1. Davi Geiger Computer VisionSeptember 2015 L1.1  Mirror Symmetry Concepts  u - vector input response v - vector mirror symmetric to u ’’

2. Davi Geiger Computer VisionSeptember 2015 L1.2 2015 L1.2  Mirror Symmetry Concepts  S - reflection matrix u - vector input response v - vector mirror symmetric to u if ’’ stencil

3. Davi Geiger Computer VisionSeptember 2015 L1.3 Measure of Mirror Symmetry Given u & v - vectors input response with angles  and  ’ respectively, i.e., where A measure of symmetry with respect to a reflection along an angle  is given by

4. Davi Geiger Computer VisionSeptember 2015 L1.4 Given two points, x,x’ and associated vectors u & v. We show that is invariant with respect to the group SE(2), rotations and translations, by applying an isometry. Translation subgroup: Rotation subgroup, SO(2): Invariance to Rotations and Translations group, SE(2)

5. Davi Geiger Computer VisionSeptember 2015 L1.5 Stability to Deformations Given a pair {(x,u);(x’,v)} and consider a small deformations/displacement so that      ’   ’ +   ’, and due to the deformations in x, x’, we also have  ’   ’ +   ’, Thus, the measure is deformed as As long as all  is small will be small

6. Davi Geiger Computer Vision September 2015 L1.6 Image-Measure of Mirror Symmetry Given image coordinates (x,y) with wavelet convolution responses For each pair (x,y) and (x’,y’) in the image, a distance d away, such that We work with a MS  (x,y) c=(c x,c y ) d (x’,y’) d 

7. Davi Geiger Computer Vision September 2015 L1.7 Hough Space for Lines and Symmetry Accumulator y x    t The parameter  can take positive and negative values y  x     

8. Davi Geiger Computer Vision September 2015 L1.8 Visualization of Symmetry Accumulator    x |Sym (  ) | y  2

9. Davi Geiger Computer VisionSeptember 2015 L1.9  (x,y) c=(c x,c y ) d (x’,y’) d Symmetry computation 

10. Davi Geiger Computer VisionSeptember 2015 L1.10 First column: shows where the structure is centered, through the box, and the final output of an example (pointwise multiplication of the second and third columns) Second column: shows where the first wavelet output is placed, through the box, (solid red line, the rest of the structure is dashed) and an example output. Third column: shows where the second wavelet output is placed, through the box, and an example output. Visualization of Symmetry Computation MS (c, d  ) = J(x,y,  J*(x’,y’,  ’ 