伊利诺伊州立大学计算机视觉PPT第9章-FittingandRegistration-Vision_Spring2012

2
Solution
x AT A AT b
1
least squares solution to Ax b
x A\b
(matlab)
Problem statement
minimize x T AT Ax s.t. x T x 1
Solution
[ v, ] eig( A T A)
– – – Try all possible Each point votes for all consistent parameters Repeatedly sample enough points to solve for parameters
2. Score the given parameters
Bad
• May not be what you want to optimize • Sensitive to outliers
– Bad matches, extra points
• Doesn’t allow you to get multiple good fits
– Detecting multiple objects, lines, etc.
2
2
y T y 2( Ap )T y ( Ap )T ( Ap )
dE 2 A T Ap 2 A T y 0 dp
Matlab: p = A \ y;
1
AT Ap AT y p AT A AT y
Modified from S. Lazebnik
Problem with “vertical” least squares
Find (a, b, c) to minimize the sum of squared perpendicular distances
ax+by+c=0
Unit normal: N=(a, b) E (a x( by d )) x y i, i
n 2 i 1 i i
E i 1 ( a xi b yi c )
proof: /PointLineDistance2-Dimensional.html
ax+by+c=0
Unit normal: N=(a, b) E (a x( by d )) x y i, i
n 2 i 1 i i
Slide modified from S. Lazebnik
Demo – part 1
Other ways to search for parameters (for when no closed form solution exists)
• Line search
1. 2. For each parameter, step through values and choose value that gives best fit Repeat (1) until no parameter changes
• Not rotation-invariant • Fails completely for vertical lines
Slide from S. Lazebnik
Slide modified from S. Lazebnik
Total least squares
If (a2+b2=1) then Distance between point (xi, yi) is |axi + byi + c|
– Number of consistent points, possibly weighted by distance
3. Choose from among the set of parameters
– Global or local maximum of scores
4. Possibly refine parameters using inliers
•
Grid search
1.
2.Propose s来自veral sets of parameters, evenly sampled in the joint set Choose best (or top few) and sample joint parameters around the current best; repeat
• Design challenges
– Design a suitable goodness of fit measure
• Similarity should reflect application goals • Encode robustness to outliers and noise
– Design an optimization method
Hough transform
P.V.C. Hough, Machine Analysis of Bubble Chamber Pictures, Proc. Int. Conf. High Energy Accelerators and Instrumentation, 1959
Given a set of points, find the curve or line that explains the data points best
• Avoid local optima • Find best parameters quickly
Fitting and Alignment: Methods
• Global optimization / Search for parameters
– Least squares fit – Robust least squares – Iterative closest point (ICP)
y=mx+b
E i 1 ( yi m xi b) 2
n
(xi, yi)
2
E i 1 x i
n
x1 1 y1 m m 1 yi Ap y b b xn 1 yn
• Hypothesize and test
– Generalized Hough transform – RANSAC
Simple example: Fitting a line
Least squares line fitting
•Data: (x1, y1), …, (xn, yn) •Line equation: yi = m xi + b •Find (m, b) to minimize
n
n E i 1 2( a xi b yi c ) 0 c
2
a n b n c i 1 xi i 1 yi a x b y n n
x1 x n E i 1 ( a ( xi x ) b( yi y )) 2 xn x
See details on Raleigh Quotient: /wiki/Rayleigh_quotient
Recap: Two Common Optimization Problems
Problem statement
minimize Ax b
Robust least squares (to deal with outliers)
General approach: minimize
u x , ;
i i i
u 2 i 1 ( yi m xi b ) 2
n
ui (xi, θ) – residual of ith point w.r.t. model parameters θ ρ – robust function with scale parameter σ
error( , data i ) 2 2. Choose params to minimize: 2 error( , data ) 2 i i – E.g., numerical optimization
3. Compute new 1.5 median error 4. Repeat (2) and (3) until convergence
x A Ax minimize xT x
non - trivial lsq solution to Ax 0
T
T
1 2..n : x v1
Least squares (global) optimization
Good
• Clearly specified objective • Optimization is easy
ax+by+c=0
Unit normal: N=(a, b) E (a x( by d )) x y i, i
n 2 i 1 i i
E i 1 ( a xi b yi c ) 2
n
Slide modified from S. Lazebnik
Total least squares
Hough Transform: Outline
1. Create a grid of parameter values 2. Each point votes for a set of parameters, incrementing those values in grid 3. Find maximum or local maxima in grid
•
Gradient descent
1. 2. Provide initial position (e.g., random) Locally search for better parameters by following gradient