1	Edge Detection Today’s readings Cipolla and Gee supplemental: Forsyth, chapter 9 Watt, 10.3-10.4
2	Edge detection Convert a 2D image into a set of curves Extracts salient features of the scene More compact than pixels
3	Origin of Edges Edges are caused by a variety of factors
4	Edge detection How can you tell that a pixel is on an edge?
5	Image gradient The gradient of an image: The gradient points in the direction of most rapid change in intensity
6	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)
7	The Sobel operator Better approximations of the derivatives exist The Sobel operators below are very commonly used
8	Effects of noise Consider a single row or column of the image Plotting intensity as a function of position gives a signal
9	Solution: smooth first
10	Derivative theorem of convolution This saves us one operation:
11	Laplacian of Gaussian Look for zero-crossings of
12	2D edge detection filters is the Laplacian operator:
13	The Canny edge detector original image (Lena)
14	The Canny edge detector
15	The Canny edge detector
16	The Canny edge detector
17	Effect of Gaussian kernel width
18	Edge detection by subtraction
19	Edge detection by subtraction
20	Edge detection by subtraction
21	Gaussian - subtraction filter
22	An edge is not a line...
23	Finding lines in an image Option 1: Search for the object at every possible position in the image What is the cost of this operation? Option 2: Use a voting scheme: Hough transform
24	Finding lines in an image Connection between image (x,y) and Hough (m,b) spaces A line in the image corresponds to a point in Hough space To go from image space to Hough space: given a set of points (x,y), find all (m,b) such that y = mx + b
25	Finding lines in an image Connection between image (x,y) and Hough (m,b) spaces A line in the image corresponds to a point in Hough space To go from image space to Hough space: given a set of points (x,y), find all (m,b) such that y = mx + b What does a point (x₀, y₀) in the image space map to?
26	Hough transform algorithm Typically use a different parameterization d is the perpendicular distance from the line to the origin q is the angle this perpendicular makes with the x axis Why?
27	Hough transform algorithm Typically use a different parameterization d is the perpendicular distance from the line to the origin q is the angle this perpendicular makes with the x axis Why? Basic Hough transform algorithm Initialize H[d, q]=0 for each edge point I[x,y] in the image for q = 0 to 180 H[d, q] += 1 Find the value(s) of (d, q) where H[d, q] is maximum The detected line in the image is given by What’s the running time (measured in # votes)?
28	Extensions Extension 1: Use the image gradient same for each edge point I[x,y] in the image compute unique (d, q) based on image gradient at (x,y) H[d, q] += 1 same same What’s the running time measured in votes? Extension 2 give more votes for stronger edges Extension 3 change the sampling of (d, q) to give more/less resolution Extension 4 The same procedure can be used with circles, squares, or any other shape
29	Extensions Extension 1: Use the image gradient same for each edge point I[x,y] in the image compute unique (d, q) based on image gradient at (x,y) H[d, q] += 1 same same What’s the running time measured in votes? Extension 2 give more votes for stronger edges Extension 3 change the sampling of (d, q) to give more/less resolution Extension 4 The same procedure can be used with circles, squares, or any other shape
30	Summary Things to take away from this lecture What is an edge and where does it come from Edge detection by differentiation Image gradients continuous and discrete filters (e.g., Sobel operator) Effects of noise on gradients Derivative theorem of convolution Derivative of Gaussian (DoG) operator Laplacian operator Laplacian of Gaussian (LoG) Canny edge detector (basic idea) Effects of varying sigma parameter Approximating an LoG by subtraction Hough Transform