10 Questions
What is the result of adding two vectors?
A vector calculated by adding corresponding components
The magnitude of the cross product of two vectors is always equal to the product of their magnitudes.
False
What are the two attributes of a vector?
Direction and Magnitude
A _______________ is a location in space.
point
Match the following image transformations with their descriptions:
Filtering = Changes the range of the image function Warping = Changes the domain of the image function
What is the formula for the cross product of two vectors a and b?
(a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k
The dot product of two vectors is a vector.
False
What is an affine space?
A set of points equipped with a set of transformations, or translations, which forms a vector space.
The cross product of two vectors can be calculated using a _______________________ matrix.
determinant
Match the following vectors with their respective directions:
i = x axis j = y axis k = z axis
Study Notes
Image Transformations
- Filtering: changes the range of the image function
- Warping: changes the domain of the image function
- Image transformations can be categorized into two types: filtering and warping
Basic Elements in Geometry
- Point: a location in space
- Scalars: objects with no geometric properties
- Vector: a quantity with two attributes, direction and magnitude
- Vector operations:
- Addition: calculated by adding corresponding components using the head-to-tail axiom
- Cross product: defined as a vector c, with its direction perpendicular to both a and b, and a magnitude equal to the area of the parallelogram that the vectors span
- Vector notation:
a = a1i + a2j + a3k, wherei,j, andkare unit vectors in the directions of x, y, and z axes
Cross Product
- Formula:
a x b = (a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k - Matrix form:
a x b = |a1 a2 a3|,b1 b2 b3| - Examples:
-
a = 4i + 3j + 7k,b = 2i + 5j + 4k, thena x b = -23i - 2j + 14k -
a x b = (a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k
-
Dot Product
- Formula:
a.b = |a||b|cos(θ) - Scalar product:
a.b = a1a2 + b1b2 + c1c2 - Examples:
-
a = a1i + b1j + c1k,b = a2i + b2j + c2k, thena.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 forms a vector space
This quiz covers geometric transformations and projective geometry in computer vision, including image transformations, affine spaces, and camera matrices. Based on Richard Szeliski's textbook.
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