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Questions and Answers
What are the two main types of image transformations?
What are the two main types of image transformations?
Filtering and Warping
What is the definition of a scalar in geometry?
What is the definition of a scalar in geometry?
An object with no geometric properties
What are the two attributes of a vector?
What are the two attributes of a vector?
Direction and Magnitude
How is the sum of two vectors calculated?
How is the sum of two vectors calculated?
What is the definition of the cross product of two vectors?
What is the definition of the cross product of two vectors?
What is the formula for the magnitude of the cross product of two vectors?
What is the formula for the magnitude of the cross product of two vectors?
What is the purpose of the 3D viewing pipeline?
What is the purpose of the 3D viewing pipeline?
What is the role of the camera matrix in computer vision?
What is the role of the camera matrix in computer vision?
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?
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?
In the cross product matrix formula, what is the determinant of the matrix representing the cross product of two vectors a and b?
In the cross product matrix formula, what is the determinant of the matrix representing the cross product of two vectors a and b?
What is the formula for the dot product of two vectors a and b, where a = a1i + b1j + c1k and b = a2i + b2j + c2k?
What is the formula for the dot product of two vectors a and b, where a = a1i + b1j + c1k and b = a2i + b2j + c2k?
What is an affine space in mathematics?
What is an affine space in mathematics?
If a = 4i + 3j + 7k and b = 2i + 5j + 4k, what is the value of a x b?
If a = 4i + 3j + 7k and b = 2i + 5j + 4k, what is the value of a x b?
What is the geometric interpretation of the cross product of two vectors?
What is the geometric interpretation of the cross product of two vectors?
If a = a1i + a2j + a3k and b = b1i + b2j + b3k, what is the formula for a x b in matrix form?
If a = a1i + a2j + a3k and b = b1i + b2j + b3k, what is the formula for a x b in matrix form?
What is the significance of the unit vectors i, j, and k in vector operations?
What is the significance of the unit vectors i, j, and k in vector operations?
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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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