CSE383 Computer Vision Lecture 12

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Questions and Answers

What are the two main types of image transformations?

Filtering and Warping

What is the definition of a scalar in geometry?

An object with no geometric properties

What are the two attributes of a vector?

Direction and Magnitude

How is the sum of two vectors calculated?

By adding corresponding components using the head-to-tail axiom Signup and view all the answers

What is the definition of the cross product of two vectors?

A vector perpendicular to both vectors, with a magnitude equal to the area of the parallelogram spanned by the two vectors Signup and view all the answers

What is the formula for the magnitude of the cross product of two vectors?

Ia x bI = IaI IbI sin(θ) Signup and view all the answers

What is the purpose of the 3D viewing pipeline?

To transform 3D objects into 2D images Signup and view all the answers

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

To project 3D points onto a 2D image Signup and view all the answers

What is the formula to find the cross product of two vectors a and b, where a = a1i + a2j + a3k and b = b1i + b2j + b3k?

(a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k Signup and view all the answers

In the cross product matrix formula, what is the determinant of the matrix representing the cross product of two vectors a and b?

la1 a2 a3/b1 b2 b3, where a = a1i + a2j + a3k and b = b1i + b2j + b3k Signup and view all the answers

What is the formula for the dot product of two vectors a and b, where a = a1i + b1j + c1k and b = a2i + b2j + c2k?

a1a2 + b1b2 + c1c2 Signup and view all the answers

What is an affine space in mathematics?

A set of points equipped with a set of transformations (bijective mappings), or translations, which form a vector space Signup and view all the answers

If a = 4i + 3j + 7k and b = 2i + 5j + 4k, what is the value of a x b?

-23i - 2j + 14k Signup and view all the answers

What is the geometric interpretation of the cross product of two vectors?

The area of the parallelogram formed by the two vectors, with direction perpendicular to the plane of the parallelogram Signup and view all the answers

If a = a1i + a2j + a3k and b = b1i + b2j + b3k, what is the formula for a x b in matrix form?

la1 a2 a3/b1 b2 b3, where a x b = (a2b3 - a3b2)i - (a3b1 - a1b3)j + (a1b2 - a2b1)k Signup and view all the answers

What is the significance of the unit vectors i, j, and k in vector operations?

They are used to represent the x, y, and z axes in Cartesian coordinates, respectively Signup and view all the answers

Study Notes

Geometric Transformations & Projective Geometry

Image Transformations

Filtering: changes the range of the image function
Warping: changes the domain of the image function

Basic Elements in Geometry

Point: a location in space
Scalar: an object with no geometric properties
Vector: a quantity with two attributes, direction and magnitude
Coordinate Reference Frames: 3D frames of reference

Vector Operations

Vector addition: the sum of two vectors is a vector, calculated by adding corresponding components using the head-to-tail axiom
Vector multiplication: the cross product of two vectors, a x b, is defined as a vector c, its direction is perpendicular to both a and b, and its magnitude is equal to the area of the parallelogram that the vectors span
Magnitude of the cross product: |a x b| = |a| |b| sin θ
Cross product formula: a x b = (a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k
Cross product matrix: a x b =(det | a1 a2 a3 | b1 b2 b3 |)

Dot Product

a.b = |a| |b| cos θ
Dot product formula: a.b = a1a2 + b1b2 + c1c2

Affine Spaces

An affine space is a "flat" space without a fixed origin and without the notion of vectors starting at a particular point
It is a set of points equipped with a set of transformations (bijective mappings), or translations, which form a vector space

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Description

This quiz covers geometric transformations and projective geometry in computer vision, including image transformations, affine spaces, and 3D viewing pipeline. Based on Richard Szeliski textbook.