 Welcome to today's lecture. So, let us have a quick look at the topics which we have been trying to learn as part of the third module. So, till now throughout this module, we have been understanding about image classification as in what is an image classification, what are the different types of techniques that are used for classifying images. Remember classification as such, it consists of two steps, one is recognizing or identifying real world objects and by real world objects, remember I am referring to urban area or it can be water body or it can be vegetation, built up area and so on. Recognizing or identifying real world objects that is the first step and the second step is labeling of the pixels to be classified. Now this is where we discussed about different algorithms of supervised and unsupervised classification with which we could label the pixels, is not it? So, now we are in the fifth lecture of the third module and today we shall be introducing the fuzzy classification. We have learnt about supervised classification, unsupervised classification and the method of assessing accuracy of classification using the error metrics or confusion metrics, the different metrics that can be extracted from a confusion metrics like the producer's accuracy or user's accuracy, overall accuracy, kappa statistics and so on, is not it? Now with all that background in mind, let us try to understand what is fuzzy classification, okay? Now in the screen in front of you, you see two images of Mumbai, okay? Towards your left side, the image is a Google Earth representation and towards the right side, you see a false color composite created using synthetic aperture radar image. So, the reason I am showing you this image is to highlight that till now, our entire discussion in image classification was focused towards something known as a hard classification, okay? Hard classification. Remember, wherever we discussed maximum likelihood classification or minimum distance to mean classification or parallel piped classification, so in all these algorithms, each pixel is definitely allocated to one of the n classes, okay? Say in this image that you see in front of you assume there are five classes. Every pixel is going to be allocated compulsorily to one of these five classes irrespective of what is the maximum reflectance that is coming out from the pixel. For example, with the maximum likelihood classification that is one of the most widely used hard classifiers in remote sensing, that is the maximum likelihood classification, each pixel is going to be allocated to the class to which it has the highest posterior probability of membership, okay? Let me reiterate and we are talking about maximum likelihood classification wherein each pixel is taken, the probability of that pixel belonging to each class is calculated and the pixel is finally assigned to the class to which it has the highest probability of being. It falls under the hard classification technique, is not it? Because all these hard classification techniques, they rely on probabilistic approach, probabilistic approach. So the standard assumptions followed in the algorithms that we have seen so far are purity of every pixel and the discrete mutually exclusive nature of each class. Now what do you mean by purity of a pixel or discrete mutually exclusive nature of each class? Let us try to understand this through this slide. You know, if you look at real world landscapes, they do not follow whatever is shown at the top, you know? There is a gradual transition of one land cover type to the other and I am using the example of coastal areas where water and land gradually blend into one another before you see land, okay? There is water body and a region where water and land are slowly trying to blend together and then there is land area, is not it? So real world landscapes, they I am going to say that they are going to transition gradually to one another and for this reason we will have pixels in a satellite image that contain proportion of materials rather than just one single class because you know the landscapes themselves are heterogeneous, okay? And limitation in spatial resolution of remote sensing images because of that it is possible that satellite images will contain something known as a mixed cells. I am going to call them as mixed pixels wherein one pixel can be representative of multiple land cover classes like the example of a coastal region wherein if you recollect the image of Mumbai region whenever we have pixels in the coastal region it may represent more than one land cover class as in a small part of it can be water, another part can be urban area and small part can be land. So, you know mixed pixels multiple classes may exist in the same pixels and the presence of these mixed pixels has been recognized as a major problem, serious problem because it tends to affect the effective utilization of satellite imagery in per pixel classifications, okay? So shown in front of you at the top is the representation for a conventional per pixel classification, okay? Where two classes are shown water and wetland and if I try to classify an image that consists of these two classes using any of the hard classification algorithms, every pixel irrespective of whether it belongs to the coastal region or not it is going to be compulsorily being classified to either water pixel or it is going to be a wetland pixel, there is nothing in between, you know? The concept of a pixel consisting of more than one classes does not exist in hard classification approaches. Now look at the diagram, second diagram that is below, this is the classical way in which fuzzy classification algorithms perform, okay? So, there is something written known as membership function in the y-axis, is it? We will define membership function but to understand assume that every pixel will be assigned a membership that shows you how much of what class is contained in that pixel. For example, assume the pixels in the coastal region, okay? Which may have a little bit of water, a little bit of wetland, so using membership function we get to classify each pixel as, for example, we can classify it by saying that 70 percentage of that one pixel consists of water and rest 30 percentage consists of wetland, okay? You are partitioning between a pixel, you are partitioning within a pixel based on its membership to other classes. In hard classification there is only 0 percent and 100 percent as shown here, isn't it? Either a true or a false, it is either completely a water pixel or either completely a wetland pixel whereas the concept of fuzzy classification allows us to define every pixel in terms of its membership to all the other classes. As I mentioned, we can use membership functions to define the percentage of pixel that belongs to each of these two classes. I am using the example of two because in the example shown here two classes are just mentioned. Okay? Now with this logic, let us try to understand more about fuzzy classification. So, it was in 1965 that the fuzzy logic was first initiated by Zaday. It is a multi-valued logic which allows intermediate values to be defined between conventional evaluations like true, false, yes, no, high, low, etc. Okay? So, fuzzy logic and probability both are, you know, two different ways, completely different ways of expressing uncertainty. In the case of fuzzy image processing, we use fuzzy set theory, okay? Whereas in probability theory, it uses the concept of subjective probability. I won't get into that now. But for our discussion in today's lecture, we are going to focus on fuzzy image processing, fuzzy set theory and the classification algorithms that are part of fuzzy set theory, okay? Now, there are three main stages in fuzzy image processing, okay? So, what I have done is I will try to explain the stages using a small flow chart. Okay? So, firstly, of course, we have to use expert knowledge to fuzzify an image, okay? The main power of fuzzy image processing lies in creating and modifying membership functions, creating and modifying membership values. Because once you transform an image into a membership plane, note that I am using the word membership plane. Once you transform an image into a membership plane, appropriate fuzzy techniques tend to modify the membership values. And this can be achieved by using, you know, fuzzy clustering techniques or fuzzy integration approaches. And, you know, understanding of fuzzy image processing as such, it is much easier if we try to study about fuzzy set theory, okay? Because, you know, fuzzy image processing, it is not a unique theory in itself, but it is a collection of different fuzzy approaches to image processing. So, here I have written that the aim of any classification algorithms that is based on fuzzy set theory is to transform an image into the membership plane, okay? If required, you can perform image defuzzification to see the result. I will be explaining these through a case study. But for now, let us try to understand the underlying logic of fuzzy set theory, okay? So, as I mentioned earlier, fuzzy set theory allows an event to belong to more than one sample space, where sharp boundaries between spaces are hardly found. If X be a collection of objects with elements noted as small x, then a fuzzy set A in X is a set of ordered pairs given by the relationship shown. Here, mu A X is nothing but the characteristic function or membership grade which can take any values between 0 to 1. Thus, nearer the values of membership function to unity, higher the grade of membership of X in A, okay? So, membership functions, they are going to have values between 0 to 1. You know, in fuzzy set theory, the operations on fuzzy sets are based on the original work of Zeta. So, I am going to write it down so that if you are interested, I would urge you to read this reference. So, whatever is being discussed here, please do not consider them as a complete collection. I am just showing you a glimpse of the operations that can be performed using fuzzy set theory. For example, we can perform fuzzy union, the union of two fuzzy sets A and B with respective membership function mu A X and mu B X is nothing but a fuzzy set C which can be written as A union B. So, the membership value will be the smallest fuzzy set containing both A and B. Let me give it some time to sink in, okay? We are trying to understand the operations in fuzzy set theory starting with fuzzy union, okay? All right? Moving on, we can also perform fuzzy intersection. The intersection of two fuzzy sets A and B with respective membership functions mu A X and mu B X is a fuzzy set C which can be written as A intersection B whose membership function is related to those of A and B by this relationship, okay? We are talking about operations in fuzzy set theory and we discussed fuzzy intersection. So, fuzzy union is possible, fuzzy intersection is possible and fuzzy complement is also possible, complement of a set, complement of a fuzzy set A in this case, okay? Here you can see not A and A. So, with the operation of union intersection with this it is easy for us to extend many of the basic identities which hold for ordinary sets to fuzzy sets, okay? Now, there are other ways of forming combinations of fuzzy sets and relating them to one another such as we can conduct algebraic sum of A and B which can be written as mu A plus B which is mu A plus mu B. We can perform algebraic product of A and B in terms of its membership function, okay? We can even carry out absolute differences of A and B. There are many operations in fuzzy set theory of which a few were discussed now, okay? Moving on, in fuzzy image processing at the end of the day membership function is the underlying power of every fuzzy model as it is capable of modeling the gradual transition from a less distinct region to another in a subtle way and membership functions they characterize the fuzziness in a fuzzy set whether the elements in the set are discrete or continuous in a graphical form for eventual use in the mathematical formalisms of fuzzy set theory. And you know there are infinite number of ways to graphically depict the membership function. The membership grades you know they themselves they can either be chosen heuristically or subjectively. The direct use of available shapes for membership function is also found effective for image enhancement. For example, you know we can have a trapezoidal membership function or triangular membership function. So, when you are trying to work with fuzzy image processing, you can either use the existing membership functions or you can create a membership function from the data themselves. So, the choice of membership function it is going to be problem dependent which requires expert knowledge and in situations where say you do not have any prior information about data variation. So, in such cases membership values can be generated from the data themselves using clustering algorithms which is what is normally done in research which is a normal practice. So, now with this background let me show you a case study where we performed fuzzy image classification using synthetic aperture radar images and specifically we were trying to use the fuzzy Gaussian maximum likelihood algorithm. We will also see what is the difference between fuzzy Gaussian maximum likelihood algorithm and Gaussian maximum likelihood algorithm. So, it is between fuzzy classification and hard classification. We will see what is the difference between both. So, for this study, the study region used is depicted here and shown are the images which were used for fuzzy classification. So, what you see here are the same study regions which were shown as part of the previous lecture. So, this study site it is located in parts of Guntur and Krishna districts of Andhra Pradesh. So, towards the left side you see synthetic aperture radar imagery after it has been subjected to pre-processing. A shows the HH polarized band, H stands for horizontal and V stands for vertical. So, B shows the HV polarized band. By now I am assuming that you are familiar with what is meant by polarization and what is meant by HH, HV and so on. And towards your right side you see an image obtained in the optical region of the electromagnetic spectrum in the visible region of the electromagnetic spectrum. So, here the land cover classes are specifically highlighted as paddy, water, sand, cotton, pallow land. So, as part of this study what we did is we tried to create a false color composite using synthetic aperture radar imagery that is HH and HV bands. And then once there was a false color composite created we tried to implement the fuzzy Gaussian maximum likelihood algorithm. Now, this algorithm it is an extension of the conventional hard per pixel Gaussian maximum likelihood algorithm. And the fuzzy representation of geographical information as I mentioned earlier it enables a new method for partitioning the spectral space. Now, in this approach the informational classes as well as the spectral classes are represented as fuzzy sets. The whole concept is that if we have a spectral space it is not partitioned by sharp boundaries. So, the gradual transition of one land cover to another is realistically represented by using fuzzy set theory by using fuzzy image processing. So, that is the whole concept but nevertheless in fuzzy Gaussian maximum likelihood algorithm we are performing a fuzzy partition of the spectral space. What you see in front of you is the fuzzy partition matrix and it is a family of fuzzy sets like f1, f2 and so on up to fm where they represent the spectral classes. And see it is the number of predefined classes in this case it is 5, 5 classes and x is nothing but a pixel measurement vector and mu f1 you know all these are membership grades, membership functions. So, this is how a fuzzy partition matrix is recorded. So, our aim is to convert an image into the fuzzy partition matrix. We need to convert an image into a new plane the membership plane. Now, assume you have five land use land cover classes. So, when you generate the fuzzy partition matrix you are going to get images that represent only one class at a time. I will show you the results but for now let us go through the steps that lead us to the results. So, come see when we are discussing these steps I am assuming that you recollect what we discussed when we were trying to understand the Gaussian maximum likelihood algorithm. So, the first step is same for both hard classification and fuzzy classification that is we need to select the training data first step is same. Second step here we are carrying out the hard Gaussian maximum likelihood classification to find the partition matrix. And from third step please note the difference. In the third step what we are going to do is we are going to determine the fuzzy mean and fuzzy covariance fuzzy mean and fuzzy covariance where instead of mean of a cluster I am going to use mu c membership in the class C cluster C mu c. Look at the difference in mean covariance and fuzzy mean and fuzzy covariance. As the fourth step we recalculate the probability values finally we determine the membership function. Now when you look at the fuzzy mean and fuzzy covariance you know these can be considered as extensions of the conventional mean and covariance matrix. Here n is the number of pixels and the step when I say determination of membership function I am actually calculating a posteriori probability of a pixel belonging to class C. Now the disadvantage here is that we are assuming normality of data. We are assuming that data are normally distributed and compared with the conventional method the fuzzy Gaussian maximum likelihood it improves remote sensing image classification in the aspects of representation of geographic information more realistically partitioning of spectral space and estimation of classification parameters. So all methods have their own pros and cons. What we see here is the fuzzy version of Gaussian maximum likelihood algorithm wherein each time we calculate the a posteriori probability of a pixel belonging to a class as given here. So what we did is we tried to perform this algorithm on the false color composite created by using synthetic aperture radar images and these are the outputs that you see in front of you output of fuzzy Gaussian maximum likelihood classification. We call it as fraction images wherein a represents phalloland, b represents cotton, c represents sand, d represents paddy and finally e represents water. Now we have to assess the accuracy isn't it? Just because we get an output we cannot be sure that it is absolutely correct. And as we discussed in one of the earlier lectures there are many ways to assess the output of a classification, accuracy of classification. It can be done visually, statistically as well by using a confusion matrix. Now typically you know whenever classification is performed using a course resolution image the output of that is compared with another satellite image which is having a very fine resolution, very fine spatial resolution. Let me be more specific isn't it? But in this case I have fraction images one for each class and to compare or to assess the classification accuracy I need another set of fraction images isn't it? So that is what you see here that is the fuzzy reference images created using the image captured in the visible region of the electromagnetic spectrum which is having better or finer spatial resolution and spectral separability. Note the words that I am using. So we call it as soft reference images showing phalloland, cotton, sand, paddy and water. For example in the fraction image that shows water only those pixels that have water will be represented. Even if there are mixels which has a small percentage of water that also will be represented but in colors between 0 to 1. Which means the images that you see in front of you are called as fraction images and the images that you obtained after fuzzy Gaussian maximum likelihood classification they need to be compared with the soft reference images. Visual comparison you have already done. Now statistically instead of creating an error matrix for fuzzy image processing there is something known as a fuzzy error matrix. I won't get into the details of that but just so that you know fuzzy error matrix can be used to tabulate and estimate the classification accuracy obtained in fuzzy image processing. All right. So in this particular lecture we were trying to understand about fuzzy set theory and how it is used to create membership functions and how we can classify images using fuzzy image processing. And we also saw a case study example wherein fuzzy Gaussian maximum likelihood algorithm was used to classify a false color composite of synthetic aperture radar images. And what you see are the fraction images that are obtained which can be used for assessing the classification accuracy using fuzzy error matrix. So let me hope that you understood the concepts which were discussed in today's lecture and I will be meeting you in the next class. Thank you.