1 / 57

Geometric Operations and Morphing

Geometric Operations and Morphing . Geometric Transformation. Operations depend on pixel’s Coordinates. Context free. Independent of pixel values. (x,y). (x’,y’). I(x,y). I’(x’,y’). (x’,y’). Example: Translation. (x,y). I(x,y). I’(x’,y’). Forward Mapping. Forward mapping:. Target.

zoltan
Download Presentation

Geometric Operations and Morphing

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Geometric Operations and Morphing

  2. Geometric Transformation • Operations depend on pixel’s Coordinates. • Context free. • Independent of pixel values. (x,y) (x’,y’) I(x,y) I’(x’,y’)

  3. (x’,y’) • Example: Translation (x,y) I(x,y) I’(x’,y’)

  4. Forward Mapping • Forward mapping: Target Source Forward Mapping

  5. Forward vs. Inverse Mapping • Forward mapping: • Problems with forward mapping due to sampling: • Holes (some target pixels are not populated) • Overlaps (some target pixels assigned few colors) Target Source Forward mapping

  6. Inverse mapping: • Each target pixel assigned a single color. • Color Interpolation is required. Source Target Inverse Mapping

  7. Example: Scaling along X • Forward mapping: • Inverse mapping: Target Source (0,0) (0,0) Target Source

  8. Interpolation • What happens when a mapping function calculates a fractional pixel address? • Interpolation: generates a new pixel by analyzing the surrounding pixels.

  9. Interpolation • Good interpolation techniques attempt to find an optimal balance between three undesirable artifacts: edge halos, blurring and aliasing. x4 scaling N.N Bilinear Bicubic

  10. Nearest Neighbor Interpolation • The assign value is taken from the pixel closest to the generated location: • Advantage: • Fast • Disadvantage: • Jagged results • Discontinues results

  11. Nearest N. Interpolation Original Image

  12. Original Image Nearest N. Interpolation

  13. Bilinear Interpolation • The assigned value is an intermediate value between the four nearest pixels:

  14. Linear Interpolation • Isolating v in the above equation: ve v vw xw x xe

  15. Bilinear Interpolation • The assign value is a weighted sum of the four nearest pixels. • Each weight is proportional to the distance from each existing pixel.

  16. N NW V NE • The bilinear interpolation is the best fit low-degree polynomial of the form: • The pixel’s boundaries are C0 continuous (continuous values across boundaries). S SE SW

  17. Bilinear example s t s z=15 z=7 v t 0 1 z=2 z=3 0

  18. Nearest N. Interpolation Bilinear Interpolation

  19. Nearest N. Interpolation Bilinear Interpolation

  20. Bicubic Interpolation • The assign value is a weighted sum of the 4x4 nearest pixels:

  21. How can we find the right coefficients? • Denote the pixel values Vpq {p,q=0..3} • The unknown coefficients are aij {i,j=0..3} s • We have a linear system of 16 equations with 16 coefficients. • The pixel’s boundaries are C1 continuous (continuous derivatives across boundaries). t

  22. N.N Bilinear Bicubic

  23. N.N

  24. Bilinear

  25. Bicubic

  26. Applying the Transformation T = …… % 2x2 transformation matrix [r,c] = size(img) % create array of destination x,y coordinates [X,Y]=meshgrid(1:c,1:r); % calculate source coordinates sourceCoor = inv(T) * [X(:) Y(:) ] ‘ ; % calculate nearest neighbor interpolation Xs = round(sourceCoor(1,:)); Ys = round(sourceCoor(2,:)); indx=find(Xs<1 | Xs>r); %out of range pixels Xs(indx)=1; Ys(indx)=1; indy=find(Ys<1 | Ys>c); %out of range pixels Xs(indy)=1; Ys(indy)=1; % calculate new image newImage = img((Xs-1).*r+Ys); newImage(indx)=0; newImage(indx)=0; newImage = reshape(newImage,r,c);

  27. Types of linear 2D transformations • Rigid(Euclidean)transformation: • Translation + Rotation (distance preserving). • Similaritytransformation: • Translation + Rotation + Uniform Scale (angle preserving). • Affinetransformation: • Translation + Rotation + Scale + Shear (parallelism preserving). • Projective transformation • Cross-ratio preserving • All above transformations are groups where Rigid  Similarity  Affine  Projective

  28. Homogeneous Coordinates • Homogeneous Coordinates is a mapping from Rn to Rn+1: • Note: (tx,ty,t) all correspond to the same non-homogeneous point (x,y). E.g. (2,3,1)(6,9,3) (4,6,2). • Inverse mapping:

  29. Some 2D Transformations • Translation : • Affine transformation: • Projective transformation:

  30. Hierarchy of Linear 2D Transformations

  31. Non Linear 2D Transformations • Non linear transformations do not necessarily preserve straight lines. • Methods: • Piecewise linear transformations • Non linear parametric mapping

  32. Non linear mapping: • Example: non linear (radial) lens distortions:

  33. Transformation Estimation • Let: x’=fx(x,y,px) ; y’=fy(x,y,py), where px and py are vector of parameters. • If the mappings are linear in px and py the parameters can be estimated using linear regression.

  34. Example: Affine transformation • Alternative representation:

  35. Given k points (P1,P2,..Pk) in 2D that have been transformed to (P'1,P'2,..,P'k) by affine transformation: • How many points uniquely define the affine (projective) transformation? • How can we find the affine transformation? • What if we have more points? • What can be done if points coordinates are inaccurate?

  36. Image Warping and Morphing • Image rectification. • Key frame animation. • Image Synthesis • Facial expression • Viewing positions Image Metamorphosis.

  37. Image Metamorphosis

  38. Cross Dissolve (pixel operations) Source Image Destination Image t cross dissolve warp + dissolve

  39. Warping + Cross Dissolve • Warp source image towards intermediate image. • Warp destination image towards intermediate image. • Cross-dissolve the two images by taking the weighted average at each pixel. source time Cross-dissolve destination warping images

  40. warp Cross-dissolve Cross-dissolve warp

  41. Image Metamorphosis • Let S,T be the source and the target images • Let G(p) be the transformation from S towards T, where G(0)=I the identity transformation • Let t[0,1] the time step to be synthesized Algorithm: • Warp S towards T: • Warp T toward S: • Cross dissolve:

  42. t sourse S(t)=G(tp){S} I(t)=(1-t)S(t)+t(T(t)) target T(t)=G((1-t)p)-1{T}

  43. Feature Based Morphing • Morph one shape into another shape • Use local features to define the geometric warping

  44. Q Q’ P P’

  45. Q Q’ P’ P

  46. Q Q’ P’ P

  47. Q Q’ P’ P

  48. Q Q’ P’ P

  49. Q Q’ P’ P

  50. Q’ Q R  R’    P P’ Source Image Dest Image One Segment Warping • [0,1] is the relative position along the segment (P’,Q’). • is the actual perpendicular distance to the segment. • (u’,v’) is the local coordinates of the segment (P’,Q’): • u’ is a unit vector parallel to Q’-P’ • v’ is the unit vector perpendicular to Q’-P’ v u u’ v’

More Related