computer vision. applications and algorithms in cv tutorial 3: grouping & fitting primitives...

Computer vision

Applications and Algorithms in CV

Tutorial 3:Grouping & Fitting

Primitives detection

Problem:In automated analysis of digital images, a sub-problem often arises of detecting simple

shapes, such as straight lines, circles or ellipses

introduction



Primitives detection

Solution?

Edge detection

introduction



Finding straight lines

An edge detector can be used as a pre-processing stage to obtain points (pixels) that are on the desired curve in the image space

Due to imperfections in either the image data or the edge detector, there may be missing points on the desired curves as well as spatial deviations between the ideal line/circle/ellipse and the noisy edge points as they are obtained from the edge detector

It is often non-trivial to group the extracted edge features to an appropriate set of lines, circles or ellipses

introduction



Grouping & Fitting methods

1. Global optimization / Search for parametersLeast squares fitTotal least squares fitRobust least squaresIterative closest point (ICP)Etc.

introduction

2. Hypothesize and testHough transformGeneralized Hough transformRANSACEtc.

2 approaches for grouping & fitting




1. Global optimization / Search for parametersLeast squares fit

introduction

Applications and Algorithms in CV Least square fitTutorial 3:

Grouping & Fitting

Least square line fitting

Data: (x1, y1), …, (xn, yn)

2 2 0T TdE

dP A Ap A y

22 1 1

2

1

1

1 2( ) ( ) ( )

1

n T T Ti ii

n n

x ym m

E x yb b

x y

Ap y y y Ap y Ap Ap

n

i ii bxmyE1

2)(

(xi, yi)

yAAApyAApA TTTT 1 Matlab: p = A \ y;

Line equation: yi = m xi + b Find (m, b) to minimize:


Grouping & Fitting


Question: Will least squares work in this case?• Fails completely for vertical lines

•No. same edge, different parametersQuestion: is LSQ invariant to rotation?

m1,b1

m2,b2




1. Global optimization / Search for parameters

Total least squares fit

introduction



Grouping & Fitting

Total least square

(xi, yi)

ax+by+c=0

Unit normal:2

1

2 2

( )

. . 1

n

i iiE ax by c

s t n a b

Find (a, b, c) to minimize the sum of squared perpendicular distances

2 2

distance from a point , to a line 0 :i i

i i

x y ax by c

ax by cd

a b

2 2

,ˆ

a bn

a b


Grouping & Fitting

0)(21

n

i ii cybxac

E1 1

n n

i ii i

a bc x y ax by

n n

2

1 12

1( ( ) ( ))

n T Ti ii

n n

x x y ya

E a x x b y yb

x x y y

p A Ap

*Solution is eigenvector corresponding to smallest eigenvalue of ATA

minimize s.t. 1T T T p A Ap p p

Total least square


Grouping & Fitting

Recap: Two Common Optimization Problems

Problem statement Solution

minimize

s.t. 1

T T

T

x A Ax

x x 1... 1

[ , ] eig( )

:

T

n nn

v A A

x v

Problem statement Solution

bAx osolution t squaresleast bAx \

2 minimize bAx bAAAx TT 1

(matlab)

LSQ & TLSQ analitic solutions


Grouping & Fitting


Least squares fit to the red points


Grouping & Fitting

Least square line fitting:: robustness to noise

Least squares fit with an outlier

Squared error heavily penalizes outliers


Grouping & Fitting

Least square line fitting:: conclusions

Good

• Clearly specified objective

• Optimization is easy (for least squares)

Bad

• Not appropriate for non-convex objectives– May get stuck in local minima

• Sensitive to outliers– Bad matches, extra points

• Doesn’t allow you to get multiple good fits– Detecting multiple objects, lines, etc.




introduction

2. Hypothesize and testHough transform


Applications and Algorithms in CV Hough TransformTutorial 3:

Grouping & Fitting

The purpose of the Hough transform is to address the problem of primitives detection by performing an explicit voting procedure over a set of parameterized image objects

Hough transform


Grouping & Fitting

A line y = mx + b can be described as the set of all points comprising it: (x1, y1), (x2, y2)...

Hough transform:: Detecting straight lines

Instead, a straight line can be represented as a point (b, m) in the parameter space

Question: does {b,m} (slope and intercept) indeed the best parameter space?


Grouping & Fitting

Hough transform:: Line parameterization

Hough transform is calculated in polar coordinates

The parameter represents the distance between the line and the origin

is the angle of the vector from the origin to this closest point


Grouping & Fitting

Using this parameterization, the equation of the line can be written as:

which can be rearranged to:

Hough transform:: Line parameterization


Grouping & Fitting

Hough transform:: Example

Question: Is there a straight line connecting the 3 dots?


Grouping & Fitting

Hough Solution:

1. For each data point, a number of lines are plotted going through it, all at different angles () (shown as solid lines)

2. Calculate For each solid line from step 1 (shown as dashed lines)

3. For each data point, organize the results in a table


point1 point2 point3


Grouping & Fitting

Hough Solution:

4. Draw the lines received for each data point in the Hough space

5. Intersection of all lines indicate a line passing through all data points (remember: each point in Hough space represent a line in the original space)



Grouping & Fitting

features votes



Grouping & Fitting

Hough transform:: other shapes


Grouping & Fitting

features

Hough transform:: effect of noise


Grouping & Fitting

features votes

Hough transform:: effect of noise


Grouping & Fitting

features votes

Hough transform:: random points


Grouping & Fitting

Random points:: Dealing with noise

Grid resolution tradeoff Too coarse: large votes obtained when too many different lines correspond to a single

bucket Too fine: miss lines because some points that are not exactly collinear cast votes for different

buckets

Increment neighboring bins Smoothing in accumulator array

Try to get rid of irrelevant features Take only edge points with significant gradient magnitude


Grouping & Fitting

Good1. Robust to outliers: each point votes separately2. Fairly efficient (often faster than trying all sets of parameters)3. Provides multiple good fits

Bad4. Some sensitivity to noise5. Bin size trades off between noise tolerance, precision, and speed/memory6. Not suitable for more than a few parameters (grid size grows exponentially)

Random points:: conclusions

RANSAC

(RANdom SAmple Consensus):Learning technique to estimate parameters of a model by random sampling of observed data

RANSAC

Applications and Algorithms in CV RANSACTutorial 3:

Grouping & Fitting

RANSAC:: algorithm

Algorithm:1. Sample (randomly) the number of points required to fit the model2. Solve for model parameters using samples 3. Score by the fraction of inliers within a preset threshold of the model

Repeat 1-3 until the best model is found with high confidence


Grouping & Fitting

RANSAC:: line fitting example

Algorithm:1. Sample (randomly) the number of points required to fit the model 2. Solve for model parameters using samples 3. Score by the fraction of inliers within a preset threshold of the model



Grouping & Fitting




6IN




Grouping & Fitting


14IN


Grouping & Fitting

:initial number of points (Typically minimum number needed to fit the model) :number of RANSAC iterations

: probability of choosing inliers at least in some iteration :probability of choosing an inliers :probability that all s points are inliers:probability that at least one point is an outlier

: probability that the algorithm never selects a set of s inliers

log 1 / log 1 sN p

RANSAC:: parameter selection


Grouping & Fitting

GoodRobust to outliersApplicable for larger number of parameters than Hough transformParameters are easier to choose than Hough transform

BadComputational time grows quickly with fraction of outliers and number

of parameters Not good for getting multiple fits

RANSAC:: conclusions

computer vision. applications and algorithms in cv tutorial 3: grouping & fitting primitives...

Documents

fitting slide

square slide

fitting total

cv tutorial

square fit tutorial

fitting recap

fit introduction slide

square line fitting