Computer VisionComputer VisionLecture #2Lecture #2

Hossam Abdelmunim1 & Aly A. Farag2

1Computer & Systems Engineering Department, Ain Shams University, Cairo, Egypt

2Electerical and Computer Engineering Department, University of Louisville, Louisville, KY, USA

ECE619/645 – Spring 2011

Geometric Primitives and Geometric Primitives and TransformationsTransformations

Geometric Primitives and Geometric Primitives and TransformationsTransformations

• 2D Point– x=(x1,x2,1)

• 2D Line– ax1+bx2+c=0

Geometric Primitives and Geometric Primitives and TransformationsTransformations

Geometric Primitives and Geometric Primitives and TransformationsTransformations

• 3D Point– x=(x1,x2,x3,1)

• 3D Line

Derive the line equation shown above.

Geometric Primitives and Geometric Primitives and TransformationsTransformations

Geometric Primitives and Geometric Primitives and TransformationsTransformations

• 3D Plane– ax+by+cz+d=0;

Derive the plane equation shown above.

Transformation MatrixTransformation MatrixTransformation MatrixTransformation Matrix

• Translation (Example in 2D)

Transformation MatrixTransformation MatrixTransformation MatrixTransformation Matrix

• Rotation Matrix (Example in 2D)

Transformation MatrixTransformation MatrixTransformation MatrixTransformation Matrix

• Scaling Matrix (Example in 3D)

Projective Transformation MatrixProjective Transformation MatrixProjective Transformation MatrixProjective Transformation Matrix

Hierarchy of Coordinate Hierarchy of Coordinate TransformationsTransformations

Hierarchy of Coordinate Hierarchy of Coordinate TransformationsTransformations

*Homogeneous Scaling, rotation, and translation

*

3D to 2D Projection3D to 2D Projection3D to 2D Projection3D to 2D Projection

What do we need?What do we need?What do we need?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.

ExampleExampleExampleExample

Geometric InterpretationGeometric InterpretationGeometric InterpretationGeometric Interpretation

Perspective v's Parallel (orthogonal) Projection

Perspective Projection Matrix Perspective Projection Matrix EquationEquation

Perspective Projection Matrix Perspective Projection Matrix EquationEquation

Para-Perspective ProjectionPara-Perspective ProjectionPara-Perspective ProjectionPara-Perspective Projection