[CV] 2. Image Formation 📸

What are geometric primitives and why are they important in Computer Vision?

• Geometric primitive: basic elements (building blocks) to describe shapes, mainly points, lines, and planes.

• Important in CV: many problems in CV involve finding their positions or relationships. E.g where a 3D point appears in an image or where 2 lines intersect

Why do parallel lines in 3D world appear to meet in 2D image?

• because of perspective projection

• A camera projects the 3D world onto a 2D image plane through a single focal point.

• Parallel lines can therefore appear to converge at vanishing point

What are homogeneous coordinates and why do we use them?

• Instead of represent 2D points with (x,y), we represent it as (wx,wy,w) where w is NOT 0

• Useful because they allow us to represent points at infinity and express transformations such as translation and projection using matrix multiplication

Why can homogeneous vectors represent the same point?

• Homogeneous coordinates are defined up to scale

• e.g (x,y,1) ~ (2x,2y,2) ~ (5x,5y,5)

• Dividing each by its last coordinate produces the same 2D point

• The direction of the homogeneous vector matters, not its magnitude!!

How do we convert between inhomogeneous, augmented, and homogeneous coordinates?

• Inhomogeneous: (x,y)

• Augmented: (x,y,1)

• Homogeneous: (wx,wy,w)

To convert homogeneous → inhomogeneous:

(x1,x2,x3) → (x1/x3 , x2/x3)

What does w=0 mean in homogeneous coordinates?

• w = 0 represents a point at infinity

• It cannot be converted into an ordinary finite 2D point

• Allows homogeneous coordinates to represent the vanishing points of parallel lines

How is 2D line represented in homogeneous coordinates?

a Cartesian line is ax + by + c = 0 and its homogeneous representation is

How do we find the line joining 2 points using homogeneous coordinates?

• Take their cross product (in the picture)

• The result is homogeneous representation of the line passing through both points

How do we find the intersection of 2 lines?

• take the cross-product of their homogeneous line vectors (in the picture)

• So same cross-product idea handles both joining points and intersecting lines

What happens when 2 parallel lines are intersected in homogeneous coordinates?

Their cross product produces a point (x,y,0) because w=0, a point at infinity, corresponding to their vanishing point

What is the line at infinity?

• It is set of all homogeneous points whose last coordinates is w=0.

• Parallel lines meet at points at infinity, and all such points lie on the line at infinity

How are homogeneous coordinates extended from 2D to 3D?

• a 3D point (x,y,z) becomes (wx,wy,wz,w)

• A plane is represented by ax + by+ cz + d= 0

• A points lies on the plane when the homogeneous inner product between plane and point is 0

What does a translation do to an object?

Translation shifts every point by a fixed vector:

x' = x+t

It changes the object’s position but preserves its orientation, size, and shape.

What does a 2D rotation do?

• Rotation : rotates points around the origin by an angle theta:

• Preserves distances and angles

What is Euclidean or rigid-body transformation?

• Euclidean transformation: combines rotation and translation:

• Moves and object without deforming it, → length and angles are preserved

• In homogeneous coordinates, rotation and translation can be combined into one matrix multiplication

Difference of translation, Euclidean, affine, similarity, and projective in terms?

What is similarity transformation?

• Similarity transformation: adds a uniform scale (s) to a rigid transformation:

• Preserves angles and shape but objects’ size can change

What is an affine transformation?

• affine transformation: allows stretching and shearing

• Parallel lines remain parallel but angles nd relative lengths may change

What is a projective transformation or homography?

• It preserves straight lines, but parallelism is NOT guaranteed.

• Therefore, parallel lines can converge to vanishing points.

What is the hierarchy of 2D transformations and their degrees of freedom (DoF)?

• Translation → 2 DoF

• Euclidean → 3 DoF

• Similarity → 4 DoF

• Affine → 6 DoF

• Projective → 8 DoF

As we move down the hierarchy, transformations become more flexible but preserve fewer geometric properties.

What does Degrees of Freedom (DoF) mean?

• DoF : the number of independent parameters needed to specify a transformation.

• More DoF → more flexibility, but generally the transformation is harder to estimate.

Do lines transform using the same matrix as points ?

How do 3D transformations differ from 2D transformations?

• The concepts are the same, but:

  • Rotation: 2 × 2 → 3 × 3 matrix

  • Translation: 2D → 3D vector

• In homogeneous coordinates, 3D transformations can be represented using 4 × 4 matrices.

• All transformations in the hierarchy preserve straight lines.

How is a projective transformation used in panorama stitching?

The basic pipeline is:

1. Find correspondences
→ 2. Estimate a homography
→ 3. Warp and blend

The homography aligns overlapping images so they can be combined into one panorama.