# invariants to affine transform what is affine transform ?

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• Invariants to affine transformWhat is affine transform?

• Why is affine transform important?Affine transform is a good approximation of projective transformProjective transform describes a perspective projection of 3-D objects onto 2-D plane by a central camera

• Affine moment invariantsTheory 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

• General construction of Affine Moment Invariants

• General construction of Affine Moment InvariantsAffine Moment Invariants

• Simple examples of the AMIs

• Graph representation of the AMIs

• Graph representation of the AMIs

• Graph representation of the AMIs

• Graph representation of the AMIs

• Graph representation of the AMIs

• Removing dependency

• Affine invariants via normalizationMany possibilities how to define normalization constraints

Several possible decompositions of affine transform

• Decomposition of the affine transformHorizontal and vertical translationScalingFirst rotationStretchingSecond rotationMirror reflection

• Normalization to partial transformsHorizontal and vertical translation -- m01 = m10 = 0Scaling -- c00 = 1First rotation -- c20 real and positiveStretching -- c20 =0 (20=02)Second rotation -- -- c21 real and positive

• Properties of the AMIs

• Application of the AMIsRecognition of distorted shapes

Image registration

• Clusters in the space of the AMIs

• Landsat TMSPOTImage registration

• Selected regions

• Matching pairs

• Image registration

• Region matching by the AMIs

• Robustness of the AMIs to distortions

• Robustness of the AMIs to distortions

• Aspect-ratio invariants

• Projective moment invariantsProjective transform describes a perspective projection of 3-D objects onto 2-D plane by a central camera

• Projective moment invariantsDo 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

• Invariants to contrast changes