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Introduction to Digital Image Processing with MATLAB® Asia Edition
McAndrew‧Wang‧Tseng
Chapter 6:
Image Geometry
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6.1 Interpolation of Data
Suppose we have a collection of four values that we wish to enlarge to eight
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6.1 Interpolation of Data x x'
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6.1 Interpolation of Data
The a and b of the linear function can be solved by
Then we can obtain the linear function
(continuous)
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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
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FIGURE 6.4
Nearest-neighbor interpolation
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FIGURE 6.5
Linear interpolation
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FIGURE 6.6
(Equation 6.1)
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6.2 Image Interpolation
Using the formula given by Equation 6.1
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FIGURE 6.7
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FIGURE 6.8
bilinear interpolation
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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.
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FIGURE 6.9 & 6.10
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14. 6.3 General Interpolation
(Equation 6.2)
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FIGURE 6.12
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16. 6.3 General Interpolation
The functions R0(u) and R1(u) are just two members of a family of possible interpolation functions
Another such function provides cubic interpolation
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FIGURE 6.14
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FIGURE 6.15
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FIGURE 6.16
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20. 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
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FIGURE 6.17
This can be implemented with a simple function
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22. 6.4 Enlargement by Spatial Filtering
We can now replace the zeros by applying a spatial filter to this matrix
nearest-neighbor
bilinear
bicubic
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23. 6.4 Enlargement by Spatial Filtering
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FIGURE 6.18
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6.5 Scaling Smaller
Making an image smaller is also called image minimization
Subsampling
e.g.
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FIGURE 6.19
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6.6 Rotation
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FIGURE 6.21
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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
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FIGURE 6.22
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FIGURE 6.23
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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
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FIGURE 6.24
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FIGURE 6.25
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FIGURE 6.26
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FIGURE 6.27
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FIGURE 6.28
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