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# Class Note for EECS 841 with Professor Potetz at KU (2)

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Date Created: 02/06/15

EECS 841 Computer Vision Brian Potetz Fall 2008 Lecture 8 Hough Transforms Suggested Reading Forsyth amp Ponce Computer Vision A Modern Approach Chapter 151 E R Davies quotMachine Vision Chapters 911 L G Shapiro G C Stockman Computer Vision Chapter 10 MarrHildreth Edge Detection Solution Instead of thresholding the rst derivative nd zero crossings of the second derivative Common 2nd Order Derivative Operator Laplacian Af wf i rri 7 7 do 397 20 Laplacian of Gaussian MarrHildreth Edge Detection Solution Instead of thresholding the rst derivative nd zero crossings of the second derivative Edges can be localized to subpixel accuracy But all contours must be closed Spaghetti Localizing Edges Thresholding V 2 can result in wide poorly localized edges Especially for high a Solution For each pixel above threshold nd the local maximum of Vfl2 along the direction of the gradient Vf MarrHildreth Canny Edge Detection Edge Detection Choice of 039 Determines Features Choice of 039 Determines Features Large 0 results in coarse large scale features Large 0 results in coarse large scale features Small 0 results in ne small scale features Small 0 results in fine small scale features n this 11quot 39 LII quot39 quot39339 LjL 1 t quotW zquot i a 39 39739 n m nllll 2 39l 39 Canny with 0 1 Canny with 0 2 Edges in Scaespace The Gaussian Pyramid G4G3gausszan 2 downsam le As sigma increases edges can combine but G3 G2g 2 downsam I never form G2 G1gaussian I 2 G1 G0 gaussian J 2 391 I I f 39i 399 J 39l V quotI quotl l I II quot i 1 l 39 39 w l39li39fltillln I il39ll 3 I T 4 X Edge Detection using Pyramids Gaussian Pyramid Frequency Composition Coarsetofine strategy Do edge detection at higher level Consider edges of finer scales only near the edges of higher scales Level 3 Laplacian Pyramid Frequency Composition The Laplacian Pyramid is Level 3 a band pass representation vice a low pass representation for the Gaussian f Horror Photo prof dmartin EECS 841 Computer Vision Brian Potetz Fall 2008 Lecture 8 Hough Transforms Suggested Reading Forsyth amp Ponce quotComputer Vision A Modern Approachquot Chapter 151 E R Davies quotMachine Vision Chapters 911 L G Shapiro G C Stockman quotComputer Vision Chapter 10 Applications of Line Detection Automated Inspection Applications of Line Detection Applications of Line Detection Automated Inspection Automated Inspection Model tting amp Object detection Model tting amp Object detection f39 39 Murdering Automated Driving quot 1 t I a rquot3939 52 g I Drive straight Steer left Steer right I39L Applications of Line Detection Hough Transforms Equation for a line y mac b Automated Inspection Suppose I know x ampy but not m amp 9 Model tting amp Object detection What is the set of all lines that pass through x ampy Automated Driving Inferring Shape from Blocks y 07 90 Hough Transforms Hough Transforms Equation for a line y mac b Equation for a line y mac b Suppose I know x ampy but not m amp 9 Suppose I know x ampy but not m amp b What is the set of all lines that pass through x ampy What is the set of all lines that pass through x ampy In parameterspace the space of mb In parameterspace the space of mb Each point in mb specifies a line in xy space Each point in mb specifies a line in xy space Givenxampy b mg y y b y bu 07 yo 07 yo III III Hough Transforms Equation fora line y mac b Suppose I knowxampy but not m amp b What is the set of all lines that passthroughx ampy In parameterspace the space of mb Each point in mb specifies a line in Xy space Givenxampy b mac 7y y b Iowa Hough Transforms Equation fora line y m b Suppose I knowx Sty but not m amp b What is the set of all lines that pass through xampy In parameterspace the space of mb Each point in mb specifies a line in Xy space Givenxampy bmz7y 111 0410 y Hough Transforms Equation fora line y mac b Suppose I knowxampy but not m amp b What is the set of all lines that passthroughx ampy In parameterspace the space of mb Each point in mb specifies a line in Xy space Givenxampy bmxiy 1211 ltZ y Hough Transforms Equation fora line y m b Suppose I knowx Sty but not m amp b What is the set of all lines that pass through xampy In parameterspace the space of mb Each point in mb specifies a line in Xy space Givenxampy b mac 7y y b Hough Transforms Equation fora line y mac b Suppose I knowxampy but not m amp b What is the set of all lines that passthroughx ampy In parameterspace the space of mb Each point in mb specifies a line in Xy space Givenxampy b mac 7y Y b Hough Transforms Equation fora line y m b Suppose I knowx Sty but not m amp b What is the set of all lines that pass through xampy In parameterspace the space of mb Each point in mb specifies a line in Xy space Givenxampy b mac 7y Y b Hough Transforms Equation fora line y mm b Suppose I know xampy but not m amp b What is the set of all lines that pass through x ampy In parameterspace the space of mb Each point in mb speci es a line in xy space Givenxampy bmx y Hough Transforms Equation for a line y mm b Suppose I know x ampy but not m amp b What is the set of all lines that pass through x ampy In parameterspace the space of mb Each point in mb speci es a line in xy space Givenxampy bmx y Hough Transform Algorithm Quantize parameter space between appropriate maximum and minimum values for c and m 2 Form an accumulator array Acm initialized to O 3 For each point above threshold in the edge map increment all the points in mbspace the accumulator along the appropriate line bmI y 4 Local maxima in the accumulator correspond to lines Reparameterization of Lines Original parameterization of lines y mac b For vertical lines m goes to in nity Reparameterization of Lines Original parameterization of lines y mac b For vertical lines m goes to in nity New parameterization of lines 1 Sin y cos 0 Reparameterization of Lines Original parameterization of lines y mac b For vertical lines m goes to in nity New parameterization of lines 1 Sin y cos 0 0 yo Reparameterization of Lines Reparameterization of Lines New parameterization of lines 70 sinQ5 y cos Q5 0 New parameterization of lines 70 Sin Q5 y cos Q5 0 pTSiDltQ5 9 pTSiDltQ5 9 rJc2 l y2 tan6yx 7 Vx2y2 tan6yx y mayo p y 0790 p 5 5 Reparameterization of Lines Reparameterization of Lines 2 Example Points 3 Example Points Reparameterization of Lines Reparameterization of Lines 4 Example Points 5 Example Points 0 20 40 60 80 100 120 140 150 0 20 40 Reparameterization of Lines rho pixels from center Two Lines Radon Transform Sum of image intensity fxy along each line at an angle I from xaxis Edge Fouhd Lines Detection Original Parameter Space Mechanics of the Hough transform Don t waste the Gradient Magnitude Initially each point over threshold gets equal say Other me hods give more weight to points with greater gradient magnitude Don t waste the Gradient Direction Edge detection methods give us direction as well as gradient magnitude Weight the curve 0 7 sin 9 higher near the edge direction Other methods allow each edge point only one vote Mechanics of the Hough transform Quantizing p space How big should the cells be Too big and we merge quite different lines Too small and noise causes lines to be missed What counts as a line Thresholding results in clusters of lines Use nonmaximum suppression to choose only the local maximum of each cluster Which points belong to each line Often we want to find line segments not in nite lines Search along each infinite line to locate segment endpoints Each line could have several segments Finding Circles by Hough Transform Equation of Circle x W y bgt2 r2 If radius is known 2D Hough Space AccumulatorArray Aab Question What is the advantage of this Hough transform over Convolution Red circles locus of the center of possible circles Finding Circles by Hough Transform Equation of Circle xi a2 yi b2 r2 If radius is known 2D Hough Space AccumulatorArray Aab Question What is the advantage ofthis Hough transform over Convolution Red circles locus of the Finding Coins Original Edges note noise Often only points in the accumulator center Of possible circles consistentwith the edge direction are incremented Finding C0ins Assuming PennySize Assuming QuarterSize Finding Coins Edges note noise Results Real World Circle Examples Crosshair indicates results of Hough transform bounding box found via motion differencing Finding Circles by Hough Transform Equation of Circle 2 2 2 xi a yi 17 r If radius is unknown 3D Hough Space AccumulatorArray Aabr What is he surface in the hough space Red circles locus of the center of possible circles Finding Circles by Hough Transform Finding Circles by Hough Transform Equation of Circle x agt2 y bgt2 r2 Equation of Circle x agt2 y b2 r2 If radius is unknown 3D Hough Space If radius is unknown 3D Hough Space AccumulatorArray Aabr AccumUIatorArray 1401919 What is the surface in the hough space Alternatively use a 2D Accumulator array Aab and increment it for each radius a In some range of pOSSIble radii Red circles locus of the Red circles locus of the center of poss39ble C39rdes39 Question Why wouldn t this falsely center Of poss39ble C39rdes39 accept ellipses b W Generalized Hough Transform Generalized Hough Transform Find contours that match a xed template Template can be any shape Works just like Hough transform for circles of known size Mogul

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