more than half the length of the original signal. This has to do with the filtering process, which is implemented by convolving the signal with a filter. The convolution ―smears‖ the signal, introducing several extra samples into the result. Multiple-Level Decomposition: The decomposition process can be iterated, with successive approximations being decomposed in turn, so that one signal is broken down into many lower resolution components. This is called the wavelet decomposition tree. Figure 26 Looking at a signal‘s wavelet decomposition tree can yield valuable information. Figure 27 Number of Levels: Since the analysis process is iterative, in theory it can be continued indefinitely. In reality, the decomposition can proceed only until the individual details consist of a single sample or pixel. In practice, you‘ll select a suitable number of levels based on the nature of the signal, or on a suitable criterion such as entropy. Wavelet Reconstruction: We‘ve learned how the discrete wavelet transform can be used to analyze or decompose, signals and images. This process is called decomposition or analysis. The other half of the story is how those components can be assembled back into the original signal without loss of information. This process is called reconstruction, or synthesis. The mathematical manipulation that effects synthesis is called the inverse discrete wavelet transforms (IDWT). To synthesize a signal in the Wavelet Toolbox, we reconstruct it from the wavelet coefficients: Figure 28 Where wavelet analysis involves filtering and down sampling, the wavelet reconstruction process consists of up sampling and filtering. Up sampling is the process of lengthening a signal component by inserting zeros between samples: Figure 29 The Wavelet Toolbox includes commands like idwt and waverec that perform single-level or multilevel reconstruction respectively on the components of one- dimensional signals. These commands have their two-dimensional analogs, idwt2 and waverec2. Reconstruction Filters: The filtering part of the reconstruction process also bears some discussion, because it is the choice of filters that is crucial in achieving perfect reconstruction of the original signal. The down sampling of the signal components performed during the decomposition phase introduces a distortion called aliasing. It turns out that by carefully choosing filters for the decomposition and reconstruction phases that are closely related (but not identical), we can ―cancel out‖ the effects of aliasing. The low- and high pass decomposition filters (L and H), together with their associated reconstruction filters (L' and H'), form a system of what is called quadrature mirror filters: Figure 30 Reconstructing Approximations and Details: We have seen that it is possible to reconstruct our original signal from the coefficients of the approximations and details. Figure 31 It is also possible to reconstruct the approximations and details themselves from their coefficient vectors. As an example, let‘s consider how we would reconstruct the first-level approximation A1 from the coefficient vector cA1. We pass the coefficient vector cA1 through the same process we used to reconstruct the original signal. However, instead of combining it with the level-one detail cD1, we feed in a vector of zeros in place of the detail coefficients vector: Figure 32 The process yields a reconstructed approximation A1, which has the same length as the original signal S and which is a real approximation of it. Similarly, we can reconstruct the first-level detail D1, using the analogous process: Figure 33 The reconstructed details and approximations are true constituents of the original signal. In fact, we find when we combine them that: A1 + D1 = S Note that the coefficient vectors cA1 and cD1—because they were produced by Down sampling and are only half the length of the original signal — cannot directly be combined to reproduce the signal. It is necessary to reconstruct the approximations and details before combining them. Extending this technique to the components of a multilevel analysis, we find that similar relationships hold for all the reconstructed signal constituents. That is, there are several ways to reassemble the original signal: Figure 34 Relationship of Filters to Wavelet Shapes: In the section ―Reconstruction Filters‖, we spoke of the importance of choosing the right filters. In fact, the choice of filters not only determines whether perfect reconstruction is possible, it also determines the shape of the wavelet we use to perform the analysis. To construct a wavelet of some practical utility, you seldom start by drawing a waveform. Instead, it usually makes more sense to design the appropriate quadrature mirror filters, and then use them to create the waveform. Let‘s see how this is done by focusing on an example. Consider the low pass reconstruction filter (L') for the db2 wavelet. Wavelet function position Figure 35 The filter coefficients can be obtained from the dbaux command: Lprime = dbaux(2) Lprime = 0.3415 0.5915 0.1585 –0.0915 If we reverse the order of this vector (see wrev), and then multiply every even sample by –1, we obtain the high pass filter H': Hprime = –0.0915 –0.1585 0.5915 –0.3415 Next, up sample Hprime by two (see dyadup), inserting zeros in alternate positions: HU =–0.0915 0 –0.1585 0 0.5915 0 –0.3415 0 Finally, convolve the up sampled vector with the original low pass filter: H2 = conv(HU,Lprime); plot(H2) Figure 36 If we iterate this process several more times, repeatedly up sampling and convolving the resultant vector with the four-element filter vector Lprime, a pattern begins to emerge: Figure 37 The curve begins to look progressively more like the db2 wavelet. This means that the wavelet‘s shape is determined entirely by the coefficients of the reconstruction filters. This relationship has profound implications. It means that you cannot choose just any shape, call it a wavelet, and perform an analysis. At least, you can‘t choose an arbitrary wavelet waveform if you want to be able to reconstruct the original signal accurately. You are compelled to choose a shape determined by quadrature mirror decomposition filters. The Scaling Function: We‘ve seen the interrelation of wavelets and quadrature mirror filters. The wavelet function is determined by the high pass filter, which also produces the details of the wavelet decomposition. There is an additional function associated with some, but not all wavelets. This is the so-called scaling function . The scaling function is very similar to the wavelet function. It is determined by the low pass quadrature mirror filters, and thus is associated with the approximations of the wavelet decomposition. In the same way that iteratively up- sampling and convolving the high pass filter produces a shape approximating the wavelet function, iteratively up-sampling and convolving the low pass filter produces a shape approximating the scaling function. Multi-step Decomposition and Reconstruction: A multi step analysis-synthesis process can be represented as: Figure 38 This process involves two aspects: breaking up a signal to obtain the wavelet coefficients, and reassembling the signal from the coefficients. We‘ve already discussed decomposition and reconstruction at some length. Of course, there is no point breaking up a signal merely to have the satisfaction of immediately reconstructing it. We may modify the wavelet coefficients before performing the reconstruction step. We perform wavelet analysis because the coefficients thus obtained have many known uses, de- noising and compression being foremost among them. But wavelet analysis is still a new and emerging field. No doubt, many uncharted uses of the wavelet coefficients lie in wait. The Wavelet Toolbox can be a means of exploring possible uses and hitherto unknown applications of wavelet analysis. Explore the toolbox functions and see what you discover. WAVELET DECOMPOSITION: Images are treated as two dimensional signals, they change horizontally and vertically, thus 2D wavelet analysis must be used for images. 2D wavelet analysis uses the same ‘mother wavelets‘ but requires an extra step at every level of decomposition. The 1D analysis filtered out the high frequency information from the low frequency information at every level of decomposition; so only two sub signals were produced at each level. In 2D, the images are considered to be matrices with N rows and M columns. At every level of decomposition the horizontal data is filtered, then the approximation and details produced from this are filtered on columns. Fig 1: Decomposition of an Image At every level, four sub-images are obtained; the approximation, the vertical detail, the horizontal detail and the diagonal detail. Below the Saturn image has been decomposed to one level. The wavelet analysis has found how the image changes vertically, horizontally and diagonally. Fig 2:2-D Decomposition of Saturn Image to level 1 To get the next level of decomposition the approximation sub-image is decomposed, this idea can be seen in figure 3. Fig 3: Saturn Image decomposed to Level 3. Only the 9 detail sub-images and the final sub-image is required to reconstruct the image perfectly. When compressing with orthogonal wavelets the energy retained is: The number of zeros in percentage is defined by: Figure: Information flow diagram in image fusion scheme employing multi-scale decomposition. The information flow diagram of wavelet- based image fusion algorithm is shown in above figure. In wavelet image fusion scheme, the source images I1 (x,y) and I2 (x,y), are decomposed into approximation and detailed coefficients at required level using DWT. The approximation and detailed coefficients of both images are combined using fusion rule Φ. The fused image (If (x, y)) could be obtained by taking the inverse discrete wavelet transform (IDWT) as: If (x, y) = IDWT [Φ{ DWT (I1 (x,y)), DWT (I2 (x,y))}] ………. (5) The fusion rule used in this project is simply averages the approximation coefficients and picks the detailed coefficient in each sub band with the largest magnitude. Entropy: Entropy of grayscale image E = entropy (I) Entropy is a statistical measure of randomness that can be used to characterize the texture of the input image. Entropy is defined as -sum (p.*log2 (p)) where p contains the histogram counts returned from imhist. By default, entropy uses two bins for logical arrays and 256 bins for uint8, uint16, or double arrays. I can be a multidimensional image. If I have more than two dimensions, the entropy function treats it as a multidimensional grayscale image and not as an RGB image. Image can be logical, uint8, uint16, or double and must be real, nonempty, and nonsparse. E is double. Entropy converts any class other than logical to uint8 for the histogram count calculation so that the pixel values are discrete and directly correspond to a bin value. INTRODUCTION TO MATLAB What Is MATLAB? MATLAB® is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation. Typical uses include 1. Math and computation 2. Algorithm development 3. Data acquisition 4. Modeling, simulation, and prototyping 5. Data analysis, exploration, and visualization 6. Scientific and engineering graphics 7. Application development, including graphical user interface building. MATLAB is an interactive system whose basic data element is an array that does not require dimensioning. This allows you to solve many technical computing problems, especially those with matrix and vector formulations, in a fraction of the time it would take to write a program in a scalar non interactive language such as C or FORTRAN. The name MATLAB stands for matrix laboratory. MATLAB was originally written to provide easy access to matrix software developed by the LINPACK and EISPACK projects. Today, MATLAB engines incorporate the LAPACK and BLAS libraries, embedding the state of the art in software for matrix computation. MATLAB has evolved over a period of years with input from many users. In university environments, it is the standard instructional tool for introductory and advanced courses in mathematics, engineering, and science. In industry, MATLAB is the tool of choice for high-productivity research, development, and analysis. MATLAB features a family of add-on application-specific solutions called toolboxes. Very important to most users of MATLAB, toolboxes allow you to learn and apply specialized technology. Toolboxes are comprehensive collections of MATLAB functions (M-files) that extend the MATLAB environment to solve particular classes of problems. Areas in which toolboxes are available include signal processing, control systems, neural networks, fuzzy logic, wavelets, simulation, and many others. The MATLAB System: The MATLAB system consists of five main parts: Development Environment: This is the set of tools and facilities that help you use MATLAB functions and files. Many of these tools are graphical user interfaces. It includes the MATLAB desktop and Command Window, a command history, an editor and debugger, and browsers for viewing help, the workspace, files, and the search path. The MATLAB Mathematical Function: This is a vast collection of computational algorithms ranging from elementary functions like sum, sine, cosine, and complex arithmetic, to more sophisticated functions like matrix inverse, matrix eigen values, Bessel functions, and fast Fourier transforms. The MATLAB Language: This is a high-level matrix/array language with control flow statements, functions, data structures, input/output, and object-oriented programming features. It allows both "programming in the small" to rapidly create quick and dirty throw-away programs, and "programming in the large" to create complete large and complex application programs. Graphics: MATLAB has extensive facilities for displaying vectors and matrices as graphs, as well as annotating and printing these graphs. It includes high-level functions for two- dimensional and three-dimensional data visualization, image processing, animation, and presentation graphics. It also includes low-level functions that allow you to fully customize the appearance of graphics as well as to build complete graphical user interfaces on your MATLAB applications. The MATLAB Application Program Interface (API): This is a library that allows you to write C and Fortran programs that interact with MATLAB. It includes facilities for calling routines from MATLAB (dynamic linking), calling MATLAB as a computational engine, and for reading and writing MAT-files. MATLAB WORKING ENVIRONMENT: MATLAB DESKTOP:- Matlab Desktop is the main Matlab application window. The desktop contains five sub windows, the command window, the workspace browser, the current directory window, the command history window, and one or more figure windows, which are shown only when the user displays a graphic. The command window is where the user types MATLAB commands and expressions at the prompt (>>) and where the output of those commands is displayed. MATLAB defines the workspace as the set of variables that the user creates in a work session. The workspace browser shows these variables and some information about them. Double clicking on a variable in the workspace browser launches the Array Editor, which can be used to obtain information and income instances edit certain properties of the variable. The current Directory tab above the workspace tab shows the contents of the current directory, whose path is shown in the current directory window. For example, in the windows operating system the path might be as follows: C:\MATLAB\Work, indicating that directory ―work‖ is a subdirectory of the main directory ―MATLAB‖; WHICH IS INSTALLED IN DRIVE C. clicking on the arrow in the current directory window shows a list of recently used paths. Clicking on the button to the right of the window allows the user to change the current directory. MATLAB uses a search path to find M-files and other MATLAB related files, which are organize in directories in the computer file system. Any file run in MATLAB must reside in the current directory or in a directory that is on search path. By default, the files supplied with MATLAB and math works toolboxes are included in the search path. The easiest way to see which directories are on the search path. The easiest way to see which directories are soon the search path, or to add or modify a search path, is to select set path from the File menu the desktop, and then use the set path dialog box. It is good practice to add any commonly used directories to the search path to avoid repeatedly having the change the current directory. The Command History Window contains a record of the commands a user has entered in the command window, including both current and previous MATLAB sessions. Previously entered MATLAB commands can be selected and re-executed from the command history window by right clicking on a command or sequence of commands. This action launches a menu from which to select various options in addition to executing the commands. This is useful to select various options in addition to executing the commands. This is a useful feature when experimenting with various commands in a work session. Using the MATLAB Editor to create M-Files: The MATLAB editor is both a text editor specialized for creating M-files and a graphical MATLAB debugger. The editor can appear in a window by itself, or it can be a sub window in the desktop. M-files are denoted by the extension .m, as in pixelup.m. The MATLAB editor window has numerous pull-down menus for tasks such as saving, viewing, and debugging files. Because it performs some simple checks and also uses color to differentiate between various elements of code, this text editor is recommended as the tool of choice for writing and editing M-functions. To open the editor , type edit at the prompt opens the M-file filename.m in an editor window, ready for editing. As noted earlier, the file must be in the current directory, or in a directory in the search path. Getting Help: The principal way to get help online is to use the MATLAB help browser, opened as a separate window either by clicking on the question mark symbol (?) on the desktop toolbar, or by typing help browser at the prompt in the command window. The help Browser is a web browser integrated into the MATLAB desktop that displays a Hypertext Markup Language(HTML) documents. The Help Browser consists of two panes, the help navigator pane, used to find information, and the display pane, used to view the information. Self-explanatory tabs other than navigator pane are used to perform a search. Fig: Image 1 Fig: Image 2 Figure: Fused image by simple average. Figure: Fused image by PCA Figure: Fused image by Wavelets CONCLUSIONS Pixel-level image fusion using wavelet transform and principal component analysis are implemented in PC MATLAB. Different image fusion performance metrics with and without reference image have been evaluated. The simple averaging fusion algorithm shows degraded performance. Image fusion using wavelets with higher level of decomposition shows better performance in some metrics while in other metrics, the PCA shows better performance. Some further investigation is needed to resolve this issue. REFERENCES 1. Gonzalo, Pajares & Jesus Manuel, de la Cruz. A wavelet-based image fusion tutorial. Pattern Recognition, 2004, 37, 1855-872. 2. Varsheny, P.K. Multi-sensor data fusion. Elec. Comm. Engg., 1997, 9(12), 245-53. 3. Mallet, S.G. A heory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Mach. Intel., 1989, 11(7), 674-93. 4. Wang, H.; Peng, J. & Wu, W. Fusion algorithm for multisensor image based on discrete multiwavelet transform. IEE Proc. Visual Image Signal Process., 2002, 149(5). 5. Mitra Jalili-Moghaddam. Real-time multi-focus image fusion using discrete wavelet transform and Laplasican pyramid transform. Chalmess University of Technology, Goteborg, Sweden, 2005. Masters thesis. 6. Daubechies, I. Ten lectures on wavelets. In Regular Conference Series in Applied Maths, Vol. 91, 1992, SIAM, Philadelphia. 7. h t t p : / / e n . w i k i p e d i a . o r g / w i k i / Principal_components_analysis. 8. Naidu, V.P.S.; Girija, G. & Raol, J.R. Evaluation of data association and fusion algorithms for tracking in the presence of measurement loss. In AIAA Conference on Navigation, Guidance and Control, Austin, USA, August 2003, pp. 11-14. 9. Arce, Gonzalo R. Nonlinear signal processing A statistical approach. Wiley- Interscience Inc. Publication, USA, 2005.

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