2d transforms Flashcards
(102 cards)
1
Q
What is the purpose of 2D transformations in visual computing?
A
To move, rotate, scale, and manipulate image coordinates and shapes efficiently using matrix operations.
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Q
A
3
Q
What type of transformation preserves distances and angles?
A
Rigid transformations.
4
Q
What is an example of a rigid transformation?
A
Rotation, reflection, or translation.
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Q
A
6
Q
What are articulated transformations?
A
Piecewise rigid transformations such as those found in robotic arms.
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Q
A
8
Q
What are non-rigid transformations?
A
Transformations that allow stretching and deformation.
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Q
A
10
Q
What is the general matrix form of a 2D transformation?
A
p’ = M · p, where M is a 2×2 matrix and p is a 2D point.
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Q
A
12
Q
What is the matrix for uniform scaling?
A
[[s, 0], [0, s]]
13
Q
What is the effect of uniform scaling?
A
It scales the object equally in both x and y directions.
14
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A
15
Q
What is the matrix for non-uniform scaling?
A
[[a, 0], [0, b]]
16
Q
What does non-uniform scaling do?
A
It scales the x and y directions differently.
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Q
A
18
Q
What is the rotation matrix for angle θ?
A
[[cosθ, -sinθ], [sinθ, cosθ]]
19
Q
What does this rotation matrix do?
A
Rotates points counterclockwise around the origin.
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Q
A
21
Q
What is the matrix for horizontal shear?
A
[[1, k], [0, 1]]
22
Q
What does horizontal shear do?
A
It shifts x-coordinates based on the y-coordinate.
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A
24
Q
What is the matrix for vertical shear?
A
[[1, 0], [k, 1]]
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What does vertical shear do?
It shifts y-coordinates based on the x-coordinate.
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What is the reflection matrix for a horizontal flip (about y-axis)?
[[-1, 0], [0, 1]]
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What is the reflection matrix for a vertical flip (about x-axis)?
[[1, 0], [0, -1]]
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Why can't translation be represented by a 2×2 matrix?
Because translation is not a linear transformation in standard 2D matrix form.
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What are homogeneous coordinates used for?
To represent translation and linear transformations together using a single 3×3 matrix.
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What does the homogeneous coordinate (x, y) become?
[x, y, 1]
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What is the homogeneous matrix for translation by (tx, ty)?
[[1, 0, tx], [0, 1, ty], [0, 0, 1]]
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What is a compound transformation?
A combination of multiple transformations applied in sequence using matrix multiplication.
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Why does the order of transformation matrices matter?
Because matrix multiplication is not commutative; changing the order changes the result.
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What are the steps to rotate a point around an arbitrary point (x₀, y₀)?
Translate to origin, rotate, translate back.
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What is the formula for rotating around a point (x₀, y₀)?
M = T⁻¹ · R · T
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What is an affine transformation?
A transformation that preserves lines and parallelism but not necessarily lengths or angles.
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What is the general 3×3 matrix form of an affine transformation in homogeneous coordinates?
[[a11, a12, t1], [a21, a22, t2], [0, 0, 1]]
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How many degrees of freedom does an affine transformation have?
6 (scale, shear, rotate, translate)
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What do similarity transformations preserve?
Shape and angles, but not size.
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What is the difference between Euclidean, similarity, and affine transformations?
Euclidean preserves distance and angles (3 DOF), similarity preserves shape (4 DOF), affine preserves parallelism (6 DOF).
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What is the column vector representation of a 2D point transformation?
p' = M · p
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What is the row vector representation of a 2D point transformation?
p'ᵀ = pᵀ · Mᵀ
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How do you convert between row and column vector conventions in matrix transformations?
Transpose the matrix and reverse the multiplication order.
61
What is a projective transformation in 2D?
A transformation represented by a 3×3 matrix that includes perspective effects and maps straight lines to straight lines.
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What type of coordinates are used for projective transformations?
Homogeneous coordinates.
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What is the general matrix form of a projective transformation?
[[m11, m12, m13], [m21, m22, m23], [m31, m32, m33]]
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What is the main difference between affine and projective transformations?
Affine transformations preserve parallelism; projective transformations do not necessarily preserve parallel lines.
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How many degrees of freedom does a 2D projective transformation have?
8 degrees of freedom (the scale of the matrix is arbitrary).
67
What additional transformation types do projective transforms include beyond affine?
They include perspective warps and mappings between non-parallel quadrilaterals.
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What is a homography?
A 3×3 matrix that represents a linear transformation in homogeneous coordinates, including translation, scale, rotation, shear, and perspective.
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What is required to compute a homography between two images?
At least 4 point correspondences.
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What is the goal of image alignment?
To compute a 2D transformation that warps one image into the coordinate system of another.
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What is the minimum number of points required to estimate an affine transform?
3 points.
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What is the minimum number of points required to estimate a projective transform (homography)?
4 points.
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What does backward mapping mean in image warping?
Using the inverse transform to map output pixel coordinates back to input image coordinates.
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What is a common issue with backward mapping?
It can result in coordinates that are out of image bounds or between pixels (non-integer values).
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What is nearest-neighbour interpolation?
A method that assigns each target pixel the value of the closest source pixel.
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What is bilinear interpolation?
A method that uses the 4 closest pixels and performs weighted averaging based on the relative distances.
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What is the formula for bilinear interpolation in two steps?
First interpolate horizontally, then vertically using weights α and β.
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What is the advantage of bilinear interpolation over nearest-neighbour?
It provides smoother, more accurate results, especially during scaling or rotation.
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What are vanishing points in projective geometry?
Points where parallel lines in 3D appear to converge in 2D perspective images.
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What does projective transformation allow that affine does not?
It allows modeling of perspective effects, such as those seen in photographs of planar scenes.
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What does the projective transform matrix apply to?
A point in homogeneous coordinates as a matrix-vector multiplication.
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What is the scale factor 's' in projective transformation equations?
A factor applied to ensure the result can be normalized back from homogeneous coordinates.
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Why are homogeneous coordinates used in 2D transformations?
They allow translations and projective transformations to be expressed using matrix multiplication.
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What is forward mapping in image warping?
Mapping each input pixel to a location in the output image using a transformation matrix.
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What is the downside of forward mapping?
It may leave gaps (unassigned pixels) in the output image.
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How is backward mapping typically implemented in practice?
By computing the inverse transform and using interpolation to get the pixel value from the input image.
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Why do projective transforms use 3×3 matrices instead of 2×2 or 2×3?
Because they can express all linear, affine, and perspective transformations in a single unified form.
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What does a homography represent in computer vision?
A plane-to-plane transformation, such as between two images of the same scene taken from different viewpoints.
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What is image stitching?
Combining multiple images into one by aligning and blending them using transformations like homographies.
