Outline
• Announcement
– Homework #1
• Local operations (continued)
– Geometric operations
• Linear filters
Announcement
• Homework #1
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Geometric Operations
• Geometric operations change the spatial
relationships among the objects in an image
– Have an image on a rubber sheet and then deform
the sheet
– A geometric operation is more general in that it
can move any point in the input image to any
point in the output image
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Geometric Operations – cont.
• Two separate algorithms
– Spatial transformation
• An algorithm that defines the spatial transformation
itself
• This specifies the motion of each pixel
– Pixel value interpolation
• Integer pixel positions can map to fractional positions
• Nearest neighbor interpolation
• Bilinear interpolation
– Implementation issues
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Spatial Transformation
• Simple transformations
– Rotation
– Scaling
– Affine transformation
• General transformations
– Specified by the motion of each pixel
• Called “optical flow”
– Specified by control points
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Applications of Geometric Operations
• Geometric calibration
– Remove the camera-induced geometric distortion
from digital images
• Image rectification
– Transform images of non-rectangular pixel
coordinates to display systems
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Applications of Geometric Operations – cont.
• Image registration
– Register similar images for purposes of
comparison
• Motion estimation and video analysis
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Image Mosaicing
• Remote sensing of earth and planets,
panoramas ....
• More on the web
– Image mosaics from CMU
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Applications of Geometric Operations – cont.
• Model-based object recognition
– Transform an object model to match the input
image
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Applications of Geometric Operations – cont.
• Map projection
– Project images for purposes of mapping
• How to produce photo-mosaic maps of the Earth,
moon, or the planets
– Cartography
• Produce two-dimensional maps of spherical or
ellipsoidal bodies
• Map properties
• Cartographic projections
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Applications of Geometric Operations – cont.
• Image morphing
– A technique that allows one object to transform
gradually into another
– Generate a movie sequence from two images
• Image interpolation
– How to generate a realistic looking
transformation
• It has tremendous commercial values
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Image Morphing
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An Example
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Image Morphing
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More Examples on the Web
• On the web
– http://www.cis.ohiostate.edu/graphics/kf/morph_example.html
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Research Problems
• How to create a model very efficiently
– Identify important features in one image and the
corresponding features in the other image
• Deformable templates
– Specify the deformation for certain effects
• Facial expression modeling
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Linear System Theory
• What is a system?
– A system is anything that accepts an input and
produces an output in response
y[n] = T{x[n]}
where x[n] is the input sequence and y[n] is the output
sequence in responses to x[n]
– How to represent a sequence?
x[n] 

 x[k ] [n  k ]
k  
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Linear System
• Linearity
–
–
–
–
y1[n] = T{x1[n]}
y2[n] = T{x2[n]}
Then
y1[n]+y2[n] = T{x1[n]+x2[n]}
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Shift-Invariant System
• Shift invariance
– y[n] = T{x[n]}
– y[n-T] = T{x[n-T]}
• LSI system
– A LSI system is completely characterized by its
impulse response h[n]
– For any other input, we can obtain the response
through convolution
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Filtering
• Closely related to convolution
• Filter examples
– Smoothing by averaging
– Smoothing by Gaussian
2
2

1
(x  y ) 

G ( x, y) 
exp  
2
2
2
2


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