1

Formation et Analyse d’ImagesSession 5

Daniela Hall

4 November 2004

2

Course Overview

• Session 1: – Homogenous coordinates and tensor notation– Image transformations– Camera models

• Session 2:– Camera models– Reflection models– Color spaces

• Session 3:– Review color spaces– Pixel based image analysis

• Session 4:– Gaussian filter operators– Scale Space

3

Course overview

• Session 5:– Contrast description– Hough transform

• Session 6:– Kalman filter– Tracking of regions, pixels, and lines

• Session 7:– Stereo vision – Epipolar geometry

• Session 8: exam

4

Session overview

1. Contrast description

2. Hough transform

5

Feature extraction in computer vision

• The goal of feature extraction is – Reduce the amount of information– Simplify the interpretation– Preserve the information necessary for further

processing (task dependent)

• The ideal feature extraction alogrithm is invariant– Invariance: the observation is the same independent of

the view point and illumination.

• Contours are good candidates for invariant feature extraction

6

Contrast

• Contrast can be caused by– Change in surface orientation (discontinuity in depth)

• useful for description of polyhedrical objects

– Limit of a shadow• useful to recognize 3D object form

– Texture on a surface• independent of surface, can be used to recover the texture

– Reflections• depends on view point and illumination

7

Contrast detection process

Image processing

Image analysisimage

intermediateimage representation

image description

8

Contrast detection methods

• Search for extremum in 1st derivative

• Search for zero crossing in 2nd derivative

9

Contrast detection

Step edge Smoothedstep edge

First derivative Second derivative(white pos, yellow negative)

10

Image processing tools

• Let s(i), f(i) be two real discrete signals

• s(i) has Ns discrete values on the interval 0<=i<=Smax

• f(i) has Nf discrete values on the interval 0<=i<=Fmax

• Pmax=min{Smax,Fmax}

• s(i)=0 for all i<0, Smax<i

• f(i)=0 for all i<0, Fmax<i

• The scalar product of s(i) and f(i) is a scalar p

1max

0

)()()()()(),(P

ii

ifisifisifisp

11

Correlation and convolution

• Correlation is a scalar product with an offset k over all values of k.

• Convolution is a correlation where one of the signals is flipped (i-k)->(Fmax-1)-(i-k)

1max1max

0

)()()()()()(

)(),()()()(kF

ki

S

ii

kifisifkisifkis

ifkisifiskc

kF

ki

S

i

kiFfisiFfkis

iFfkisifiskr1max

max

1max

0max

max

))()1(()()1()(

))1((),()()()(

12

Contrast detection

• Roberst contrast detector

• Sobel contrast detector

• Difference filters

13

Roberts contrast detector

• Consists of two masks m1(i,j)= and m2(i,j)=1 0

0 -1

0 1-1 0

4)

),(

),((tan),(

),(),(),(

),(),(),(

1

21

22

21

1

0

1

0

jiE

jiEji

jiEjiEjiE

LkmLjkipmpjiEk L

nnnCompute correlation

for n=1,2:Contrast magnitude

Contrast direction

14

Roberst contrast detector

• The detector is very sensitive to high frequency noise.

• This sensitivity can be reduced by a low pass filtering.

15

Sobel detector

• very popular detector (Duda-Hart 1972)

m0(i,j)= m90(i,j)=

• This filter can be seen as a convolution of two components:

m0(i,j)= * m90(i,j)= *

1 0 -12 0 -21 0 -1

1 2 10 0 0-1 -2 -1

121

-1 0 1-101

1 2 1

16

Sobel detector

• As for Roberst the amplitude and orientation are computed by:

4)

),(

),((tan),(

),(),(),(

),(),(),(

0

901

290

20

1

0

1

0

jiE

jiEji

jiEjiEjiE

LkmLjkipmpjiEk L

nnnCompute correlation

for n=1,2:Contrast magnitude

Contrast direction

17

Examples

Original image Roberts detector

Sobel detector Smoothing bybinomial filters7

Differencefilter -1 0 1

Difference filter -1 1

18

Contrast detection by derivatives

• Steps:– Smoothing: suppression of noise– Computation of gradient magnitude and orientation– Local extremum detection by hysteresis thresholding– Edge segment extraction

19

Smoothing by binomial filters

• Binomial filters are the best approximation of a discrete Gaussian with integer values.

• Binomial filters are generated by n convolutions of [1 1]

• Binomial series: The binomial series is composed of the coefficients of the polynom

!)!(

!

)(

,

2/

2/,

mmn

n

m

nb

yxbyx

nm

n

nm

mmnnm

n

20

The binomial series• The coefficients of the binomial series can

be generated by Pascal’s triangle

1 1 1

1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 11 8 29 56 70 56 29 8 1

Level(n)012345678

Sum1248163264128256

Variance0¼½¾15/46/47/42

21

Binomial filters

• n convolutions of [1 1]

• Gain: 2n

• Variance: var{bn(m)}=n var{b1(m)} = n/4

• discrete Gaussian with sigma = 1: 1 4 6 4 1

• sigma= 2: 1 8 29 56 70 56 29 8 1

• sigma=0.5: 1 2 1

22

Discrete Gaussian filter

• discrete Gaussian with sigma = 1: 1 4 6 4 1

• Gain: 24=16

G(x,1.0) |0.004|0.054|0.242|0.399|0.242|0.054|0.004|

Example of sampling a 1D Gaussian (sigma=1, R=3) session 4

16 G(x,1.0) |0.064|0.864|3.872|6.384|3.872|0.864|0.064|

23

Contrast detection by derivatives

• Steps:– Smoothing: suppression of noise– Computation of gradient magnitude and orientation– Local extremum detection by hysteresis thresholding– Edge segment extraction

24

Difference filters

• For a discrete signal lim does not exist

• This is equivalent to s(n)*[1/2, 0, ½]

• This is equivalent to s(n)*[1,-1]• To compute derivatives of a discrete signal, do

convolution with the Gaussian derivative kernel (more details in session 4)

})()(

{lim)(

0 dt

dttsts

dt

tdsdt

2

)2()()(:2

nsns

dn

ndsdn

1

)1()()(:1

nsns

dn

ndsdn

25

Difference filters

• Alternative method to convolution with Gaussian derivative kernel are the difference filters.

• Smoothing followed by a difference filter is equivalent to the Sobel operator.

-1 0 1 -1 0 1

26

Examples

Original image Roberts detector

Sobel detector Smoothing bybinomial filters

Differencefilter -1 0 1

Difference filter -1 1

27

Examples

Original image Roberts detector

Sobel detector Smoothing bybinomial filters7

Differencefilter -1 0 1

Difference filter -1 1

28

Contrast detection by derivatives

• Steps:– Smoothing: suppression of noise– Computation of gradient magnitude E(i,j) and direction theta– Local extremum detection and hysteresis thresholding– Edge segment extraction

29

Local extremum detection

• Find direction of maximum gradient. Without loss of generality, let the direction be (1, 0)T

• Check if E(i,j) is a maximum in this direction• This means:

– Find contrast points C(i,j) that fulfill

else0

),(),(),(

and 0),(1

),( jjiiEjiEjjiiE

jiEjiC

30

Hysteresis thresholding

• Problem: some detectors produce weak edges which should be eliminated.

• Solution: hysteresis thresholding• Properties: eliminates weak edges below a low

threshold, but not if they are connected to an edge above a high threshold.

weak edge

strongedge

C(i,j) C(i,j) afterchaining

31

Hysteresis thresholding

• Uses a high and a low threshold– low threshold th0: sensitive to noise, but

produces contour candidates– high threshold th1: indicates strong edges

1. Compute map F(i,j) with two thresholds th0, th1

else0

th1j)E(i, th0if1

|j)E(i,| th1if2

),( jiF

32

Hysteresis filtering

2. Computing connected components of edge elements

An edge is selected when it contains at least one strong edge element (with label 2).

Edges consisting of only weak edge elements are discarded.

33

Hysteresis filtering

• The thresholding and chaining can be computed in one step.

• Start with the high threshold image.

• For each strong edge continue adding (8 or 4) connected pixels with E(i,j)>=th0.

34

Example

original image Contourimage

Low threshold image

High thresholdimage

35

Contrast detection methods

• Search for extremum in 1st derivative: ok

• Search for zero crossing in 2nd derivative

36

Contrast detection by 2nd derivatives

• The Laplacien is a isotropic filter that is used for the contrast detection.– Lap(i,j)= Lxx(i,j) + Lyy(i,j)– with Lxx(i,j)= I(i,j) * Gxx(sigma)

• As for Gaussians there exists a integer approximation of the Laplacian.– Lxx(i,j) : [ 1 -1] * [1 -1] = [-1 2 -1] – Lxx(i,j)+Lyy(i,j):

* + * =0 -1 0-1 4 -10 -1 0

0 0 0 0 1 00 0 0

-1 2 -1

0 0 0 0 1 00 0 0

-12-1

37

Gradient norm and Laplacian

38

Example

image Lap

Differenceimage Lx

Differenceimage Ly

39

ExampleDifference image Lx Laplacian

40

Example

image Lap

Differenceimage Lx

Differenceimage Ly

41

Contour detection by 2nd derivatives

• In theorie:– Contours can be detected more precisely using

zero-crossing of 2nd derivative. – Sub-pixel precision is possible by interpolation.– zero crossing of 2nd derivatives produce closed

contours.

• In real life:– The zero crossing of the 2nd derivative detect

also many small unstable regions.

42

Detection of zero crossing

• The extrema of the 1st derivative correspond to zero crossing of the 2nd derivative.

• To generate a contour image C(i,j) proceed as follows:– consider 4-connected neighbors (N,S,W,E)

else0

0||)()(or

0|WE|)()(1

),( NSSsignNsign

EsignWsignjiC

43

Detection of zero crossing

• For a stabler detection add a term that is related to the strength of the edge.

• Disadvantage: this measure may break a closed contour. The contours are no longer closed.

else0

thres|S- N|0||)()(or

thres|E-W |0|WE|)()(1

),( NSSsignNsign

EsignWsignjiC

44

Session overview

1. Contrast description

2. Hough transform

45

Hough transform

• The Hough transform is a standard tool in image analysis that allows recognition of global patterns in an image space by recognition of local patterns in a transformed parameter space.

• It is particularly useful when the patterns one is looking for are sparsely digitized, have ``holes'' and/or the pictures are noisy.

• The basic idea of this technique is to find curves that can be parameterized like straight lines, polynomials, circles, etc., in a suitable parameter space.

Hough59 P.V.C. Hough, Machine Analysis of Bubble Chamber Pictures, International Conference on High Energy Accelerators and Instrumentation, CERN, 1959.

46

Hough transform

• Although the transform can be used in higher dimensions the main use is in two dimensions to find, e.g. straight lines, centres of circles with a fixed radius, parabolas y = ax2 + bx + c with constant c, etc.

• As an example consider the detection of straight lines in an image. We assume them parameterized in the form: y=mx+c

• Each image point (x,y) gives rise to a certain number of possible values for m and c.

• This set of values forms a line c=-mx+y in the Hough space (m,c)• When the contour points are aligned, the corresponding lines in the

Hough space have the same intersection point (m0,c0). • A point in hough space indicates a line in the image space.

source: http://rkb.home.cern.ch/rkb/AN16pp/node122.html

47

Hough transform

• Computation:– Initialise histogram h(m,c) (this represents the Hough

space)

– for each image point (i,j)• if C(i,j)=1, then compute all possible couples (m,c) that pass

through the point (i,j).

• increment all corresponding cells by 1.

– A local maximum (m0,c0) in h(m,c) indicates that the image points are aligned to the line y=m0 x+c0

48

Hough transform

• Problems:– Care has to be taken when one quantizes the parameter

space. • When the bins table are chosen too fine, the local maximum

can be dispersed over several bins.

• When the quantization is not fine enough, on the other hand, nearly parallel lines which are close together will lie in the same bin.

– The coefficients m,c are not uniform in orientation• Solution: use a different parameterization (p,theta):

x sin(theta)-y cos(theta) + p =0

49

Hough transform

– Use a different parameterization (p,theta): x sin(theta)-y cos(theta) + p =0

– p is the distance from the origin and theta the angle with the normal

– Here each point in the image space corresponds to a sinusoidal curve in the Hough space. The intersection of the sinusoidal curves in the Hough space correspond to a line in the image space.

50

Hough transform

• Computation:– Initialise histogram h(p,theta) (this represents the

Hough space)

– for each image point (i,j)• if C(i,j)=1, then

– for all 0<theta<2pi compute p=-i sin(theta)+j cos(theta)

– set h(p,theta) = h(p,theta)+1

– A local maximum (p0,theta0) in h(p,theta) indicates that the image points are aligned to the line

p0=-x sin(theta0)+y cos(theta)

51

Generalised Hough transform

• You can compute the hough transform for any pattern that can be parameterized, such as circles, lines, parabolas.

• For circles:(x-a)2 + (y-b)2 = r2

• We use a hough space h(a,b,r).• Each image point corresponds to a cone in space (a,b,r)• For a fixed r, each image point corresponds to a circle in hough space.• Algorithm:

– for each image point and each radius r>0 compute the corresponding circle in hough space. Increment all cells.

– After filling the histogram, locate the local maximum (a0,b0,r0). This maximum corresponds to the circle in the image with center (a0, b0) and radius r0.