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Chapter 6: Image Geometry 6.1 Interpolation of Data

Chapter 6: Image Geometry 6.1 Interpolation of Data. Suppose we have a collection of four values that we wish to enlarge to eight. The a and b of the linear function can be solved by Then we can obtain the linear function. (continuous). 6.1 Interpolation of Data.

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Chapter 6: Image Geometry 6.1 Interpolation of Data

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  1. Chapter 6: Image Geometry6.1 Interpolation of Data • Suppose we have a collection of four values that we wish to enlarge to eight • The a and b of the linear function can be solved by • Then we can obtain the linear function (continuous)

  2. 6.1 Interpolation of Data • In digital (discrete), none of the points coincide exactly with an original xj, except for the first and last • We have to estimate function values based on the known values of nearby f (xj) • Such estimation of function values based on surrounding values is • called interpolation • Nearest-neighbor interpolation

  3. FIGURE 6.5 • Linear interpolation (Equation 6.1)

  4. 6.2 Image Interpolation • Using the formula given by Equation 6.1 bilinear interpolation

  5. 6.2 Image Interpolation • Function imresize • Where A is an image of any type, k is a scaling factor, and ’method’ is either ’nearest’ or ’bilinear’, etc.

  6. FIGURE 6.9 & 6.10

  7. 6.3 General Interpolation • Generalized interpolation function: • R0(u) Nearest-neighbor interpolation (Equation 6.2)

  8. 6.3 General Interpolation • R1(u) Linear interpolation • Cubic interpolation

  9. FIGURE 6.14

  10. FIGURE 6.15

  11. FIGURE 6.16

  12. 6.4 Enlargement by Spatial Filtering • If we merely wish to enlarge an image by a power of two, there is a quick and dirty method that uses linear filtering • e.g. • zero-interleaved

  13. FIGURE 6.17 • This can be implemented with a simple function

  14. 6.4 Enlargement by Spatial Filtering • We can now replace the zeros by applying a spatial filter to this matrix nearest-neighbor bilinear bicubic

  15. 6.4 Enlargement by Spatial Filtering

  16. FIGURE 6.18

  17. 6.5 Scaling Smaller • Making an image smaller is also called image minimization • Subsampling • e.g.

  18. FIGURE 6.19

  19. 6.6 Rotation

  20. FIGURE 6.21

  21. 6.6 Rotation • In Figure 6.21, the filled circles indicate the original position, and the open circles point their positions after rotation • We must ensure that even after rotation, the points remain in that grid • To do this we consider a rectangle that includes the rotated image, as shown in Figure 6.22

  22. FIGURE 6.22

  23. FIGURE 6.23

  24. 6.6 Rotation • The gray value at (x”, y”) can be found by interpolation, using surrounding gray values. This value is then the gray value for the pixel at (x’, y’) in the • rotated image

  25. FIGURE 6.25

  26. 6.7 Anamorphosis

  27. FIGURE 6.27

  28. FIGURE 6.28

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