Limitations of Thresholding

Transcription

1 Limitations of Thresholding Wh can we segment images much better b ee than through thresholding processes? We might improve results b considering image contet: Surface Coherence

2 Gradient.illusion.arp.jpg Aha! Humans are suckers for contet!

3 b Adrian Pingstone, based on the original created b Edward H. Adelson

4 b Adrian Pingstone, based on the original created b Edward H. Adelson

5 Chapter 3 in Machine Vision b Jain et al.

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7 Note on Performance Assessment In real-life, we use two or even three separate sets of test data:. A training set, for tuning the algorithm 2. A validation set for tuning the performance score 3. An unseen test set to get a final performance score on the tuned algorithm

8 Image Transformations 0

9 Outline Gre-level transformations Histogram equalization Geometric transformations Affine transformations Interpolation Warping and morphing

10 Gre-level Transformations Start with I, I : R n Z n 256 or I : R Change the image gre level in each piel b a fied mapping f: R R : R I 2 (, ) = f(i (,)) I 2 I

11 Linear contrast stretching f is a linear function: f() = + We must preserve the range of gre level values as {0,,, 255}. 3

12 =.0, =-40 5

13 =.0, =40 7

14 =0.5, =0 20

15 =2.0, =0 2

16 Non-linear Gamma Correction Non-linear gre-level transformations are useful too Gamma correction adjusts for differences between camera sensitivit and the human ee f() = A A=255 (- ) ensures that the gre scale range is unchanged 23

17 =

18 =0.5 25

19 =2 27

20 =4 28

21 For eample, CRT s would have gamma = 2.5, so preappl a gamma = /

22 Histogram Equalization Tries to use all the gre levels equall often The gre level histogram is then flat Use the cumulative histogram for f 30

23 % of piels Cumulative histogram grascales 3

24 Histogram Equalization function eqim = histeq(image) [X,Y] = size(image); % Construct the cumulative histogram for i=0:255 cumhist(i) = sum(sum(image <= i))/(x*y); end % Use it to transform the gre level in each piel. eqim = fi(255*cumhist(image)); 32

25 Equalized Image Original Equalized 33

26 Equalized Histograms Original Original Grascales Histogram after equalization Grascales Cumulative histogram after equalization 34

27 Unassessed Eercise Write matlab functions for linear gre-level transformations and gamma correction. Tr moderate and etreme settings on an image and observe the effects the have. Plot the gre-level histograms before and after the gre-level transformations. 35

28 Geometric Transformations Change the location of image features 36

29 Geometric Transformations I is the original image, I 2 is the warped image. I 2 (, ) = I (, ) (, ) = T(, ) (, ) T (, ) I I 2 37

30 Affine Transformations = a + b + t = c + d + t ' ' a c b d t t Translation, scaling, rotation, shear. 38

31 Translation a c b d 0 0, t t W H / /

32 Scaling a c b d , t t W H / /

33 Stretch a c b d t W /, 0 t 0 4 4

34 Rotation a c b d cos sin sin cos, / 4, t t ( W (cos ( W sin ) H H (cos sin ) / 2 )) / 2 42

35 Shear (along ais) a c b d s 0, s t t 0 sh / 2 43

36 Homogeneous Coordinates ' ' t d c t b a t t d c b a ' ' = a + b + t = c + d + t Remember:

37 Homogeneous Coordinates ' ' t t a c 0 b d Subsequent operations are inserted here, b pre-multipling 45

38 Implementation Alwas use the inverse transformation: function warpedimage = transform(image, T) [W,H] = size(image); warpedimage = zeros(w,h); invt = invert(t); for =:W for =:H warpedimage(,)=image(invt(,)); end end 46

39 Interpolation Usuall, (, ) = T - (, ) are not integer coordinates. We estimate I (, ) b interpolation from surrounding positions. 47

40 Nearest Neighbour Interpolation 2 T - 2 Take the value at the nearest grid point: I (, ) = I 2 ( 2, 2 ) 48

41 Bilinear interpolation 2 Weighted average from neighbouring grid points: 2 I (, ) = I ( 2, 2 ) + ( ) I ( 2, ) + ( ) I (, 2 ) + ( ) ( ) I (, ) =, =. 49

42 Higher order interpolation Quadratic interpolation fits a bi-quadratic function to a 33 neighbourhood of grid points Cubic interpolation fits a bi-cubic function to a 44 neighbourhood. 50

43 interp2 The matlab function interp2 does nearest neighbour, linear and cubic interpolation. Look it up and tr it out! 5

44 Interpolation comparison Nearest neighbour Bilinear Cubic 52

45 Warps Polnomial transformations, e.g., quadratic transformations: = a 0 + a + a 2 + a a 4 + a 5 2 = b 0 + b + b 2 + b b 4 + b 5 2 Affine transformations map lines to lines. The are first-order polnomial transformations. Higher-order polnomials can bend lines. 53

46 Quadratic warp eample 54

47 55 Control point warps Moves some piels to specified locations. Interpolate the displacement at intermediate positions. Find a polnomial warp P that maps: ), ( ), ( : ), ( ), ( ), ( ), ( ' ' ' 2 ' ' ' m m m m

48 56 Control point warps A = X P Least squares estimate of P is (m >= 6): P = (X T X) - X T A ' ' ' 2 ' 2 ' ' : : : : : : : : b a b a b a b a b a b a m m m m m m m m

49 Eample Two piel displacement 57

50 Five Piel Displacement 58

51 Etreme warp 59

52 Overdetermined 60

53 Applications Special effects Film industr Computer games Image registration Medical imaging Securit 6

54 Image Morphing How do we get the i-th image in the sequence? 62

55 Landmarks 63

56 Image Registration Determines the transformation that aligns two similar images 64

57 Image Similarit We find the transformation that maimises the similarit of the images. A simple similarit measure is: S( I, I 2 ) I ( I ( ) I 2 ( )) Man others are used including the crosscorrelation and mutual information

58 Registration eample 66

59 Registration We have to search for the transformation the minimizes S. Simple in the D eample. For more comple transformations, we use numerical optimization (see the matlab function fminunc for eample). 67

60 Feature-Based Image Metamorphosis Beier & Neel Siggraph 92 (Quick overview)

61 Eplicit Correspondences Individual face Average face

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