16 Questions
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
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
What is the formula for the magnitude of the cross product of two vectors?
Ia x bI = IaI IbI sin(θ)
What is the purpose of the 3D viewing pipeline?
To transform 3D objects into 2D images
What is the role of the camera matrix in computer vision?
To project 3D points onto a 2D image
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
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
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
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
If a = 4i + 3j + 7k and b = 2i + 5j + 4k, what is the value of a x b?
-23i - 2j + 14k
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
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
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
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
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.
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