Computer Vision Lecture #2 Hossam Abdelmunim 1 & Aly A. Farag 2 1 Computer & Systems Engineering Department, Ain Shams University, Cairo, Egypt 2 Electerical and Computer Engineering Department, University of Louisville, Louisville, KY, USA ECE619/645 – Spring 2011

Geometric Primitives and Transformations 2D Point – x=(x1,x2,1) 2D Line – ax1+bx2+c=0

Geometric Primitives and Transformations 3D Point – x=(x1,x2,x3,1) 3D Line Derive the line equation shown above.

Geometric Primitives and Transformations 3D Plane –ax+by+cz+d=0; Derive the plane equation shown above.

Transformation Matrix Translation (Example in 2D)

Transformation Matrix Rotation Matrix (Example in 2D)

Transformation Matrix Scaling Matrix (Example in 3D)

Projective Transformation Matrix

Hierarchy of Coordinate Transformations *Homogeneous Scaling, rotation, and translation *

3D to 2D Projection

What do we need? we need to specify how 3D primitives (points) are projected onto the image plane. We can do this using a linear 3D to 2D projection matrix.

ExampleExample

Geometric Interpretation Perspective v's Parallel (orthogonal) Projection

Perspective Projection Matrix Equation

Para-Perspective Projection