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)
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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
• Nearest-neighbor
interpolation
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FIGURE 6.5
• Linear interpolation
(Equation 6.1)
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6.2 Image Interpolation
• Using the formula given by Equation 6.1
bilinear interpolation
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6.2 Image Interpolation
•
f ( x' , y ' )  f ( x  1, y  1)   (1   ) f ( x  1, y ) 
(1   ) f ( x, y  1)  (1   )(1   ) f ( x, y )
• 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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6.3 General Interpolation
• Generalized interpolation function:
(Equation 6.2)
• R0(u) Nearest-neighbor interpolation
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6.3 General Interpolation
• R1(u) Linear interpolation
• 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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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
•
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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
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bilinear
bicubic
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 6.25
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6.7 Anamorphosis
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FIGURE 6.27
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FIGURE 6.28
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