5.4 Generalized Hough Transform - cs.cmu.edu

Generalized Hough Transform16-385 Computer Vision

Hough Circles

Finding Circles by Hough Transform

Equation of Circle:

222 )()( rbyax ii =−+−

If radius is known:

),( baAAccumulator Array

(2D Hough Space)

-5 -4 -3 -2 -1 0 1 2 3 4

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-5 -4 -3 -2 -1 0 1 2 3 4

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a

b

(x, y)

Image space Parameter space

(x� a)2 + (y � b)2 = r

2 (x� a)2 + (y � b)2 = r

2

parameters

variables

parameters

variables

-5 -4 -3 -2 -1 0 1 2 3 4

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-5 -4 -3 -2 -1 0 1 2 3 4

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a

b

(x, y)

(x� a)2 + (y � b)2 = r

2 (x� a)2 + (y � b)2 = r

2

parameters

variables

parameters

variables

-5 -4 -3 -2 -1 0 1 2 3 4

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-5 -4 -3 -2 -1 0 1 2 3 4

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a

b

(x� a)2 + (y � b)2 = r

2 (x� a)2 + (y � b)2 = r

2

parameters

variables

parameters

variables

-5 -4 -3 -2 -1 0 1 2 3 4

-4

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-5 -4 -3 -2 -1 0 1 2 3 4

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(x� a)2 + (y � b)2 = r

2 (x� a)2 + (y � b)2 = r

2

a

b

parameters

variables

parameters

variables

-5 -4 -3 -2 -1 0 1 2 3 4

-4

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-5 -4 -3 -2 -1 0 1 2 3 4

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a

b

(x� a)2 + (y � b)2 = r

2 (x� a)2 + (y � b)2 = r

2

parameters

variables

parameters

variables

Finding Circles by Hough Transform

Equation of Circle:

222 )()( rbyax ii =−+−

If radius is not known: 3D Hough Space!

Use Accumulator array ),,( rbaA

What is the surface in the hough space?

Finding Circles by Hough Transform

Equation of Circle:

222 )()( rbyax ii =−+−

If radius is not known: 3D Hough Space!

Use Accumulator array ),,( rbaA

What is the surface in the hough space?

Using Gradient InformationGradient information can save lot of computation:

Edge Location

Edge Direction

Need to increment only one point in accumulator!!

iφ),( ii yx

Assume radius is known:

φ

φ

sincosrybrxa

−=

−=

-5 -4 -3 -2 -1 0 1 2 3 4

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-5 -4 -3 -2 -1 0 1 2 3 4

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a

b

(x, y,�)

(x� a)2 + (y � b)2 = r

2 (x� a)2 + (y � b)2 = r

2

parameters

variables

parameters

variables

-5 -4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

1

2

3

4

-5 -4 -3 -2 -1 0 1 2 3 4

-4

-3

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-1

1

2

3

4

a

b

(x, y,�)

(x� a)2 + (y � b)2 = r

2 (x� a)2 + (y � b)2 = r

2

parameters

variables

parameters

variables

Pennie Hough detector Quarter Hough detector

Pennie Hough detector Quarter Hough detector

Can you use Hough Transforms for other objects, beyond lines and circles?

Generalized Hough Transform

Generalized Hough TransformFind Object Center given edges

Create Accumulator Array

Initialize:

For each edge point

For each entry in table, compute:

Increment Accumulator:

Find Local Maxima in

),( cc yxA

),(0),( cccc yxyxA ∀=

),,( iii yx φ

1),(),( += cccc yxAyxA

),( cc yxA

ik

ikic

ik

ikic

ryy

rxx

α

α

sin

cos

+=

+=

),( cc yx ),,( iii yx φ

r̄ik

Do you have to use edge detectors to vote in Hough Space?

c

Template

A.Train phase:

1.Get features

2.Store all displacements of feature from center

B.Test phase:

1.Get features & lookup displacements

2.Vote for center location

Template

A.Train phase:

1.Get features

2.Store all displacements of feature from center

B.Test phase:

1.Get features & lookup displacements

2.Vote for center location

Test image

A.Train phase:

1.Get features

2.Store all displacements of feature from center

B.Test phase:

1.Get features & lookup displacements

2.Vote for center location

Application of Hough Transforms

Detecting shape features

F. Jurie and C. Schmid, Scale-invariant shape features for recognition of object categories, CVPR 2004

Original images

Laplacian circles Hough-like circles

Which feature detector is more consistent?

Robustness to scale and clutter

Object detection

training image

visual codeword withdisplacement 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

Index displacements by “visual codeword”

1. get featuresTrain phase

2. store displacementsTrain phase

Test phase

The Hough transform …

Deals with occlusion well?

Detects multiple instances?

Robust to noise?

Good computational complexity?

Easy to set parameters?