Review: Linear Systems We define a system as a unit that converts an input function into an output function. Independent variable System operator Bahadir K. Gunturk

Linear Systems Let where fi(x) is an arbitrary input in the class of all inputs {f(x)}, and gi(x) is the corresponding output. If Then the system H is called a linear system. A linear system has the properties of additivity and homogeneity. Bahadir K. Gunturk

Linear Systems The system H is called shift invariant if for all fi(x) {f(x)} and for all x0. This means that offsetting the independent variable of the input by x0 causes the same offset in the independent variable of the output. Hence, the input-output relationship remains the same. Bahadir K. Gunturk

Linear Systems The operator H is said to be causal, and hence the system described by H is a causal system, if there is no output before there is an input. In other words, A linear system H is said to be stable if its response to any bounded input is bounded. That is, if where K and c are constants. Bahadir K. Gunturk

Linear Systems (x-a) (a) A unit impulse function, denoted (a), is defined by the expression (x-a) (a) a x Bahadir K. Gunturk

Linear Systems A unit impulse function, denoted (a), is defined by the expression Then Bahadir K. Gunturk

Linear Systems The term is called the impulse response of H. From the previous slide It states that, if the response of H to a unit impulse [i.e., h(x, )], is known, then response to any input f can be computed using the preceding integral. In other words, the response of a linear system is characterized completely by its impulse response. Bahadir K. Gunturk

Linear Systems If H is a shift-invariant system, then and the integral becomes This expression is called the convolution integral. It states that the response of a linear, fixed-parameter system is completely characterized by the convolution of the input with the system impulse response. Bahadir K. Gunturk

Linear Systems Convolution of two functions is defined as In the discrete case Bahadir K. Gunturk

Linear Systems In the 2D discrete case is a linear filter. Bahadir K. Gunturk

Convolution Example 1 -1 2 2 3 1 1 -1 2 h f Rotate From C. Rasmussen, U. of Delaware Bahadir K. Gunturk

Convolution Example Step 1 1 1 1 2 2 2 3 -1 2 1 2 1 3 3 -1 -1 1 2 2 1 h 1 3 2 2 1 1 1 -1 3 2 1 4 2 5 -1 -2 1 f f*h From C. Rasmussen, U. of Delaware Bahadir K. Gunturk

Convolution Example Step 2 1 1 1 2 2 2 3 -1 2 1 2 1 3 3 -1 -1 1 2 2 1 h 1 3 2 2 1 1 1 3 2 1 -2 4 2 5 4 -2 -1 3 f f*h From C. Rasmussen, U. of Delaware Bahadir K. Gunturk

Convolution Example Step 3 1 1 1 2 2 2 3 -1 2 1 2 1 3 3 -1 -1 1 2 2 1 h 1 3 2 2 1 1 1 3 2 1 -2 4 3 4 5 -1 -3 3 f f*h From C. Rasmussen, U. of Delaware Bahadir K. Gunturk

Convolution Example Step 4 1 1 1 2 2 2 3 -1 2 1 2 1 3 3 -1 -1 1 2 2 1 h 1 3 2 2 1 1 1 3 2 1 -2 6 1 4 -2 5 -3 -3 1 f f*h From C. Rasmussen, U. of Delaware Bahadir K. Gunturk

Convolution Example Step 5 1 1 1 2 2 2 3 -1 2 1 2 1 3 3 -1 -1 1 2 2 1 h 1 3 2 2 1 3 2 1 2 2 4 9 -2 5 -1 4 1 -1 -2 2 f f*h From C. Rasmussen, U. of Delaware Bahadir K. Gunturk

Convolution Example Step 6 1 1 1 2 2 2 3 -1 2 1 2 1 3 3 -1 -1 1 2 2 1 h 1 3 2 2 2 3 2 1 2 2 6 4 9 -2 5 -2 2 3 -2 -2 1 f f*h From C. Rasmussen, U. of Delaware Bahadir K. Gunturk

and so on… Convolution Example From C. Rasmussen, U. of Delaware Bahadir K. Gunturk

Example = * Bahadir K. Gunturk

Example = * Bahadir K. Gunturk

MATLAB Review your matrix-vector knowledge Matlab help files are helpful to learn it Exercise: f = [1 2; 3 4] g = [1; 1] g = [1 1] g’ z = f * g’ n=0:10 plot(sin(n)); plot(n,sin(n)); title(‘Sinusoid’); xlabel(‘n’); ylabel(‘Sin(n)’); n=0:0.1:10 plot(n,sin(n)); grid; figure; subplot(2,1,1); plot(n,sin(n)); subplot(2,1,2); plot(n,cos(n)); Bahadir K. Gunturk

MATLAB Some more built-ins a = zeros(3,2) b = ones(2,4) c = rand(3,3) %Uniform distribution help rand help randn %Normal distribution d1 = inv(c) d2 = inv(rand(3,3)) d3 = d1+d2 d4 = d1-d2 d5 = d1*d2 d6 = d1.*d3 e = d6(:) Bahadir K. Gunturk

MATLAB Image processing in Matlab x=imread(‘cameraman.tif’); figure; imshow(x); [h,w]=size(x); y=x(0:h/2,0:w/2); imwrite(y,’man.tif’); % To look for a keyword lookfor resize Bahadir K. Gunturk

MATLAB M-file Save the following as myresize1.m function [y]=myresize1(x) % This function downsamples an image by two [h,w]=size(x); for i=1:h/2, for j=1:w/2, y(i,j) = x(2*i,2*j); end Compare with myresize2.m function [y]=myresize2(x) for i=0:h/2-1, for j=0:w/2-1, y(i+1,j+1) = x(2*i+1,2*j+1); Compare with myresize3.m function [y]=myresize3(x) % This function downsamples an image by two y = x(1:2:end,1:2:end); We can add inputs/outputs function [y,height,width]=myresize4(x,factor) % Inputs: % x is the input image % factor is the downsampling factor % Outputs: % y is the output image % height and width are the size of the output image y = x(1:factor:end,1:factor:end); [height,width] = size(y); Bahadir K. Gunturk

Try MATLAB f=imread(‘saturn.tif’); figure; imshow(f); [height,width]=size(f); f2=f(1:height/2,1:width/2); figure; imshow(f2); [height2,width2=size(f2); f3=double(f2)+30*rand(height2,width2); figure;imshow(uint8(f3)); h=[1 1 1 1; 1 1 1 1; 1 1 1 1; 1 1 1 1]/16; g=conv2(f3,h); figure;imshow(uint8(g)); Bahadir K. Gunturk

EE 7730 Edge Detection

Detection of Discontinuities Matched Filter Example >> a=[0 0 0 0 1 2 3 0 0 0 0 2 2 2 0 0 0 0 1 2 -2 -1 0 0 0 0]; >> figure; plot(a); >> h1 = [-1 -2 2 1]/10; >> b1 = conv(a,h1); figure; plot(b1); Bahadir K. Gunturk

Detection of Discontinuities Point Detection Example: Apply a high-pass filter. A point is detected if the response is larger than a positive threshold. The idea is that the gray level of an isolated point will be quite different from the gray level of its neighbors. Threshold Bahadir K. Gunturk

Detection of Discontinuities Point Detection Detected point Bahadir K. Gunturk

Detection of Discontinuities Line Detection Example: Bahadir K. Gunturk

Detection of Discontinuities Line Detection Example: Bahadir K. Gunturk

Detection of Discontinuities Edge Detection: An edge is the boundary between two regions with relatively distinct gray levels. Edge detection is by far the most common approach for detecting meaningful discontinuities in gray level. The reason is that isolated points and thin lines are not frequent occurrences in most practical applications. The idea underlying most edge detection techniques is the computation of a local derivative operator. Bahadir K. Gunturk

Origin of Edges Edges are caused by a variety of factors surface normal discontinuity depth discontinuity surface color discontinuity illumination discontinuity Edges are caused by a variety of factors Bahadir K. Gunturk

Profiles of image intensity edges Bahadir K. Gunturk

Image gradient The gradient of an image: The gradient points in the direction of most rapid change in intensity The gradient direction is given by: The edge strength is given by the gradient magnitude Bahadir K. Gunturk

Edge Detection The gradient vector of an image f(x,y) at location (x,y) is the vector The magnitude and direction of the gradient vector are is also used in edge detection in addition to the magnitude of the gradient vector. Bahadir K. Gunturk

The discrete gradient How can we differentiate a digital image f[x,y]? Option 1: reconstruct a continuous image, then take gradient Option 2: take discrete derivative (finite difference) Bahadir K. Gunturk

Effects of noise Consider a single row or column of the image Plotting intensity as a function of position gives a signal Bahadir K. Gunturk

Solution: smooth first Bahadir K. Gunturk Look for peaks in

Derivative theorem of convolution This saves us one operation: Bahadir K. Gunturk

Laplacian of Gaussian Consider Laplacian of Gaussian operator Bahadir K. Gunturk Zero-crossings of bottom graph

2D edge detection filters Laplacian of Gaussian Gaussian derivative of Gaussian is the Laplacian operator: Bahadir K. Gunturk

The Canny edge detector original image (Lena) Bahadir K. Gunturk

The Canny edge detector norm of the gradient Bahadir K. Gunturk

The Canny edge detector thresholding Bahadir K. Gunturk

The Canny edge detector thinning (non-maximum suppression) Bahadir K. Gunturk

Non-maximum suppression Check if pixel is local maximum along gradient direction requires checking interpolated pixels p and r Bahadir K. Gunturk

Predicting the next edge point Assume the marked point is an edge point. Then we construct the tangent to the edge curve (which is normal to the gradient at that point) and use this to predict the next points (here either r or s). Bahadir K. Gunturk

Non-maximum suppression Bahadir K. Gunturk

Hysteresis The threshold used to find starting point may be large in following the edge. This leads to broken edge curves. The trick is to use two thresholds: A large one when starting an edge chain, a small one while following it. Bahadir K. Gunturk

Edge detection by subtraction original Bahadir K. Gunturk

Edge detection by subtraction smoothed (5x5 Gaussian) Bahadir K. Gunturk

Edge detection by subtraction Why does this work? smoothed – original (scaled by 4, offset +128) Bahadir K. Gunturk filter demo

Gaussian - image filter delta function Bahadir K. Gunturk Laplacian of Gaussian

Edge Detection Bahadir K. Gunturk

Edge Detection Bahadir K. Gunturk

Edge Detection Bahadir K. Gunturk

Edge Detection Bahadir K. Gunturk

Edge Detection Bahadir K. Gunturk

Edge Detection The Laplacian of an image f(x,y) is a second-order derivative defined as Bahadir K. Gunturk

Edge Detection Bahadir K. Gunturk

Corners contain more edges than lines. A point on a line is hard to match. Bahadir K. Gunturk

Corners contain more edges than lines. A corner is easier Bahadir K. Gunturk

Corner Detector Locate points where intensity is varying in two directions. Bahadir K. Gunturk

Reading Chris Harris and Mike Stephens. A combined corner and edge detector. In M. M. Matthews, editor, Proceedings of the 4th ALVEY vision conference, pages 147--151, University of Manchester, England, September 1988. Bahadir K. Gunturk