Multi-stable Perception. Necker Cube
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1 Multi-stable Perception Necker Cube
2 Spinning dancer illusion, Nobuuki Kaahara
3
4 Fitting and Alignment Computer Vision Szeliski 6.1 James Has Acknowledgment: Man slides from Derek Hoiem, Lana Lazebnik, and Grauman&Leibe 2008 AAAI Tutorial
5 Project 2 due Monda
6 Review Fitting: find the parameters of a model that best fit the data Alignment: find the parameters of the transformation that best align matched points
7 Review: Fitting and Alignment Design challenges Design a suitable goodness of fit measure Similarit should reflect application goals Encode robustness to outliers and noise Design an optimization method Avoid local optima Find best parameters quickl
8 Fitting and Alignment: Methods Global optimization / Search for parameters Least squares fit Robust least squares Iterative closest point (ICP) Hpothesize and test Hough transform RANSAC
9 Review: Hough Transform 1. Create a grid of parameter values 2. Each point votes for a set of parameters, incrementing those values in grid 3. Find maimum or local maima in grid
10 Review: Hough transform P.V.C. Hough, Machine Analsis of Bubble Chamber Pictures, Proc. Int. Conf. High Energ Accelerators and Instrumentation, 1959 Given a set of points, find the curve or line that eplains the data points best m = m + b Hough space b Slide from S. Savarese
11 Review: Hough transform m b m Slide from S. Savarese b
12 Incorporating image gradients Recall: when we detect an edge point, we also know its gradient direction But this means that the line is uniquel determined! Modified Hough transform: For each edge point (,) θ = gradient orientation at (,) ρ = cos θ + sin θ H(θ, ρ) = H(θ, ρ) + 1 end
13 Hough Transform How would we find circles? Of fied radius Of unknown radius Of unknown radius but with known edge orientation
14 Hough transform for circles Conceptuall equivalent procedure: for each (,,r), draw the corresponding circle in the image and compute its support r Is this more or less efficient than voting with features?
15 Hough Transform How would we find circles? Of fied radius Of unknown radius Of unknown radius but with known edge orientation
16 Hough transform for circles image space Hough parameter space r (, ) ri (, ) (,) (, ) ri (, )
17 Hough transform conclusions Good Robust to outliers: each point votes separatel Fairl efficient (much faster than tring all sets of parameters) Provides multiple good fits Bad Some sensitivit to noise Bin size trades off between noise tolerance, precision, and speed/memor Can be hard to find sweet spot Not suitable for more than a few parameters grid size grows eponentiall Common applications Line fitting (also circles, ellipses, etc.) Object instance recognition (parameters are affine transform) Object categor recognition (parameters are position/scale)
18 RANSAC (RANdom SAmple Consensus) : Fischler & Bolles in 81. Algorithm: 1. Sample (randoml) the number of points required to fit the model 2. Solve for model parameters using samples 3. Score b the fraction of inliers within a preset threshold of the model Repeat 1-3 until the best model is found with high confidence
19 RANSAC Line fitting eample Algorithm: 1. Sample (randoml) the number of points required to fit the model (#=2) 2. Solve for model parameters using samples 3. Score b the fraction of inliers within a preset threshold of the model Repeat 1-3 until the best model is found with high confidence Illustration b Savarese
20 RANSAC Line fitting eample Algorithm: 1. Sample (randoml) the number of points required to fit the model (#=2) 2. Solve for model parameters using samples 3. Score b the fraction of inliers within a preset threshold of the model Repeat 1-3 until the best model is found with high confidence
21 RANSAC Line fitting eample N I 6 Algorithm: 1. Sample (randoml) the number of points required to fit the model (#=2) 2. Solve for model parameters using samples 3. Score b the fraction of inliers within a preset threshold of the model Repeat 1-3 until the best model is found with high confidence
22 RANSAC Algorithm: N I Sample (randoml) the number of points required to fit the model (#=2) 2. Solve for model parameters using samples 3. Score b the fraction of inliers within a preset threshold of the model Repeat 1-3 until the best model is found with high confidence
23 How to choose parameters? Number of samples N Choose N so that, with probabilit p, at least one random sample is free from outliers (e.g. p=0.99) (outlier ratio: e ) Number of sampled points s Minimum number needed to fit the model Distance threshold Choose so that a good point with noise is likel (e.g., prob=0.95) within threshold Zero-mean Gaussian noise with std. dev. σ: t 2 =3.84σ 2 N log 1 p/ log1 1 e s proportion of outliers e s 5% 10% 20% 25% 30% 40% 50% For p = 0.99 modified from M. Pollefes
24 RANSAC conclusions Good Robust to outliers Applicable for larger number of model parameters than Hough transform Optimization parameters are easier to choose than Hough transform Bad Computational time grows quickl with fraction of outliers and number of parameters Not good for getting multiple fits Common applications Computing a homograph (e.g., image stitching) Estimating fundamental matri (relating two views)
25 How do we fit the best alignment?
26 Alignment Alignment: find parameters of model that maps one set of points to another Tpicall want to solve for a global transformation that accounts for *most* true correspondences Difficulties Noise (tpicall 1-3 piels) Outliers (often 50%) Man-to-one matches or multiple objects
27 Parametric (global) warping T p = (,) p = (, ) Transformation T is a coordinate-changing machine: p = T(p) What does it mean that T is global? Is the same for an point p can be described b just a few numbers (parameters) For linear transformations, we can represent T as a matri p = Tp ' T '
28 Common transformations original Transformed translation rotation aspect affine perspective Slide credit (net few slides): A. Efros and/or S. Seitz
29 Scaling Scaling a coordinate means multipling each of its components b a scalar Uniform scaling means this scalar is the same for all components: 2
30 Scaling Non-uniform scaling: different scalars per component: X 2, Y 0.5
31 Scaling Scaling operation: ' ' a b Or, in matri form: ' ' a 0 0 b scaling matri S
32 2-D Rotation (, ) (, ) = cos() - sin() = sin() + cos()
33 2-D Rotation This is eas to capture in matri form: ' ' cos sin sin cos Even though sin() and cos() are nonlinear functions of, is a linear combination of and is a linear combination of and What is the inverse transformation? Rotation b For rotation matrices R 1 T R R
34 Basic 2D transformations Translate Rotate Shear Scale 1 1 ' ' cos sin sin cos ' ' s s 0 0 ' ' t t 1 f e d c b a Affine Affine is an combination of translation, scale, rotation, shear
35 Affine Transformations Affine transformations are combinations of Linear transformations, and Translations Properties of affine transformations: Lines map to lines Parallel lines remain parallel Ratios are preserved Closed under composition 1 f e d c b a ' ' f e d c b a or
36 Projective Transformations Projective transformations are combos of Affine transformations, and Projective warps Properties of projective transformations: ' ' w' Lines map to lines Parallel lines do not necessaril remain parallel Ratios are not preserved Closed under composition Models change of basis Projective matri is defined up to a scale (8 DOF) a d g b e h c f i w
37 2D image transformations (reference table) Szeliski 2.1
38 Eample: solving for translation A 1 A 2 A 3 B 1 B 2 B 3 Given matched points in {A} and {B}, estimate the translation of the object A i A i B i B i t t
39 Eample: solving for translation A 1 A 2 A 3 B 1 B 2 B 3 Least squares solution A i A i B i B i t t (t, t ) 1. Write down objective function 2. Derived solution a) Compute derivative b) Compute solution 3. Computational solution a) Write in form A=b b) Solve using pseudo-inverse or eigenvalue decomposition A n B n A n B n A B A B t t
40 Eample: solving for translation A 1 A 5 B 4 A 2 A B 1 3 (t, t ) A 4 B 2 B 3 B 5 Problem: outliers RANSAC solution 1. Sample a set of matching points (1 pair) 2. Solve for transformation parameters 3. Score parameters with number of inliers 4. Repeat steps 1-3 N times B i B i A i A i t t
41 Eample: solving for translation B 4 A 1 B 5 B 6 A 2 A B 1 3 (t, t ) A 4 B 2 B 3 A 5 A 6 Problem: outliers, multiple objects, and/or man-to-one matches Hough transform solution 1. Initialize a grid of parameter values 2. Each matched pair casts a vote for consistent values 3. Find the parameters with the most votes 4. Solve using least squares with inliers B i B i A i A i t t
42 Eample: solving for translation (t, t ) Problem: no initial guesses for correspondence A i A i B i B i t t
43 What if ou want to align but have no prior matched pairs? Hough transform and RANSAC not applicable Important applications Medical imaging: match brain scans or contours Robotics: match point clouds
Photo by Carl Warner
Photo b Carl Warner Photo b Carl Warner Photo b Carl Warner Fitting and Alignment Szeliski 6. Computer Vision CS 43, Brown James Has Acknowledgment: Man slides from Derek Hoiem and Grauman&Leibe 2008 AAAI
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