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Digital Image Processing Hough Transform for Object Recognition Hough Transform • Elegant method for direct object recognition • Edges need not be connected • Complete object need not be visible • Key Idea: Edges VOTE for the possible model Image and Parameter Spaces y mx c y Equation of Line: y mx c Find: (m, c) Consider point: ( xi , yi ) ( xi , yi ) x yi mx i c or c xi m yi Image Space m Parameter space also called Hough Space (m, c) c Parameter Space Line Detection by Hough Transform y Algorithm: • Quantize Parameter Space (m, c) (m, c) • Create Accumulator Array A(m, c) x • Set A(m, c) 0 m, c Parameter Space A(m, c) • For each image edge ( xi , yi ) increment: 1 1 1 1 A(m, c) A(m, c) 1 1 1 2 • If (m, c) lies on the line: 1 1 1 1 c xi m yi 1 1 • Find local maxima in A(m, c) Better Parameterization NOTE: m y Large Accumulator ( xi , yi ) More memory and computations Improvement: (Finite Accumulator Array Size) x Line equation: w x cos y sin Image Space Here 0 2 w 0 w wm ax Given points ( xi , yi ) find ( w, ) ? Hough Space Sinusoid Hough Space The basic idea Each straight line in this image can be described by an equation Each white point if considered in isolation could lie on an infinite number of straight lines The basic idea Each straight line in this image can be described by an equation Each white point if considered in isolation could lie on an infinite number of straight lines In the Hough transform each point votes for every line it could be on The lines with the most votes win How do we represent lines? Any line can be represented by two numbers f Here we will represent the w yellow line by (w,f) In other words we define it using - a line from an agreed origin - of length w - at angle f to the horizontal f Hough space w Since we can use (w,f) to represent any line in the image space We can represent any line f0 in the image space as a point in the plane defined by (w,f) This is called Hough space f180 w=0 w=100 How does a point in image space vote? w x cos(f ) y sin(f ) f f0 w f180 w=0 w=100 How do multiple points prefer one line? One point in image space corresponds to a sinusoidal curve in image space Two points correspond to two curves in Hough space The intersection of those two curves has “two votes”. This intersection represents the straight line in image space that passes through both points Hough Transform Create fand w for all possible lines Create an array A indexed by fand w for each point (x,y) for each angle f w = x*cos(f)+ y*sin(f) A[f,w] = A[f,w]+1 end end where A > Threshold return a line Image space Votes Horizontal axis is θ, vertical is rho. Image votes space Mechanics of the Hough transform Difficulties How many lines? how big should the Count the peaks in cells be? (too big, and the Hough array we merge quite Treat adjacent peaks different lines; too as a single peak small, and noise causes lines to be Which points belong missed) to each line? Search for points close to the line Solve again for line and iterate Real World Example Original Edge Found Lines Detection Parameter Space Another example f w Finding Circles by Hough Transform Equation of Circle: ( xi a ) 2 ( yi b) 2 r 2 If radius is known: (2D Hough Space) Accumulator Array A(a, b) b a Hough Space Finding Circles by Hough Transform Equation of Circle: ( xi a ) 2 ( yi b) 2 r 2 If radius is not known: 3D Hough Space! Use Accumulator array A(a, b, r ) Real World Circle Examples Finding Coins Original Edges (note noise) Finding Coins (Continued) Penn Quarters y Finding Coins (Continued) Note that because the quarters and penny are different sizes, a different Hough transform (with separate accumulators) was used for each circle size. Coin finding sample images from: Vivek Kwatra Hough Transform: Comments • Works on Disconnected Edges • Relatively insensitive to occlusion • Effective for simple shapes (lines, circles, etc) • Trade-off between work in Image Space and Parameter Space

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computer vision, image processing, Pattern Recognition, machine vision, Prentice Hall, artificial intelligence, Object recognition, computer vision syndrome, image data, Image Analysis

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posted: | 3/24/2011 |

language: | English |

pages: | 25 |

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