An edge is not a line... Edge Detection. Finding lines in an image. Finding lines in an image. How can we detect lines?

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1 Edge Detection An edge is not a line... original image Cann edge detector Compute image derivatives if gradient magnitude > τ and the value is a local maimum along gradient direction piel is an edge candidate gradient magnitude How can we detect lines? Finding lines in an image Finding lines in an image b Option 1: Search for the line at ever possible position/orientation What is the cost of this operation? image space b 0 m 0 m Hough space Option : Use a voting scheme: Hough transform Connection between image (,) 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 (,), find all (m,b) such that = m + b 1

2 Finding lines in an image b Hough transform algorithm Tpicall use a different parameterization 0 0 image space Connection between image (,) 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 (,), find all (m,b) such that = m + b What does a point ( 0, 0 ) in the image space map to? A: the solutions of b = - 0 m + 0 this is a line in Hough space m Hough space d is the perpendicular distance from the line to the origin θ is the angle this perpendicular makes with the ais Wh? dea keep an accumulator arra (Hough space) and let each edge piel contribute to it Line candidates are the maima in the accumulator arra Hough transform algorithm Tpicall use a different parameterization d is the perpendicular distance from the line to the origin θ is the angle this perpendicular makes with the ais Tpical Hough Transform Basic Hough transform algorithm nitialize H[d, θ]=0 For each edge point [,] in the image For θ = 0 to 180 H[d, θ] += 1 where point is now a sinusoid in Hough space Find the value(s) of (d, θ) where H[d, θ] is maimum The detected line in the image is given b What s the running CS68, Jana time Kosecka (measured in # t )?

3 Radon transform Etensions Etension 1: Use the image gradient 1. same. for each edge point [,] in the image compute unique (d, θ) based on image gradient at (,) H[d, θ] += 1 3. same 4. Same 1. Etension give more votes for stronger edges The same procedure can be used with circles, squares, or an other shape Hough Transform for Curves The H.T. can be generalized to detect an curve that can be epressed in parametric form: Y = f(, a1,a, ap) a1, a, ap are the parameters The parameter space is p-dimensional The accumulating arra is LARGE! H.T. Summar H.T. is a voting scheme points vote for a set of parameters describing a line or curve. The more votes for a particular set the more evidence that the corresponding curve is present in the image. Can detect MULTPLE curves in one shot. Computational cost increases with the number of parameters describing the curve. 3

4 Line fitting Line Fitting ρ θ second moment matri associated with each connected component v 1 - eigenvector of A Non-ma suppressed gradient magnitude Edge detection, non-maimum suppression (traditionall Hough Transform issues of resolution, threshold selection and search for peaks in Hough space) Connected components on edge piels with similar orientation - group piels with common orientation Line fitting lines determined from eigenvalues and eigenvectors of A Candidate line segments - associated line qualit Corner detection Corners contain more edges than lines. Finding Corners A point on a line is hard to match. ntuition: Right at corner, gradient is ill defined. Near corner, gradient has two different values. 4

5 We look at matri: Sum over a small region, the hpothetical corner Formula for Finding Corners C = Gradient with respect to, times gradient with respect to First, consider case where: C = λ1 = 0 0 λ This means all gradients in neighborhood are: (k,0) or (0, c) or (0, 0) (or off-diagonals cancel). What is region like if: 1. l1 = 0?. l = 0? 3. l1 = 0 and l = 0? 4. l1 > 0 and l > 0? Matri is smmetric General Case: So, to detect corners From Linear Algebra, it follows that because C is smmetric: λ C = R R 0 λ Filter image. Compute magnitude of the gradient everwhere. We construct C in a window. Use Linear Algebra to find λ1 and λ. f the are both big, we have a corner. With R a rotation matri. So ever case is like one on last slide. 5

6 Point Feature Etraction Harris Corner Detector - Eample Compute eigenvaluesof G f smalest eigenvalue σ of G is bigger than τ - mark piel as candidate feature point Alternativel feature qualit function (Harris Corner Detector) 6