invariants to affine transform
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Invariants to affine transform. What is affine transform ?. Why is affine transform important?. Affine transform is a good approximation of projective transform Projective transform describes a perspective projection of 3-D objects onto 2-D plane by a central camera. Affine moment invariants. - PowerPoint PPT PresentationTRANSCRIPT
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Invariants to affine transform
What is affine transform?
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Why is affine transform important?
• Affine transform is a good approximation of projective transform• Projective transform describes a perspective projection of 3-D objects onto 2-D plane by a central camera
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Affine moment invariants
• Theory of algebraic invariants (Hilbert, Schur, Gurewich)• Tensor algebra, Group theory (Lenz, Meer)• Algebraic invariants revised (Reiss, Flusser & Suk, Mamistvalov)• Image normalization (Rothe et al.)• Graph theory (Flusser & Suk)• Hybrid approaches
All methods lead to the same invariants
Many ways how to derive them
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General construction of Affine Moment Invariants
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General construction of Affine Moment Invariants
Affine Moment Invariants
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Simple examples of the AMI’s
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Graph representation of the AMI’s
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Graph representation of the AMI’s
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Graph representation of the AMI’s
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Graph representation of the AMI’s
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Graph representation of the AMI’s
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Removing dependency
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Affine invariants via normalization
Many possibilities how to define normalization constraints
Several possible decompositions of affine transform
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Decomposition of the affine transform
• Horizontal and vertical translation
• Scaling• First rotation• Stretching• Second rotation• Mirror reflection
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Normalization to partial transforms
• Horizontal and vertical translation -- m01 = m10 = 0
• Scaling -- c00 = 1
• First rotation -- c20 real and positive
• Stretching -- c20 =0 (μ20=μ02)
• Second rotation -- -- c21 real and positive
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Properties of the AMI’s
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Application of the AMI’s
• Recognition of distorted shapes
• Image registration
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Landmark-based robot navigation
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Clusters in the space of the AMI’s
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Landsat TM SPOT
Image registration
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Selected regions
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Matching pairs
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Image registration
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Region matching by the AMI’s
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Robustness of the AMI’s to distortions
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Robustness of the AMI’s to distortions
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Aspect-ratio invariants
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Projective moment invariants
Projective transform describes a perspective projection of 3-D objects onto 2-D plane by a central camera
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Projective moment invariants
• Do not exist using any finite set of moments
• Do not exist using infinite set of (all) moments
• Exist formally as infinite series of moments of both positive and negative indexes
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Invariants to contrast changes