1. 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)

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