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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)

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Image Interpolation

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  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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