Singularity Detection

Task1: Singularity Detection in Stair Case Signal

clcclear all;close all;t=-10:.1:10;% generating f(t)x=@(t) 1*(t>=0 & t<2) + 2*(t>=2 & t<4)+3*(t>=4 & t<=10);figure(1)subplot(3,3,1)plot(t,x(t));xlim([0 10]);title('Stair Case signal')xlabel('Time(sec)')ylabel('Amplitude') % Generating si(t) % phi(t)= gaussian theta= 1/sqrt(2*pi*.1)*exp(-(t.^2)/.2);figure(2)subplot(3,1,1)plot(t,theta);title('Gaussian Pulse')xlabel('Time(sec)')ylabel('Amplitude') psi1=-1.*diff(theta)/.1;t1=-10:.1:(10-.1);subplot(3,1,2)plot(t1,psi1);title('First Derivative')xlabel('Time(sec)')ylabel('Amplitude') psi2=diff(theta,2)/.1;t3=-10:.1:(10-.2);subplot(3,1,3)plot(t3,psi2)title('Second Derivative')xlabel('Time(sec)')

ylabel('Amplitude') y1=conv(x(t),psi1);t2=-20:.1:(20-.1);figure(1)subplot(3,1,2)plot(t2,y1)title('Output showing point of singularities')%xlim([0 10]);xlabel('Time(sec)')ylabel('Amplitude') y2=conv(x(t),psi2);t3=-20:.1:(20-.2);subplot(3,1,3)plot(t3,y2)title('Output showing point of singularities')%xlim([0 10]);xlabel('Time(sec)')ylabel('Amplitude')

Fig: Input Stair Case Signal And Output showing Singularities

Fig: Gaussian Pulse and its first and second derivatives

Task2: Singularity Detection in Sine wave

clc;clear all;close all;ts=.0001; t=-5:ts:5; x=sin(4.*t);% x=sin(4*2*pi*.t);figure(1)subplot(3,1,1)plot(t,x)title('Sine Wave')xlabel('Time(sec)')ylabel('Amplitude') % generating psitheta= (1/sqrt(2*pi*.001))*exp(-(t.^2)/.000002);subplot(3,1,2)plot(t,theta)title('Gaussian Pulse')xlabel('Time(sec)')ylabel('Amplitude')

psi1=-1.*diff(theta)/ts;y1=conv(x,psi1);t1= -10:ts:(10-ts);subplot(3,1,3)plot(t1,y1)title('Output showing point of singularities')xlabel('Time(sec)')ylabel('Amplitude')

Fig: Input Sine Wave And Output showing Singularities

Task3: Singularity Detection in Lenna Image

clc;clear all;close all;i=imread('lenna.tif');figure()imshow(i);title('Original Image')

% generating psit=linspace(-5,5,256);theta= 1/sqrt(2*pi*.0001)*exp(-(t.^2)/.0002);psi1=-1.*diff(theta)/(t(2)-t(1));% horizontal singularity detectionX=zeros(256,510); for j=1:256 X(j,:) = conv(i(j,:),psi1);end figure()imshow(uint8(X))title('Image showing Horizontal Singularities for First Order derivative') % Vertical singularity detectionX1=zeros(510,256); for j=1:256 X1(:,j) = conv(i(:,j),psi1);end figure()imshow(uint8(X1))title('Image showing Vertical Singularities for First Order derivative')%n=2psi2=diff(theta,2)/(t(2)-t(1));% horizontal singularity detectionX2=zeros(256,509); for j=1:256 X2(j,:) = conv(i(j,:),psi2);end figure()imshow(uint8(X2))title('Image showing Horizontal Singularities for Second Order derivative') % Vertical singularity detectionX3=zeros(509,256); for j=1:256

X3(:,j) = conv(i(:,j),psi2);end figure()imshow(uint8(X3))title('Image showing Vertical Singularities for Second Order derivative')

Task4: Singularity Detection in Lenna Image with high Variance

clc;clear all;close all;i=imread('lenna.tif');figure()imshow(i);title('Original Image') % generating psit=linspace(-5,5,256);theta= 1/sqrt(2*pi*.01)*exp(-(t.^2)/.2);psi1=-1.*diff(theta)/(t(2)-t(1));% horizontal singularity detectionX=zeros(256,510); for j=1:256 X(j,:) = conv(i(j,:),psi1);end figure()

imshow(uint8(X))title('Image showing Horizontal Singularities for First Order derivative') % Vertical singularity detectionX1=zeros(510,256); for j=1:256 X1(:,j) = conv(i(:,j),psi1);end figure()imshow(uint8(X1))title('Image showing Vertical Singularities for First Order derivative')%n=2psi2=diff(theta,2)/(t(2)-t(1));% horizontal singularity detectionX2=zeros(256,509); for j=1:256 X2(j,:) = conv(i(j,:),psi2);end figure()imshow(uint8(X2))title('Image showing Horizontal Singularities for Second Order derivative') % Vertical singularity detectionX3=zeros(509,256); for j=1:256 X3(:,j) = conv(i(:,j),psi2);end figure()imshow(uint8(X3))title('Image showing Vertical Singularities for Second Order derivative')