Prof. Kristen Grauman

Transcription

1 Fitting Prof. Kristen Grauman UT Austin

2 Fitting Want to associate a model with observed features [Fig from Marszalek & Schmid, 2007] For eample, the model could be a line, a circle, or an arbitrary shape. Kristen Grauman

3 Fitting: Main idea Choose a parametric model to represent a set of features Membership criterion is not local Can t tell whether a point belongs to a given model just by looking at that point Three main questions: What model represents this set of features best? Which of several model instances gets which feature? How many model instances are there? Computational compleity is important It is infeasible to eamine every possible set of parameters and every possible combination of features Slide credit: L. Lazebnik

4 Eample: Line fitting Why fit lines? Many objects characterized by presence of straight lines Wait, why aren t we done just by running edge detection? Kristen Grauman

5 Difficulty of line fitting Kristen Grauman Etra edge points (clutter), multiple models: which points go with which line, if any? Only some parts of each line detected, and some parts are missing: how to find a line that bridges missing evidence? Noise in measured edge points, orientations: how to detect true underlying parameters?

6 Voting It s not feasible to check all combinations of features by fitting a model to each possible subset. Voting is a general technique where we let the features vote for all models that are compatible with it. Cycle through features, cast votes for model parameters. Look for model parameters that t receive a lot of votes. Noise & clutter features will cast votes too, but typically their votes should be inconsistent i t with the majority of good features. Kristen Grauman

7 Fitting lines: Hough transform Given points that belong to a line, what is the line? How many lines are there? Which points belong to which lines? Hough Transform is a voting technique that can be used to answer all of these questions. Main idea: 1. Record vote for each possible line on which each edge point lies. 2. Look for lines that get many votes. Kristen Grauman

8 Finding lines in an image: Hough space y b b 0 image space m 0 m Hough (parameter) space Connection between image (,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 (,y), find all (m,b) such that y = m + b Slide credit: Steve Seitz

9 Finding lines in an image: Hough space y b y 0 0 image space m Hough (parameter) space Connection between image (,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 (,y), find all (m,b) such that y = m + b What does a point ( 0,y 0 ) in the image space map to? Answer: the solutions of b = - 0 m + y 0 this is a line in Hough space Slide credit: Steve Seitz

10 Finding lines in an image: Hough space y ( 1, y 1 ) y 0 ( 0, y 0 ) b 0 image space b = 1m + y 1 m Hough (parameter) space What are the line parameters for the line that contains both ( 0, y 0 ) and ( 1, y 1 )? It is the intersection of the lines b = 0 m + y 0 and b = 1 m + y 1

11 Finding lines in an image: Hough algorithm y b image space m Hough (parameter) space How can we use this to find the most likely parameters (m,b) for the most prominent line in the image space? Let each edge point in image space vote for a set of possible parameters in Hough space Accumulate votes in discrete set of bins; parameters with the most votes indicate line in image space.

12 Polar representation for lines Issues with usual (m,b) parameter space: can take on infinite values, undefined for vertical lines. Image ro ows [0,0] y d Image columns d : perpendicular p distance from line to origin : angle the perpendicular makes with the -ais cos y sin d Point in image space sinusoid segment in Hough space

13 Hough line demo htt // di i 1 it/ i hi/ lid /i 2001/j a/hough.html

14 Hough transform algorithm Using the polar parameterization: cos y sin d H: accumulator array (votes) Basic Hough transform algorithm 1. Initialize H[d, ]= for each edge point I[,y] in the image d for = [ min to ma ] // some quantization d cos y 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 by d cos y sin Time compleity (in terms of number of votes per pt)? Source: Steve Seitz

15 Original image Canny edges Vote space and top peaks Kristen Grauman

16 Kristen Grauman Showing longest segments found

17 Impact of noise on Hough y d Image space edge coordinates Votes What difficulty does this present for an implementation?

18 Impact of noise on Hough Image space edge coordinates Votes Here, everything appears to be noise, or random edge points, but we still see peaks in the vote space.

19 Etensions Etension 1: Use the image gradient 1. same 2. for each edge point I[,y] in the image = gradient at (,y) d cos y sin H[d, ] += 1 3. same 4. same (Reduces degrees of freedom) Etension 2 give more votes for stronger edges Etension 3 change the sampling of (d, ) to give more/less resolution Etension 4 The same procedure can be used with circles, squares, or any other shape

20 Etensions Etension 1: Use the image gradient 1. same 2. for each edge point I[,y] in the image compute unique (d, ) based on image gradient at (,y) H[d, ] += 1 3. same 4. same (Reduces degrees of freedom) Etension 2 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 be used with circles, squares, or any other shape Source: Steve Seitz

21 Hough transform for circles Circle: center (a,b) and radius r ( i a ) ( yi b ) r For a fied radius r, unknown gradient direction b Kristen Grauman Image space Hough space a

22 Hough transform for circles Circle: center (a,b) and radius r ( i a ) ( yi b ) r For a fied radius r, unknown gradient direction Intersection: most votes for center occur here. Kristen Grauman Image space Hough space

23 Hough transform for circles Circle: center (a,b) and radius r ( i a ) ( yi b ) r For an unknown radius r, unknown gradient direction r? Image space a Hough space b Kristen Grauman

24 Hough transform for circles Circle: center (a,b) and radius r ( i a ) ( yi b ) r For an unknown radius r, unknown gradient direction r b Image space a Hough space Kristen Grauman

25 Hough transform for circles Circle: center (a,b) and radius r ( i a ) ( yi b ) r For an unknown radius r, known gradient direction θ Image space Hough space Kristen Grauman

26 Hough transform for circles For every edge piel (,y) : For each possible radius value r: end end For each possible gradient direction θ: // or use estimated gradient at (,y) a = r cos(θ) // column b = y + r sin(θ) // row H[a,b,r] br]+=1 1 Time compleity per edgel? Check out online demo :

27 Eample: detecting circles with Hough Original Edges Votes: Penny Note: a different Hough transform (with separate accumulators) g ( p ) was used for each circle radius (quarters vs. penny).

28 Eample: detecting circles with Hough Combined Original detections Edges Votes: Quarter Coin finding sample images from: Vivek Kwatra

29 Eample: iris detection Gradient+threshold Hough space Ma detections (fied radius) Hemerson Pistori and Eduardo Rocha Costa Kristen Grauman

30 Eample: iris detection An Iris Detection Method Using the Hough Transform and Its Evaluation for Facial and Eye Movement, by Hideki Kashima, Hitoshi Hongo, Kunihito Kato, Kazuhiko Yamamoto, ACCV Kristen Grauman

31 Voting: practical tips Minimize irrelevant tokens first Choose a good grid / discretization Too fine? Too coarse Vote for neighbors, also (smoothing in accumulator array) Use direction of edge to reduce parameters by 1 To read back which points voted for winning peaks, keep tags on the votes. Kristen Grauman

32 Pros Hough transform: pros and cons All points are processed independently, so can cope with occlusion, gaps Some robustness to noise: noise points unlikely to contribute tib t consistently tl to any single bin Can detect multiple instances of a model in a single pass Cons Compleity of search time increases eponentially with the number of model parameters Non-target shapes can produce spurious peaks in parameter space Quantization: can be tricky to pick a good grid size Kristen Grauman

33 Generalized Hough Transform What if we want to detect arbitrary shapes? Intuition: Displacement vectors Ref. point Model image Novel image Vote space N th l d di t Now suppose those colors encode gradient directions Kristen Grauman

34 Generalized Hough Transform Define a model shape by its boundary points and a reference point. Offline procedure: a At each boundary point, θ compute displacement θ p 1 p 2 vector: r = a p i. Model shape θ θ Store these vectors in a table indeed by gradient orientation θ. Kristen Grauman [Dana H. Ballard, Generalizing the Hough Transform to Detect Arbitrary Shapes, 1980]

35 Generalized Hough Transform Detection procedure: For each edge point: Use its gradient orientation θ to inde into stored table Use retrieved r vectors to vote for reference point θ θ θ p 1 θ Novel image θ Kristen Grauman θ θ Assuming translation is the only transformation here, i.e., orientation and scale are fied.

36 Generalized Hough for object detection Instead of indeing displacements by gradient orientation, inde by matched local patterns. training image visual codeword with displacement vectors B. Leibe, A. Leonardis, and B. Schiele, Combined Object Categorization and Segmentation with an Implicit Shape Model, ECCV Workshop on Statistical Learning in Computer Vision 2004 Source: L. Lazebnik

37 Generalized Hough for object detection Instead of indeing displacements by gradient orientation, inde by visual codeword test image B. Leibe, A. Leonardis, and B. Schiele, Combined Object Categorization and Segmentation with an Implicit Shape Model, ECCV Workshop on Statistical Learning in Computer Vision 2004 Source: L. Lazebnik

38 Recap so far: grouping and fitting Grouping/segmentation useful to make a compact representation and merge similar features associate features based on defined similarity measure and clustering objective Fitting problems require finding any supporting evidence for a model, even within clutter and missing features. associate features with an eplicit model Voting approaches, such as the Hough transform, make it possible to find likely model parameters without searching all combinations of features. Hough transform approach for lines circles arbitrary shapes Hough transform approach for lines, circles,, arbitrary shapes defined by a set of boundary points, recognition from patches.

39 The alignment problem as fitting We have so far considered how to fit a model to image evidence e.g., a line to edge points In alignment, we will fit the parameters of some transformation according to a set of matching feature pairs ( correspondences ). i ' i T Kristen Grauman

40 Feature-based alignment 2D transformations Affine fit RANSAC

41 Parametric (global) warping Eamples of parametric warps: translation rotation aspect affine perspective Source: Alyosha Efros

42 Parametric (global) warping T p = (,y) p = (,y ) Transformation T is a coordinate-changing g machine: p = T(p) What does it mean that T is global? Is the same for any point p can be described by just a few numbers (parameters) Let s represent T as a matri: p = Mp ' y' M y Source: Alyosha Efros

43 Scaling Scaling a coordinate means multiplying each of its components by a scalar Uniform scaling means this scalar is the same for all components: 2 Source: Alyosha Efros

44 Scaling Non-uniform scaling: different scalars per component: X 2, Y 0.5 Source: Alyosha Efros

45 Scaling Scaling operation: ' a y' by Or, in matri form: ' a 0 y' 0 b y scaling matri S Source: Alyosha Efros

46 What transformations can be represented with a 22 matri? p 2D Scaling? s * ' s 0 ' y s y s y * ' y s y y 0 ' 2D Rotate around (0,0)? y * sin * cos ' sin cos ' y y y * cos * sin ' sin cos y y cos sin ' 2D Sh? 2D Shear? y sh * ' sh 1 ' y sh y y * ' y sh y y 1 ' Source: Alyosha Efros

47 What transformations can be represented with a 22 matri? 2D Mirror about Y ais? ' y' y ' y' y 2D Mirror over (0,0)? ' y' y ' y' y 2D Translation? ' t y' y t y NO! Source: Alyosha Efros

48 2D Affine Transformations Affine transformations are combinations of Linear transformations, and Translations Parallel lines remain parallel

49 Feature-based alignment 2D transformations Affine fit RANSAC

50 Image alignment Two broad approaches: Direct (piel-based) alignment Search for alignment where most piels agree Feature-based alignment Search for alignment where etracted features agree Can be verified using piel-based alignment

51 An aside: Least Squares Eample Say we have a set of data points (X1,X1 ), (X2,X2 ), (X3,X3 ), etc. (e.g. person s height vs. weight) We want a nice compact formula (a line) to predict X s We want a nice compact formula (a line) to predict Xs from Xs: Xa + b = X We want to find a and b How many (X,X ) pairs do we need? ' 1 X 1 b a X ' 1 1 X 1 a X A=B What if the data is noisy? ' 2 X 2 b a X ' 2 1 X 2 b X A B ' ' 3 ' 2 ' X X X b a X X X 2 min B A X b X overconstrained Source: Alyosha Efros

52 Fitting an affine transformation Assuming we know the correspondences, how do we get the transformation? ), ( i i y ), ( i i y t t y m m m m y i i i i

53 Fitting an affine transformation Assuming we know the correspondences, how do we get the transformation? ), ( i i y ), ( i i y m m t t y m m m m y i i i i i i i i i i y m m y y t t 2 1

54 Fitting an affine transformation m m 2 1 i i i i i i y m m y y t t 2 1 How many matches (correspondence pairs) do we need to solve for the transformation parameters? Once we have solved for the parameters, how do we compute the coordinates of the corresponding point for? ) ( y for? Where do the matches come from? ), ( new new y

55 What are the correspondences?? We can use local invariant features! Compare content in local patches, find best matches.

56 Fitting an affine transformation Affine model approimates perspective projection of planar objects. Figures from David Lowe, ICCV 1999

57 Feature-based alignment 2D transformations Affine fit RANSAC

58 Outliers Outliers can hurt the quality of our parameter estimates, e.g., an erroneous pair of matching points from two images an edge point that is noise, or doesn t belong to the line we are fitting.

59 Outliers affect least squares fit

60 Outliers affect least squares fit

61 RANSAC RANdom Sample Consensus Approach: we want to avoid the impact of outliers, so let s look for inliers, and use those only. Intuition: if an outlier is chosen to compute the current fit, then the resulting line won t have much support from rest of the points.

62 RANSAC loop: RANSAC: General form 1. Randomly select a seed group of points on which h to base transformation estimate (e.g., a group of matches) 2. Compute transformation from seed group 3. Find inliers to this transformation 4. If the number of inliers is sufficiently large, re-compute estimate of transformation on all of the inliers Keep the transformation with the largest number of inliers

63 RANSAC for line fitting Repeat N times: Draw s points uniformly at random Fit line to these s points Find inliers to this line among the remaining points (i.e., points whose distance from the line is less than t) If there are d or more inliers, accept the line and refit using all inliers Lana Lazebnik

64 RANSAC for line fitting eample Source: R. Raguram Lana Lazebnik

65 RANSAC for line fitting eample Least squares fit Source: R. Raguram Lana Lazebnik

66 RANSAC for line fitting eample 1. Randomly select minimal subset of points Source: R. Raguram Lana Lazebnik

67 RANSAC for line fitting eample 1. Randomly select minimal subset of points 2. Hypothesize a model Source: R. Raguram Lana Lazebnik

68 RANSAC for line fitting eample 1. Randomly select minimal subset of points 2. Hypothesize a model 3. Compute error function Source: R. Raguram Lana Lazebnik

69 RANSAC for line fitting eample 1. Randomly select minimal subset of points 2. Hypothesize a model 3. Compute error function 4. Select points consistent with model Source: R. Raguram Lana Lazebnik

70 RANSAC for line fitting eample 1. Randomly select minimal subset of points 2. Hypothesize a model 3. Compute error function 4. Select points consistent with model 5. Repeat hypothesize and verify loop Source: R. Raguram Lana Lazebnik

71 RANSAC for line fitting eample Source: R. Raguram 1. Randomly select minimal subset of points 2. Hypothesize a model 3. Compute error function 4. Select points consistent with model 5. Repeat hypothesize and verify loop 79 Lana Lazebnik

72 RANSAC for line fitting eample Uncontaminated sample Source: R. Raguram 1. Randomly select minimal subset of points 2. Hypothesize a model 3. Compute error function 4. Select points consistent with model 5. Repeat hypothesize and verify loop 80 Lana Lazebnik

73 RANSAC for line fitting eample 1. Randomly select minimal subset of points 2. Hypothesize a model 3. Compute error function 4. Select points consistent with model 5. Repeat hypothesize and verify loop Source: R. Raguram Lana Lazebnik

74 That is an eample fitting a model (line) What about fitting a transformation (translation)?

75 RANSAC eample: Translation Putative matches Source: Rick Szeliski

76 RANSAC eample: Translation Select one match, count inliers

77 RANSAC eample: Translation Select one match, count inliers

78 RANSAC eample: Translation Find average translation vector

79 RANSAC pros and cons Pros Simple and general Applicable to many different problems Often works well in practice Cons Lots of parameters to tune Doesn t work well for low inlier ratios (too many iterations, or can fail completely) Can t always get a good initialization of the model based on the minimum number of samples Lana Lazebnik

80 Summary: grouping and fitting Grouping/segmentation useful to make a compact representation and merge similar features associate features based on defined similarity measure and clustering objective Fitting problems require finding any supporting evidence for a model, even within clutter and missing features. associate features with an eplicit model Voting approaches efficiently find model parameters Hough transform approach for lines, circles,, arbitrary shapes defined by a set of boundary points, recognition from patches. Alignment: solve for transformation relating two images Useful for spatial verification of putative local feature matches

81