Differential Geometry Computer Vision #8

Differential Geometry 1. Curvature of curve 2. Curvature of surface 3. Application of curvature

Parameterization of curve 1. curve -- s arc length a(s) = ( x(s), y(s) ) 2. tangent of a curve a’(s) = ( x’(s), y’(s) ) 3. curvature of a curve a”(s) = ( x”(s), y”(s) ) |a”(s)| -- curvature s a(s)

Example (circle) 1. Arc length, s 2. coordinates 3. tangent 4. curvature

Definition of curvature The normal direction (n) toward the empty side.

Corner model and its signatures b a c d s=0 a bc a bcda arc length d ab c s=0

Gaussian filter and scale space a a a b c d e b c d e f g h i j k + + +

Curvature of surfaces normal sectionnon-normal section normal curvature Principal directions and principal curvatures

Principal curvatures plane: all directions sphere: all directions cylinder: ellipsoid: hyperboloid:

Gaussian curvature and mean curvature

Parabolic points Parabolic point elliptic point hyperbolic point F.Klein used the parabolic curves for a peculiar investigation. To test his hypothesis that the artistic beauty of a face was based on certain mathematical relation, he has all the parabolic curves marked out on the Apollo Belvidere. But the curves did not possess a particularly simpler form, nor did they follow any general law that could be discerned.

Lines of curvature Principal directions, which gives the maximum and the minimal normal curvature. Principal direction curves along principal directions PD

Lines of curvature

Curvature primal sketches along lines of curvature

Important formula 1. Surface 2. surface normal 3. the first fundamental form 4. the second fundamental form

Y Z X

Summary 1. curvature of curve 2. curvature of surface –Gaussian curvature –mean curvature

Surface Description #2 (Extended Gaussian Image)

Topics 1.Gauss map 2.Extended Gaussian Image 3.Application of EGI

Gauss map Let S ⊂ R3 be a surface with an orientation N. The map N: S→R3 takes its values in the unit sphere The map N: S→S3 is called the Gauss map. 1D gauss map 2D

Characteristics of EGI  EGI is the necessary and the sufficient condition for the congruence of two convex polyhedra.  Ratio between the area on the Gaussian sphere and the area on the object is equal to Gaussian curvature.  EGI mass on the sphere is the inverse of Gaussian curvature.  Mass center of EGI is at the origin of the sphere  An object rotates, then EGI of the object also rotates. However, both rotations are same.

Relationship between EGI and Gaussian curvature object Gaussian sphere large small large (K: small) (K: large) small large

Gaussian curvature and EGI maps u Since and exist on the tangential plane at, u we can represent them by a linear combination of and

Implementation of EGI Tessellation of the unit sphere all cells should have the same area have the same shape occur in a regular pattern geodesic dome based on a regular polyhedron semi-regular geodesic dome

Example of EGI CylinderEllipsoid side view top view

Determination of attitude using EGI viewing direction EGI table

The complex EGI(CEGI) Normal distance and area of a 3-D object are encoded as a complex weight. Pn k associated with the surface normal n k such that:

(note: The weight is shown only for normal n 1 for clearly.) The complex EGI(CEGI) Gauss mapping Origin (a) Cube (b) CEGI of cube

Bin picking system based on EGI Photometric stereo segmentation Region selection Photometric stereo EGI generation EGI matching Grasp planning Needle map isolated regions target region precise needle map EGI object attitude

Lookup table for photometric stereo Calibration Hand-eye calibration

Photometric Stereo Set-up

Bin-Picking System

Summary 1. Gauss map 2. Extended Gaussian Image 3. Characteristics of EGI congruence of two convex polyhedra EGI mass is the inverse of Gaussian curvature mass center of EGI is at the origin of the sphere 4. Implementation of EGI Tessellation of the unit sphere Recognition using EGI 5. Complex EGI 6. Bin-picking system based on EGI 7. Read Horn pp pp