Image GeometryDigital Image Processing
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Interpolation of DataSuppose we have a collection of four values that we
wish to enlarge to eight:
Interpolated results x-axis => y-axis
Note that none of both counterparts exactly coincide, except for the first and last. Therefore, an estimation (i.e., interpolation) function is needed for all mapped values.
or
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Interpolation of Data
• The a and b of the linear function can be solved by
• Then we can obtain the linear function
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Popular Linear Interpolation Methods
Nearest-neighbor interpolation Linear interpolation
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Calculation of Linear Interpolation
Ch6-p.124
f(x1), F, f(x2) are co-linear
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Image InterpolationConsider a 4x4
image is enlarged to an 8x8 image
Only the corner pixels are aligned
Interplation is needed
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Bilinear Interpolation
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Image Resize – imresize()• Function imresize
• Where A is an image of any type, k is a scaling factor, and ’method’ is either ’nearest’ or ’bilinear’, etc.
Ch6-p.127
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General Interpolation
Ch6-p.129
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Cubic Interpolation
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Bi-cubic Interpolation
Ch6-p.131
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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 filteringWe may expand the image size and fill the interleaved
rows and columns with 0s. e.g.
Replace the interleaved 0s with interpolated values by either nearest-neighbor, bilinear, bi-cubic interpolation methods.
Ch6-p.132
Enlargement Example (1/4)Create a 4x4 matrix (later expanded to 8x8)
The 8x8 zero-interleaved matrix can be obtained by:
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Enlargement Example (2/4)The enlarged 8x8 matrix
The interpolation methods could be either
nearest-neighbor bilinear bicubic
Ch6-p.133
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Enlargement Example (3/4)Nearest-neighbor
Bilinear
Ch6-p.134
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Enlargement Example (4/4)Bi-cubic
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Scaling Smaller• Making an image smaller is also called image
minimization• Subsampling example
Nearest-neighbor minimization
Bi-cubic interpolation for minimization
Drawing a circle (points away from (512,512) approximately 256 pixels are considered on the circle)
Default: nearest neighbor
Bi-cubic minimization
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Rotation
• The filled circles indicate the original position, and the open circles point their positions after rotation• Issues:• Ensure points are on the grid after rotation• Points in the rotated image need to be
interpolated
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Rotate with nearest-neighbor interpolation
Rotate with bi-cubic interpolation
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Anamorphosis – morphing
The Ambassadors (1533) by Hans Holbein
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