 Assistant Professor, Electronics and Communication Engineering, Walchen Institute of Technology, Solaabur. Today, we will discuss image enhancement using frequency domain. Learning outcome, at the end of this session, students will be able to apply image enhancement algorithms in frequency domain. In continuation with the previous class, we will discuss Butterworth low pass filter and Gaussian low pass filter. So, let us see the Butterworth low pass filter. The transfer function of a Butterworth low pass filter of order n with cutoff frequency at a distance d 0 from the origin is defined here. If this is a transfer function, that means this is a transfer function and it is represented using three different ways. So, this is an image representation of the transfer function. This is a 3D representation of a transfer function, same transfer function and this is a coordinate representation of the same transfer function. So, just these are three representations of this transfer function and here we see the equation of Butterworth low pass filter is 1 divided by 1 plus d of u v, where this is the distance of a pixel from the center that is 0 0 and this is what the d 0, which is the radius of this circle, that means this is if I change this radius, then it will affect the performance of a Butterworth low pass filter. Also we can change the value of n, that is order of filter in this case. So, we have two parameters to vary in this Butterworth low pass filter. One is d 0, that is radius of a circle and the order of a filter and its 3D representation if you look at this is as compared to earlier ideal low pass filter, we see that this function smoothly in non-linear way it increasing and it forms shape of a bell. So, this is a origin, u is the one of the variable in frequency domain, v is another variable and this is a h of u v that is filter function. This is also filter function, but it is represented as a image and this is a another representation of a filter function in coordinate system. So, here on x axis we have d of u v and on y axis we have h of u v and we see that there are different curves we are getting with radius d 0 having value n equal to 1, 2, 3, 4. So, if I change the order of a function, the behavior of the filter changes even not only with order we can change this radius also to change its behavior. Let us see the effect of this Butterworth low pass filter on a image. For example, there are six figures here, first one this is a original image, then this is a second image with result of filtering with Butterworth filter of order 2 and cutoff radius 5. So, order is 2 and radius is 5. Here we are in this example we are keeping order fixed of 2 and we are changing the radius. So, we can have two parameters to change here order and radius. So, this is with order 2 and radius 15, this is with order 2, radius 30, this is radius 80, this is radius 230. If you look at the results of this Butterworth low pass filter, we see that with very small radius we get a blurred image and as we go on increasing the radius we see that the image gets more enhanced and enhanced. Then the next frequency domain filter that is called Gaussian low pass filter we are using and the transfer function of a Gaussian low pass filter is given as h of u v is equal to e raised to minus d square u v divided by 2 d 0 square where d 0 is the radius of a circle here if you look at this image and this d of u v is the distance of a particular pixel from the origin and we see the 3D representation which is somewhat similar to the Butterworth low pass filter but it is the top response of the Butterworth filter is different than this Gaussian filter here we see a notch here. So, and here we see the filter function this is not completely white but it is having different intensities and see the effect of changing the radius. Here in this Gaussian low pass filter we can change only the radius we do not have the power as we did with the Butterworth low pass filter. So, here if I change the value of radius from 10 to 20 to 40 we see that the response of a filter also changes and it affects the output of a Gaussian low pass filter. Let us see the effect of Gaussian low pass filter with these 3 images. The first one is a result of filtering with ideal low pass filter that means here this original image is not shown here but this is a result of processing original image with ideal low pass filter with radius 15 and the same thing that means is processed with Butterworth filter of order 2 and radius 15 and the same image is also filtered using Gaussian filter of radius 15 and we see that this out of these 3 filters that is ideal low pass filter, Butterworth low pass filter and Gaussian low pass filter somewhat we see that the performance of ideal low pass filter is not that much good as compared to this Butterworth and Gaussian. So this way because this is a non-linear function Butterworth and Gaussian are these non-linear function whereas this ideal low pass filter transfer function is a linear one. So let us have a question what will be the transfer function of Gaussian high pass filter? You pause the video and answer the question. So naturally the high pass filters high pass filters either it is a Gaussian or Butterworth or ideal we have the if you subtract the transfer function of the ideal low pass filter or Gaussian low pass filter from 1 we get the high pass filter. So this is H of UV is the Gaussian high pass filter that is 1 minus e raise to this function is already we have discussed which is a transfer function of Gaussian low pass filter. So subtract Gaussian low pass filter from 1 we get a Gaussian high pass filter and as usual this is what the representation of this Gaussian high pass filter with 3 representations image 3D and coordinate system same thing. And we see the result of this Gaussian high pass filter if you look at this is the original image and this is the result of processing input image with Gaussian high pass filter of radius 15 this is with radius 30 and this is with radius 80. So again you look at with radius 15 we do not see any enhancement but with radius 30 we see that there is a more enhancement as compared to this radius 80 also. So this is more appropriate what we can say radius for getting enhanced image. Then we can similarly we can have a Butterworth high pass filter. So this is what the Butterworth high pass filter transfer function and it is a representation in 3 different ways and we will see the result of this Butterworth high pass filter we can see here this is the original image this is a Butterworth high pass filter with radius 15 this is with 30 and this is with 80. And again we see that with radius 30 is more appropriate for Butterworth filter as well. References digital image processing by Raphael C. Gonzales and Titchery Woods by Tata McRoyle education thank you very much.