IMAGE FUSION A PROJECT REPORT

					                                            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.