Elaborazione delle Immagini Informazione Multimediale. Raffaella Lanzarotti

Transcription

1 Elaborazione delle Immagini Informazione Multimediale Raffaella Lanzarotti

2 HOUGH TRANSFORM Paragraph of the book at link: szeliski.org/book/drafts/szeliskibook_ _draft.pdf Thanks to Kristen Grauman 2

3 Borders are not lines How to locate lines? 3

4 Difficulties Extra border points (clutter): which points corresponds to straight lines? Missing. only parts of the line is visible, other are missing: how to fill in holes?

5 Hough Transform Again: Which points correspond to lines and which to other things? Given the points corresponding to the line, which is the line? Hoe many lines are there? Which points correspond to which line? The Hough transform is a voting technique that allows to answer to these questions. IDEA: 1. For each border point store the votes for each possible line passing trough the point. 2. Look for the lines that received the most votes.

6 Find lines in an image: Hough space y b b 0 x m 0 Image Space Hough space or parameter space m A line in an image corresponds to a point in the Hough space

7 Find lines in an image: Hough space y b y 0 x 0 x Image space m Hough space (parameters) mapping a point (x 0, y 0 ): A point in an image is mapped to a line in the Hough space, the solution of b = -x 0 m + y 0

8 Find lines in an image: Hough space y y 0 (x 0, y 0 ) (x 1, y 1 ) b x 0 Image space x b = x 1 m + y 1 m Hough space (parameters) Which are the parameters of the line passing through (x 0, y 0 ) and (x 1, y 1 )? It is the intersection of the lines in the Hough space: b = x 0 m + y 0 b = x 1 m + y 1

9 Find lines in an image: Hough space y b Image space x m Hough space (parameters) How to use this evidence to look for the most probable parameters (m,b)? Each edge point votes for a couple of parameters (m,b) compatible with it (lines passing through that point) The votes are cumulated in s discrete matrix The parameters that have accumulated to the most votes indicate the most probable lines in the image

10 Polar representation of a line Problems with the representation (m,b): m is infinitive for vertical lines. [0,0] θ Columns x : perpendicular distance of the line to the origin Rows y θ : angle between the x axis and the perpendicular to the line = x cos y sin Point in the image space à Sinusoid in the Hough space

11 Hough Transform 11

12 Point transformation 12

13 Transformation of aligned points What does it happen if in the image there are aligned points? In the parameters space, the curves corresponding to the aligned points intersects in a point: it corresponds to the couple of parameters of the line! 13

14 Transformation of aligned points 14

15 Implementation of the Hough transform Let consider a discretized plane of the parameters (, ) This corresponds to a matrix H(m,n) The intervals of the parameters have to be determined on the basis of the image characteristics. 15

16 Implementation of the Hough transform Typically, max apple apple max, /2 apple apple 2 where max =0.5 (NR 2 + NC 2 ) 1/2 being (NR, NC) the image dimensions. Quantization levels: max(nr, NC) 16

17 Implementation of the Hough transform 17

18 Algorithm 1.Initialize H(m,n) to 0 2.For each pointp 2 F, P =(x, y) 1.For each n varying in ( 1.evaluate 2.determine the index m corresponding to (n) 3.increment H(m,n) 2.end 3.end 4.Detect the local maxima on H /2, /2) with step d (n) =x cos( n )+y sin( n ) 18

19 Example 1 point 19

20 Example 2 points 20

21 Example ρ>0, θ>0 21

22 Example ρ>0, θ<0 22

23 Example ρ<0, θ>0 23

24 Example ρ<0, θ<0 24

25 Example Dashed line 25

26 Example Text with different alignments 26

27 Example polygon 27

28 Example polygon, without noise 28

29 Example polygon, without noise: Matrix of accumulation 29

30 Example polygon, with noise 30

31 Example polygon, with noise 31

32 Example polygon, with noise: Inverse transform Threshold:

33 Example: Find lines in real scenes 33

34 Cont. Threshold: 101 Threshold: 140 Threshold:

35 Variants Weight the point votes with the corresponding gradient magnitude Change the sampling (, ) to vary the resolution Reduce the search field on the basis of the characteristics of the searched object (e.g. reduce the possible ) The same procedure can be used to find other parametric shapes as circles, squares, parabolas, 35

36 HOUGH TRANSFORM FOR CIRCLES 36

37 Hough transform to look for other shapes Look for circumferences exploiting the circumference equation: (x a) 2 +(y b) 2 = c 2 Simple case: the parameter space is (a,b) fixing the radius of the circumferences c General case: the parameter space is (a,b,c) Generalization proposed by Ballard to look for objects with any shape 37

38 Hough transform for circumferences Circumference: center (a,b) and radius r ( xi a) + ( yi b) = r FIRST CASE: radius r given, look for (a, b): each edge point votes for all the circumference centers passing through it b Image space Hough space a

39 Hough transform for circumferences Circumference: center (a,b) and radius r ( xi a) + ( yi b) = r FIRST CASE: radius r given, look for (a, b): each edge point votes for all the circumference centers passing through it Intersection: most voted center Image space Hough space

40 Hough transform for circumferences Circumference: center (a,b) and radius r ( xi a) + ( yi b) = r SECOND CASE: both the radius and the center are not known r? Image space a Hough space b Kristen Grauman

41 Hough transform for circumferences Circumference: center (a,b) and radius r ( xi a) + ( yi b) = r SECOND CASE: both the radius and the center are not known r b Image space a Hough space Kristen Grauman

42 Algorithm: Hough transform for circumferences For each edge point (x,y) : For each possible value of a: For each possible value of b: r = p (x a) 2 +(y b) 2 end end H[a,b,r] += 1

43 Observation From the trigonometry we have that, if we know the coordinates of (x,y) and r, we can determine (a,b) as follows: a (x,y) = cos =sin r a = x r cos(θ) b = y + r sin(θ) (a,b) x

44 Alternative algorithm for the Hough transform for circumference For each edge point (x,y) : For each possible value of r: For each direction θ: end end a = x r cos(θ) % columns b = y + r sin(θ) % rows H[a,b,r] += 1 Time complexity? Check out online demo :

45 Ex: localization of circumference with Hough Original Edges Votes: Penny NB: parameter space: (a,b,c), that is the radius can vary

46 Ex: localization of circumference with Hough Original Edges Votes: Penny NB: parameter space: (a,b), the radius is fixed in correspondence to Penny

47 EX: Iris detection Gradient+threshold Hough space (fixed radius) Max detections Hemerson Pistori and Eduardo Rocha Costa hough-circles.html

48 Ex: Iris detection An Iris Detection Method Using the Hough Transform and Its Evaluation for Facial and Eye Movement, by Hideki Kashima, Hitoshi Hongo, Kunihito Kato, Kazuhiko Yamamoto, ACCV 2002.

49 Voting: practical tricks Minimize false edges Choose a good discretization Too detailed? Too rough Vote also for the neighborhood (smoothing the accumulator)

50 Hough transform: pros and the cons Pros It is robust to occlusions and holes: all points are processed separately It is robust to noise: noisy points have low probability to be aligned It can localize more parametric object in a single step Cons The computational complexity grows exponentially with the model parameter number Quantization: the behavior is strongly dependent on quantization step