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Rilevamento di bordi e angoli, Hough Transform

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(1)

Computer Vision - A Modern Approach Set: Linear Filters

Slides by D.A. Forsyth

Gradients and edges

• Points of sharp change in an

image are interesting:

– change in reflectance

– change in object

– change in illumination

– noise

• Sometimes called edge points

• General strategy

– determine image gradient

– now mark points where

gradient magnitude is

particularly large wrt

neighbours (ideally, curves of

such points).

(2)

There are three major issues:

1) The gradient magnitude at different scales is different; which should

we choose?

2) The gradient magnitude is large along thick trail; how

do we identify the significant points?

(3)

Computer Vision - A Modern Approach Set: Linear Filters

Slides by D.A. Forsyth

Smoothing and Differentiation

• Issue: noise

– smooth before differentiation

– two convolutions to smooth, then differentiate?

– actually, no - we can use a derivative of Gaussian filter

• because differentiation is convolution, and convolution is

associative

(4)

The scale of the smoothing filter affects derivative estimates, and also

the semantics of the edges recovered.

(5)

Computer Vision - A Modern Approach Set: Linear Filters

Slides by D.A. Forsyth

The Laplacian of Gaussian

• Another way to detect an

extremal first derivative is to

look for a zero second

derivative

• Appropriate 2D analogy is

rotation invariant

– the Laplacian

• Bad idea to apply a Laplacian

without smoothing

– smooth with Gaussian, apply

Laplacian

– this is the same as filtering

with a Laplacian of Gaussian

filter

• Now mark the zero points

where there is a sufficiently

large derivative, and enough

contrast

(6)

sigma=2

sigma=4

(7)

Computer Vision - A Modern Approach Set: Linear Filters

Slides by D.A. Forsyth

We still have unfortunate behaviour

at corners

(8)

We wish to mark points along the curve where the magnitude is biggest.

We can do this by looking for a maximum along a slice normal to the curve

(non-maximum suppression). These points should form a curve. There are

then two algorithmic issues: at which point is the maximum, and where is the

next one?

(9)

Computer Vision - A Modern Approach Set: Linear Filters

Slides by D.A. Forsyth

Non-maximum

suppression

At q, we have a

maximum if the

value is larger

than those at

both p and at r.

Interpolate to

get these

values.

(10)

Predicting

the next

edge point

Assume the

marked point is an

edge point. Then

we construct the

tangent to the edge

curve (which is

normal to the

gradient at that

point) and use this

to predict the next

points (here either

r or s).

(11)

Computer Vision - A Modern Approach Set: Linear Filters

Slides by D.A. Forsyth

Remaining issues

• Check that maximum value of gradient value is

sufficiently large

– drop-outs? use hysteresis

• use a high threshold to start edge curves and a low threshold to

continue them.

(12)

Notice

• Something nasty is happening at corners

• Scale affects contrast

(13)

Computer Vision - A Modern Approach Set: Linear Filters

(14)

fine scale

high

(15)

Computer Vision - A Modern Approach Set: Linear Filters

Slides by D.A. Forsyth

coarse

scale,

high

(16)

coarse

scale

low

(17)

Computer Vision - A Modern Approach Set: Linear Filters

Slides by D.A. Forsyth

Orientation representations

• The gradient magnitude is

affected by illumination

changes

– but it’s direction isn’t

• We can describe image patches

by the swing of the gradient

orientation

• Important types:

– constant window

• small gradient mags

– edge window

• few large gradient mags in

one direction

– flow window

• many large gradient mags

in one direction

– corner window

• large gradient mags that

swing

(18)

Computer Vision - A Modern Approach Set: Linear Filters

Slides by D.A. Forsyth

Representing Windows

• Types

– constant

• small eigenvalues

– Edge

• one medium, one small

– Flow

• one large, one small

– corner

• two large eigenvalues

H

 

I

 

I

T

window

(19)

Computer Vision - A Modern Approach Set: Linear Filters

(20)
(21)

Computer Vision - A Modern Approach Set: Linear Filters

(22)
(23)

Computer Vision - A Modern Approach Set: Linear Filters

Slides by D.A. Forsyth

Harris Corner detector

• Le regioni uniformi hanno tutti I gradienti uniformi

• I bordi lineari hanno tutti i gradienti allineati

• gli angoli sono I punti di congiunzione di due bordi e

quindi avra` I gradienti attorno allineati lungo due direttrici

fondamentali

(24)

Harris Corner detector

• Tutti e due gli auovalori di H sono > 0

• Rilevatore di angoli Harris: gli angoli sono massimi locali

della funzione R=det(H)-kTr(H)

2

(25)

Computer Vision - A Modern Approach Set: Linear Filters

(26)
(27)

Computer Vision - A Modern Approach Set: Linear Filters

Slides by D.A. Forsyth

(28)
(29)

Computer Vision - A Modern Approach Set: Linear Filters

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(31)

Computer Vision - A Modern Approach Set: Linear Filters

Slides by D.A. Forsyth

(32)
(33)

Computer Vision - A Modern Approach Set: Linear Filters

Slides by D.A. Forsyth

Normalized Cut

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asso

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cut

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