CSE 383 Computer Vision: Lecture 12 Part 2

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Questions and Answers

What is the camera considered as in the context of geometric transformations?

A 2D to 2D transformation

A 2D to 3D transformation

A 3D to 3D transformation

A 3D to 2D transformation (correct)

What is the purpose of introducing a third coordinate (W) in homogeneous coordinates?

To handle scaling and rotation transformations

To handle all transformations (translation, scaling, rotation) as multiplications (correct)

To reduce the dimensionality of the coordinate system

To handle translation transformations

What is the perspective projection represented by in the context of camera geometry?

A 3D to 3D transformation

A 3D to 2D transformation (correct)

A 2D to 2D transformation

A 2D to 3D transformation

What is the role of the camera matrix in the context of computer vision?

To transform a 3D world point to a 2D image point Signup and view all the answers

What is the relationship between the 3D world point and the 2D image point in the context of camera geometry?

The 3D world point is mapped to a single 2D image point Signup and view all the answers

What is the significance of the W coordinate in homogeneous coordinates?

It is always set to 1 Signup and view all the answers

What is the advantage of using homogeneous coordinates in computer vision?

They allow translations to be handled as a multiplication Signup and view all the answers

What is the purpose of the pinhole camera matrix in computer vision?

To transform a 3D world point to a 2D image point Signup and view all the answers

What is the relationship between Cartesian coordinates and homogeneous coordinates?

Homogeneous coordinates can be obtained by adding 1 to Cartesian coordinates Signup and view all the answers

What is the purpose of homogeneous coordinates in computer vision?

To enable projective transformations Signup and view all the answers

What is the equation for the perspective projection matrix when Z = 1 and f = 1?

[1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1] Signup and view all the answers

What is the relationship between the camera matrix P and the image point X?

X = P * X Signup and view all the answers

What is the general form of the camera matrix P?

[I 0; 0 1; 0 0 1; 0 0 0 f] Signup and view all the answers

What is the equation for the image coordinate x in terms of X?

x = fX/Z Signup and view all the answers

What is the purpose of the pinhole camera model?

To model a real-world camera's behavior Signup and view all the answers

What is the relationship between the homogeneous coordinates (X, Y, Z) and the Cartesian coordinates (x, y)?

(x, y) = (X/Z, Y/Z) Signup and view all the answers

Study Notes

Geometric Transformations & Projective Geometry

The Camera as a Coordinate Transformation

A camera is a mapping from 3D object to 2D image, involving 3D to 2D transform and 2D to 2D transform (image warping)
Camera transformation can be viewed as a coordinate transformation from 3D world to 2D image

Homogeneous Coordinates

Homogeneous coordinates allow translations to be handled as multiplications, enabling uniform treatment of scaling, rotations, and translations
Each point is given a third coordinate (X, Y, W), allowing translations to be handled as multiplications
In practice, W = 1, so this step can be considered as mapping the point from 3D space onto the plane W = 1
Homogeneous coordinates can be converted to Cartesian coordinates by dividing the triple by W

Projective Transformation

A projective transformation is a transformation from 3D world to 2D image
The image plane is a projection of a point P on the image plane

Pinhole Camera Matrix

The pinhole camera matrix P is a 3x4 matrix that transforms 3D world coordinates to 2D image coordinates
When Z = 1 and f = 1, the perspective projection matrix P is given by:
- 1 0 0 0
- 0 1 0 0
- 0 0 1 0
The pinhole camera matrix can be generalized to arbitrary focal length using the equation:
- x = PX, where X is the 3D world coordinate and x is the 2D image coordinate

General Pinhole Camera Matrix

The general pinhole camera matrix is a 3x4 matrix that transforms 3D world coordinates to 2D image coordinates
The matrix can be written in terms of homogeneous coordinates as:
- x = [I I 0] [X Y Z W]^T
The camera matrix can be generalized to arbitrary focal length using the equation:
- x = (fX + px) / (Z + pz)

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Description

This quiz covers geometric transformations and projective geometry in computer vision, including camera coordinates, homogeneous coordinates, and projective transformation.