Image Transformations

Transcription

1 Image Transformations

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

3 Gre-level transformations Changes the image gre level in each piel b a fied mapping f: : I (, ) = f(i (,)) I I

4 Linear contrast stretching f is a linear function: f() = + We must preserve the range of gre level values as {0,,, 55}. 4

5 =.0, =-40 6

6 =.0, =40 8

7 =0.5, =0

8 =.0, =0

9 Non-linear Gamma Correction Some non-linear gre-level transformations are useful. Gamma correction adjusts for differences between camera sensitivit and the human ee. f() = A A=55 (- ) ensures that the gre scale range is unchanged. 4

10 =0.5 6

11 = 8

12 Histogram Equalization Tries to use all the gre levels equall often. The gre level histogram is then flat. Use the cumulative histogram for f. 0

13 % of piels Cumulative histogram grascales

14 Histogram Equalization function eqim = histeq(image) [X,Y] = size(image); % Construct the cumulative histogram for i=0:55 cumhist(i) = sum(sum(image <= i))/(x*y); end % Use it to transform the gre level in each piel. eqim = fi(55*cumhist(image));

15 Equalized Image 3

16 Equalized histograms grascales Histogram before equalization grascales Cumulative histogram after equalization 4

17 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. 5

18 Geometric Transformations Change the location of image features 6

19 Geometric Transformations I is the original image, I is the warped image. I (, ) = I (, ) (, ) = T(, ) (, ) T (, ) I I 7

20 Affine Transformations = a + b + t = c + d + t ' ' a c b d t t Translation, scaling, rotation, shear. 8

21 Translation a c b d 0 0, t t W H / / 4 4 9

22 Scaling a c b d 0 0, t t W H / / 30

23 Stretch a c b d t W /, 0 t 0 4 3

24 Rotation a c b d cos sin sin cos, / 4, t t ( W (cos ( W sin ) H H (cos sin ) / )) / 3

25 Shear (along ais) a c b d s 0, s t t 0 sh / 33

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

27 Homogeneous Coordinates ' ' t t a c 0 b d Subsequent operations are inserted here, b pre-multipling 35

28 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 36

29 Interpolation Usuall, (, ) = T - (, ) are not integer coordinates. We estimate I (, ) b interpolation from surrounding positions. 37

30 Nearest Neighbour Interpolation T - Take the value at the nearest grid point: I (, ) = I (, ) 38

31 Bilinear interpolation Weighted average from neighbouring grid points: I (, ) = I (, ) + ( ) I (, ) + ( ) I (, ) + ( ) ( ) I (, ) =, =. 39

32 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. 40

33 interp The matlab function interp does nearest neighbour, linear and cubic interpolation. Look it up and tr it out! 4

34 Interpolation comparison Nearest neighbour Bilinear Cubic 4

35 Warps Polnomial transformations, e.g., quadratic transformations: = a 0 + a + a + a 3 + a 4 + a 5 = b 0 + b + b + b 3 + b 4 + b 5 Affine transformations map lines to lines. The are first-order polnomial transformations. Higher-order polnomials can bend lines. 43

36 Quadratic warp eample 44

37 45 Control point warps Moves some piels to specified locations. Interpolate the displacement at intermediate positions. Find a polnomial warp P that maps: ), ( ), ( : ), ( ), ( ), ( ), ( ' ' ' ' ' ' m m m m

38 46 Control point warps A = X P Least squares estimate of P is (m >= 6): P = (X T X) - X T A ' ' ' ' ' ' : : : : : : : : b a b a b a b a b a b a m m m m m m m m

39 Eample Two piel displacement 47

40 Five Piel Displacement 48

41 Etreme warp 49

42 Overdetermined 50

43 Applications Special effects Film industr Computer games Image registration Medical imaging Securit 5

44 Image Morphing How do we get the i-th image in the sequence? 5

45 Landmarks 53

46 Image Registration Determines the transformation that aligns two similar images 54

47 Image Similarit We find the transformation that maimises the similarit of the images. A simple similarit measure is: S( I, I ) I ( I ( ) I ( )) Man others are used including the crosscorrelation and mutual information. 55

48 Registration eample 56

49 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). 57

50 Feature-Based Image Metamorphosis Beier & Neel Siggraph 9 (Quick overview)

51 Eplicit Correspondences Individual face Average face

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