Example: Line fitting. Difficulty of line fitting. Fitting lines. Fitting lines. Fitting lines. Voting 9/22/2009

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1 Histograms in Matla Fitting: Voting and the Hough Transform Tuesda, Sept Kristen Grauman UT-Austin a = A(:); % reshapes matri A into vector, columns first H = hist(a(:), 10); %t takes a histogram from the A s values, into 10 uniforml sized ins H = histc(a(:), [1:N]); % counts values within the ins having specified edges Last time: segmentation Segmentation to find oject oundaries or midlevel regions, tokens. Bottom-up segmentation via clustering General choices -- features, affinit functions, and clustering algorithms Grouping also useful for quantization, can create new feature summaries Teton histograms for teture within local region Eample clustering methods K-means Graph cuts, normalized cuts Tradeoffs Review: graph-ased clustering Assuming we use a full connected graph, what is the time compleit of computing the affinities for a graph cuts-ased segmentation? Eample affinit measure: X(i) is position of node i F(i) is a feature vector for node i ased on color, teture, etc. This affinit measure limits connections to spatiall close piels. Now: Fitting Want to associate a model with oserved features [Fig from Marszalek & Schmid, 007] For eample, the model could e a line, a circle, or an aritrar shape. Fitting Choose a parametric model to represent a set of features Memership criterion is not local Can t tell whether a point elongs to a given model just looking at that point Three main questions: s What model represents this set of features est? Which of several model instances gets which feature? How man model instances are there? Computational compleit is important It is infeasile to eamine ever possile set of parameters and ever possile comination of features Source: L. Lazenik 1

2 Eample: Line fitting Wh fit lines? Man ojects characterized presence of straight lines Wait, wh aren t we done just running edge detection? Difficult of line fitting Etra edge points (clutter), multiple models: which points go with which line, if an? Onl some parts of each line detected, and some parts are missing: how to find a line that ridges missing evidence? Noise in measured edge points, orientations: how to detect true underling parameters? Fitting lines Given points that elong to a line, what is the line? Fitting lines Given points that elong to a line, what is the line? How man lines are there? Which points elong to which lines? Assuming all the points that elong to a line are known, can solve for line parameters that ield minimal error. Forsth & Ponce Voting It s not feasile to check all cominations of features fitting a model to each possile suset. Voting is a general technique where we let the features vote for all models that are compatile with it. Ccle through features, cast votes for model parameters. Look for model parameters that receive a lot of votes. Noise & clutter features will cast votes too, ut tpicall their votes should e inconsistent with the majorit of good features. Ok if some features not oserved, as model can span multiple fragments. Fitting lines Given points that elong to a line, what is the line? How man lines are there? Which points elong to which lines? Hough Transform is a voting Hough Transform is a voting technique that can e used to answer all of these questions. Main idea: 1. Record vote for each possile line on which each edge point lies.. Look for lines that get man votes.

3 Finding lines in an image: Finding lines in an image: 0 0 image space m 0 m Hough (parameter) space 0 image space m Hough (parameter) space Connection etween image (,) and Hough (m,) spaces A line in the image corresponds to a point in To go from image space to : given a set of points (,), find all (m,) such that = m + Connection etween image (,) and Hough (m,) spaces A line in the image corresponds to a point in To go from image space to : given a set of points (,), find all (m,) such that = m + What does a point ( 0, 0 ) in the image space map to? Slide credit: Steve Seitz Answer: the solutions of = - 0 m + 0 this is a line in Slide credit: Steve Seitz Finding lines in an image: Finding lines in an image: Hough algorithm ( 1, 1 ) 0 ( 0, 0 ) = 1 m image space m Hough (parameter) space image space m Hough (parameter) space What are the line parameters for the line that contains oth ( 0, 0 ) and ( 1, 1 )? It is the intersection of the lines = 0 m + 0 and = 1 m + 1 How can we use this to find the most likel parameters (m,) for the most prominent line in the image space? Let each edge point in image space vote for a set of possile parameters in Accumulate votes in discrete set of ins; parameters with the most votes indicate line in image space. Polar representation for lines Issues with usual (m,) parameter space: can take on infinite values, undefined for vertical lines. Hough transform algorithm Using the polar parameterization: cos sin = d H: accumulator arra (votes) [0,0] d d : perpendicular distance from line to origin : angle the perpendicular makes with the -ais cos sin = d Basic Hough transform algorithm d 1. Initialize H[d, ]=0. for each edge point I[,] in the image for = 0 to 180 // some quantization d = cos sin H[d, ] += 1 3. Find the value(s) of (d, ) where H[d, ] is maimum 4. The detected line in the image is given d = cos sin Hough line demo Point in image space sinusoid segment in Time compleit (in terms of numer of votes per pt)? Source: Steve Seitz 3

4 Eample: Hough transform for straight lines Eample: Hough transform for straight lines Square : Circle : d edge coordinates Votes Bright value = high vote count Black = no votes Eample: Hough transform for straight lines Impact of noise on Hough d Showing longest segments found edge coordinates Votes What difficult does this present for an implementation? 4

5 Impact of noise on Hough edge coordinates Votes Here, everthing appears to e noise, or random edge points, ut we still see peaks in the vote space. Etensions Etension 1: Use the image gradient 1. same. for each edge point I[,] in the image = gradient at (,) d = cos sin H[d, ] += 1 3. same 4. same (Reduces degrees of freedom) Etension give more votes for stronger edges Etension 3 change the sampling of (d, ) to give more/less resolution Etension 4 The same procedure can e used with circles, squares, or an other shape Etensions Etension 1: Use the image gradient 1. same. for each edge point I[,] in the image compute unique (d, ) ased on image gradient at (,) H[d, ] += 1 3. same 4. same (Reduces degrees of freedom) Etension give more votes for stronger edges (use magnitude of gradient) Etension 3 change the sampling of (d, ) to give more/less resolution Etension 4 The same procedure can e used with circles, squares, or an other shape Circle: center (a,) and radius r ( i a) + ( i ) For a fied radius r, unknown gradient direction a Source: Steve Seitz Circle: center (a,) and radius r ( i a) + ( i ) For a fied radius r, unknown gradient direction Circle: center (a,) and radius r ( i a) + ( i ) For an unknown radius r, unknown gradient direction r Intersection : most votes for center occur here. a 5

6 Circle: center (a,) and radius r ( i a) + ( i ) For an unknown radius r, unknown gradient direction Circle: center (a,) and radius r ( i a) + ( i ) For an unknown radius r, known gradient direction r a For ever edge piel (,) : For each possile radius value r: For each possile gradient direction : // or use estimated gradient a = r cos() = + r sin() H[a,,r] += 1 end end Check out online demo : Eample: detecting circles with Hough Crosshair indicates results of Hough transform, ounding o found via motion differencing. Eample: detecting circles with Hough Eample: detecting circles with Hough Original Edges Votes: Penn Comined Original detections Edges Votes: Quarter Note: a different Hough transform (with separate accumulators) was used for each circle radius (quarters vs. penn). Coin finding sample images from: Vivek Kwatra 6

7 Eample: iris detection Eample: iris detection Gradient+threshold (fied radius) Ma detections Hemerson Pistori and Eduardo Rocha Costa An Iris Detection Method Using the Hough Transform and Its Evaluation for Facial and Ee Movement, Hideki Kashima, Hitoshi Hongo, Kunihito Kato, Kazuhiko Yamamoto, ACCV 00. Voting: practical tips Minimize irrelevant tokens first (take edge points with significant gradient magnitude) Choose a good grid / discretization Too fine? Too coarse Vote for neighors, also (smoothing in accumulator arra) Utilize direction of edge to reduce free parameters 1 To read ack which points voted for winning peaks, keep tags on the votes. Hough transform: pros and cons Pros All points are processed independentl, so can cope with occlusion Some roustness to noise: noise points unlikel to contriute consistentl to an single in Can detect multiple instances of a model in a single pass Cons Compleit of search time increases eponentiall with the numer of model parameters Non-target shapes can produce spurious peaks in parameter space Quantization: hard to pick a good grid size Generalized Hough transform What if want to detect aritrar shapes defined oundar points and a reference point? p 1 a p At each oundar point, compute displacement vector: r = a p i. For a given model shape: store these vectors in a tale indeed gradient orientation. [Dana H. Ballard, Generalizing the Hough Transform to Detect Aritrar Shapes, 1980] Generalized Hough transform To detect the model shape in a new image: For each edge point Inde into tale with its gradient orientation Use retrieved r vectors to vote for position of reference point Peak in this is reference point with most supporting edges Assuming translation is the onl transformation here, i.e., orientation and scale are fied. 7

8 Eample Eample Sa we ve alread stored a tale of displacement vectors as a function of edge orientation for this model shape. displacement vectors for model points model shape Adapted from Lana Lazenik Eample Eample Now we want to look at some edge points detected in a new image, and vote on the position of that shape. range of voting locations for test point range of voting locations for test point Eample Eample votes for points with = Recall: displacement vectors for model points 8

9 Eample Eample votes for points with = votes for points with = range of voting locations for test point Application in recognition Instead of indeing displacements gradient orientation, inde visual codeword Application in recognition Instead of indeing displacements gradient orientation, inde visual codeword training image visual codeword with displacement vectors test image B. Leie, A. Leonardis, and B. Schiele, Comined Oject Categorization and Segmentation with an Implicit Shape Model, ECCV Workshop on Statistical Learning in Computer Vision 004 Source: L. Lazenik B. Leie, A. Leonardis, and B. Schiele, Comined Oject Categorization and Segmentation with an Implicit Shape Model, ECCV Workshop on Statistical Learning in Computer Vision 004 Source: L. Lazenik Summar Grouping/segmentation useful to make a compact representation and merge similar features associate features ased on defined similarit measure and clustering ojective Fitting prolems require finding an supporting evidence for a model, even within clutter and missing i features. associate features with an eplicit model Net Thursda 9/4: Deformale contours Pset : teture + clustering Out toda, due 10/6 Voting approaches, such as the Hough transform, make it possile to find likel model parameters without searching all cominations of features. Hough transform approach for lines, circles,, aritrar shapes defined a set of oundar points, recognition from patches. 9