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# Image Interpolation

Image Interpolation. Introduction What is image interpolation? Why do we need it? Interpolation Techniques 1D zero-order, first-order, third-order 2D zero-order, first-order, third-order Directional interpolation* Interpolation Applications Digital zooming (resolution enhancement)

## Image Interpolation

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### Presentation Transcript

1. Image Interpolation • Introduction • What is image interpolation? • Why do we need it? • Interpolation Techniques • 1D zero-order, first-order, third-order • 2D zero-order, first-order, third-order • Directional interpolation* • Interpolation Applications • Digital zooming (resolution enhancement) • Image inpainting (error concealment) • Geometric transformations EE465: Introduction to Digital Image Processing

2. Introduction • What is image interpolation? • An image f(x,y) tells us the intensity values at the integral lattice locations, i.e., when x and y are both integers • Image interpolation refers to the “guess” of intensity values at missing locations, i.e., x and y can be arbitrary • Note that it is just a guess (Note that all sensors have finite sampling distance) EE465: Introduction to Digital Image Processing

3. Introduction (Con’t) • Why do we need image interpolation? • We want BIG images • When we see a video clip on a PC, we like to see it in the full screen mode • We want GOOD images • If some block of an image gets damaged during the transmission, we want to repair it • We want COOL images • Manipulate images digitally can render fancy artistic effects as we often see in movies EE465: Introduction to Digital Image Processing

4. Scenario I: Resolution Enhancement Low-Res. High-Res. EE465: Introduction to Digital Image Processing

5. Scenario II: Image Inpainting Non-damaged Damaged EE465: Introduction to Digital Image Processing

6. Scenario III: Image Warping EE465: Introduction to Digital Image Processing

7. Image Interpolation • Introduction • What is image interpolation? • Why do we need it? • Interpolation Techniques • 1D zero-order, first-order, third-order • 2D zero-order, first-order, third-order • Directional interpolation* • Interpolation Applications • Digital zooming (resolution enhancement) • Image inpainting (error concealment) • Geometric transformations EE465: Introduction to Digital Image Processing

8. 1D Zero-order (Replication) f(n) n f(x) x EE465: Introduction to Digital Image Processing

9. 1D First-order Interpolation (Linear) f(n) n f(x) x EE465: Introduction to Digital Image Processing

10. Linear Interpolation Formula Basic idea: the closer to a pixel, the higher weight is assigned f(n) f(n+a) f(n+1) a 1-a f(n+a)=(1-a)f(n)+af(n+1), 0<a<1 Note: when a=0.5, we simply have the average of two EE465: Introduction to Digital Image Processing

11. Numerical Examples f(n)=[0,120,180,120,0] Interpolate at 1/2-pixel f(x)=[0,60,120,150,180,150,120,60,0], x=n/2 Interpolate at 1/3-pixel f(x)=[0,20,40,60,80,100,120,130,140,150,160,170,180,…], x=n/6 EE465: Introduction to Digital Image Processing

12. 1D Third-order Interpolation (Cubic) f(n) n f(x) x Cubic spline fitting EE465: Introduction to Digital Image Processing

13. From 1D to 2D • Just like separable 2D transform (filtering) that can be implemented by two sequential 1D transforms (filters) along row and column direction respectively, 2D interpolation can be decomposed into two sequential 1D interpolations. • The ordering does not matter (row-column = column-row) • Such separable implementation is not optimal but enjoys low computational complexity EE465: Introduction to Digital Image Processing

14. Graphical Interpretationof Interpolation at Half-pel row column f(m,n) g(m,n) EE465: Introduction to Digital Image Processing

15. Numerical Examples a b c d zero-order first-order a a b b a a b b c c d d c c d d a (a+b)/2 b (a+c)/2(a+b+c+d)/4 (b+d)/2 c (c+d)/2 d EE465: Introduction to Digital Image Processing

16. Numerical Examples (Con’t) Col n+1 Col n X(m,n) X(m,n+1) row m b a 1-a Y 1-b row m+1 X(m+1,n) X(m+1,n+1) Q: what is the interpolated value at Y? Ans.: (1-a)(1-b)X(m,n)+(1-a)bX(m+1,n) +a(1-b)X(m,n+1)+abX(m+1,n+1) EE465: Introduction to Digital Image Processing

17. Bicubic Interpolation* EE465: Introduction to Digital Image Processing

18. Edge blurring Jagged artifacts Limitation with bilinear/bicubic Jagged artifacts Edge blurring Z X Z X EE465: Introduction to Digital Image Processing

19. Directional Interpolation* Step 1: interpolate the missing pixels along the diagonal a b Since |a-c|=|b-d| x x has equal probability of being black or white c d black or white? Step 2: interpolate the other half missing pixels a Since |a-c|>|b-d| x d b x=(b+d)/2=black c EE465: Introduction to Digital Image Processing

20. Image Interpolation • Introduction • What is image interpolation? • Why do we need it? • Interpolation Techniques • 1D zero-order, first-order, third-order • 2D zero-order, first-order, third-order • Directional interpolation* • Interpolation Applications • Digital zooming (resolution enhancement) • Image inpainting (error concealment) • Geometric transformations EE465: Introduction to Digital Image Processing

21. Pixel Replication low-resolution image (100×100) high-resolution image (400×400) EE465: Introduction to Digital Image Processing

22. Bilinear Interpolation low-resolution image (100×100) high-resolution image (400×400) EE465: Introduction to Digital Image Processing

23. Bicubic Interpolation low-resolution image (100×100) high-resolution image (400×400) EE465: Introduction to Digital Image Processing

24. Edge-Directed Interpolation(Li&Orchard’2000) low-resolution image (100×100) high-resolution image (400×400) EE465: Introduction to Digital Image Processing

25. Image Demosaicing(Color-Filter-Array Interpolation) Bayer Pattern EE465: Introduction to Digital Image Processing

26. Image Example Advanced CFA Interpolation Ad-hoc CFA Interpolation EE465: Introduction to Digital Image Processing

27. Error Concealment damaged interpolated EE465: Introduction to Digital Image Processing

28. Image Inpainting EE465: Introduction to Digital Image Processing

29. Geometric Transformation Widely used in computer graphics to generate special effects MATLAB functions: griddata, interp2, maketform, imtransform EE465: Introduction to Digital Image Processing

30. Basic Principle • (x,y)  (x’,y’) is a geometric transformation • We are given pixel values at (x,y) and want to interpolate the unknown values at (x’,y’) • Usually (x’,y’) are not integers and therefore we can use linear interpolation to guess their values MATLAB implementation: z’=interp2(x,y,z,x’,y’,method); EE465: Introduction to Digital Image Processing

31. Rotation y y’ x’ θ x EE465: Introduction to Digital Image Processing

32. MATLAB Example z=imread('cameraman.tif'); % original coordinates [x,y]=meshgrid(1:256,1:256); % new coordinates a=2; for i=1:256;for j=1:256; x1(i,j)=a*x(i,j); y1(i,j=y(i,j)/a; end;end % Do the interpolation z1=interp2(x,y,z,x1,y1,'cubic'); EE465: Introduction to Digital Image Processing

33. Rotation Example θ=3o EE465: Introduction to Digital Image Processing

34. Scale a=1/2 EE465: Introduction to Digital Image Processing

35. Affine Transform parallelogram square EE465: Introduction to Digital Image Processing

36. Affine Transform Example EE465: Introduction to Digital Image Processing

37. Shear parallelogram square EE465: Introduction to Digital Image Processing

38. Shear Example EE465: Introduction to Digital Image Processing

39. Projective Transform B’ B A A’ C’ D C D’ square quadrilateral EE465: Introduction to Digital Image Processing

40. Projective Transform Example [-4 2; -8 -3; -3 -5; 6 3] [ 0 0; 1 0; 1 1; 0 1] EE465: Introduction to Digital Image Processing

41. Polar Transform EE465: Introduction to Digital Image Processing

42. Iris Image Unwrapping  r EE465: Introduction to Digital Image Processing

43. Use Your Imagination r -> sqrt(r) http://astronomy.swin.edu.au/~pbourke/projection/imagewarp/ EE465: Introduction to Digital Image Processing

44. Free Form Deformation Seung-Yong Lee et al., “Image Metamorphosis Using Snakes and Free-Form Deformations,”SIGGRAPH’1985, Pages 439-448 EE465: Introduction to Digital Image Processing

45. Application into Image Metamorphosis EE465: Introduction to Digital Image Processing

46. Summary of Image Interpolation • A fundamental tool in digital processing of images: bridging the continuous world and the discrete world • Wide applications from consumer electronics to biomedical imaging • Remains a hot topic after the IT bubbles break EE465: Introduction to Digital Image Processing

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