Image Interpolation Techniques in Digital Processing

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)

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

FIGURE 6.5 • Linear interpolation (Equation 6.1)

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

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.

FIGURE 6.9 & 6.10

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

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

FIGURE 6.14

FIGURE 6.15

FIGURE 6.16

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

FIGURE 6.17 • This can be implemented with a simple function

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

6.4 Enlargement by Spatial Filtering

FIGURE 6.18

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

FIGURE 6.19

6.6 Rotation

FIGURE 6.21

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

FIGURE 6.22

FIGURE 6.23

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

FIGURE 6.25

6.7 Anamorphosis

FIGURE 6.27

FIGURE 6.28

Image Interpolation Techniques in Digital Processing