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1 Photo b Carl Warner
2 Photo b Carl Warner
3 Photo b Carl Warner
4 Fitting and Alignment Szeliski 6. Computer Vision CS 43, Brown James Has Acknowledgment: Man slides from Derek Hoiem and Grauman&Leibe 2008 AAAI Tutorial
5 Project 2 questions?
6 This section: correspondence and alignment Correspondence: matching points, patches, edges, or regions across images
7 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
8 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
9 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
10 Review: Hough Transform. 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
11 Review: Hough transform P.V.C. Hough, Machine Analsis of Bubble Chamber Pictures, Proc. Int. Conf. High Energ Accelerators and Instrumentation, 959 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
12 Review: Hough transform m b m Slide from S. Savarese b
13 Hough Transform How would we find circles? Of fied radius Of unknown radius
14 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)
15 RANSAC (RANdom SAmple Consensus) : Fischler & Bolles in 8. Algorithm:. 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 -3 until the best model is found with high confidence
16 RANSAC Line fitting eample Algorithm:. 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 -3 until the best model is found with high confidence Illustration b Savarese
17 RANSAC Line fitting eample Algorithm:. 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 -3 until the best model is found with high confidence
18 RANSAC Line fitting eample N I 6 Algorithm:. 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 -3 until the best model is found with high confidence
19 RANSAC Algorithm: N I 4. 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 -3 until the best model is found with high confidence
20 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 p/ log e s proportion of outliers e s 5% 0% 20% 25% 30% 40% 50% modified from M. Pollefes
21 RANSAC conclusions Good Robust to outliers Applicable for larger number of objective function 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)
22 How do we fit the best alignment?
23 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 -3 piels) Outliers (often 50%) Man-to-one matches or multiple objects
24 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 '
25 Common transformations original Transformed translation rotation aspect affine perspective Slide credit (net few slides): A. Efros and/or S. Seitz
26 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
27 Scaling Non-uniform scaling: different scalars per component: X 2, Y 0.5
28 Scaling Scaling operation: ' ' a b Or, in matri form: ' ' a 0 0 b scaling matri S
29 2-D Rotation (, ) (, ) = cos() - sin() = sin() + cos()
30 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 T R R
31 Basic 2D transformations Translate Rotate Shear Scale ' ' cos sin sin cos ' ' s s 0 0 ' ' 0 0 t t f e d c b a Affine Affine is an combination of translation, scale, rotation, shear
32 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 f e d c b a 0 0 ' ' f e d c b a or
33 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
34 2D image transformations (reference table) Szeliski 2.
35 Eample: solving for translation A A 2 A 3 B 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
36 Eample: solving for translation A A 2 A 3 B B 2 B 3 Least squares solution A i A i B i B i t t (t, t ). 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
37 Eample: solving for translation A A 5 A 2 A 3 (t, t ) B 4 B A 4 B 2 B 3 B 5 Problem: outliers RANSAC solution. Sample a set of matching points ( pair) 2. Solve for transformation parameters 3. Score parameters with number of inliers 4. Repeat steps -3 N times B i B i A i A i t t
38 Eample: solving for translation B 4 A B 5 B 6 A 2 A 3 (t, t ) B A 4 B 2 B 3 A 5 A 6 Problem: outliers, multiple objects, and/or man-to-one matches Hough transform solution. 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
39 Eample: solving for translation (t, t ) Problem: no initial guesses for correspondence A i A i B i B i t t
40 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
41 Iterative Closest Points (ICP) Algorithm Goal: estimate transform between two dense sets of points. Initialize transformation (e.g., compute difference in means and scale) 2. Assign each point in {Set } to its nearest neighbor in {Set 2} 3. Estimate transformation parameters e.g., least squares or robust least squares 4. Transform the points in {Set } using estimated parameters 5. Repeat steps 2-4 until change is ver small
42 Eample: aligning boundaries. Etract edge piels p.. p n and q.. q m 2. Compute initial transformation (e.g., compute translation and scaling b center of mass, variance within each image) 3. Get nearest neighbors: for each point p i find corresponding match(i) = argmin dist(pi, qj) j 4. Compute transformation T based on matches 5. Warp points p according to T 6. Repeat 3-5 until convergence p q
43 Eample: solving for translation (t, t ) Problem: no initial guesses for correspondence A i A i B i B i t t ICP solution. Find nearest neighbors for each point 2. Compute transform using matches 3. Move points using transform 4. Repeat steps -3 until convergence
44 Algorithm Summar Least Squares Fit closed form solution robust to noise not robust to outliers Robust Least Squares improves robustness to noise requires iterative optimization Hough transform robust to noise and outliers can fit multiple models onl works for a few parameters (-4 tpicall) RANSAC robust to noise and outliers works with a moderate number of parameters (e.g, -8) Iterative Closest Point (ICP) For local alignment onl: does not require initial correspondences
45 Object Instance Recognition. Match kepoints to object model 2. Solve for affine transformation parameters 3. Score b inliers and choose solutions with score above threshold A A 2 Matched kepoints A 3 Affine Parameters # Inliers This Class Choose hpothesis with ma score above threshold
46 N piels Overview of Kepoint Matching. Find a set of distinctive kepoints A A 2 A 3 2. Define a region around each kepoint N piels f A e.g. color d( f A, f B ) T f B e.g. color K. Grauman, B. Leibe 3. Etract and normalize the region content 4. Compute a local descriptor from the normalized region 5. Match local descriptors
47 Finding the objects (overview) Input Image Stored Image. Match interest points from input image to database image 2. Matched points vote for rough position/orientation/scale of object 3. Find position/orientation/scales that have at least three votes 4. Compute affine registration and matches using iterative least squares with outlier check 5. Report object if there are at least T matched points
48 Affine Object Model Accounts for 3D rotation of a surface under orthographic projection
49 Affine Object Model Accounts for 3D rotation of a surface under orthographic projection What is the minimum number of matched points that we need? f e d c b a f e d c b a
50 Finding the objects (SIFT, Lowe 2004). Match interest points from input image to database image 2. Get location/scale/orientation using Hough voting In training, each point has known position/scale/orientation wrt whole object Matched points vote for the position, scale, and orientation of the entire object Bins for,, scale, orientation Wide bins (0.25 object length in position, 2 scale, 30 degrees orientation) Vote for two closest bin centers in each direction (6 votes total) 3. Geometric verification For each bin with at least 3 kepoints Iterate between least squares fit and checking for inliers and outliers 4. Report object if > T inliers (T is tpicall 3, can be computed to match some probabilistic threshold)
51 Eamples of recognized objects
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