Lecture 9 - Projective Geometry Flashcards
(28 cards)
What are Vanishing Points?
Vanishing Points: Where all parallel lines meet at a single point. Multiple sets of parallel lines will give multiple vanishing
points.
- Can go to infinity.
- They may lie outside the image boundary (they dont need to converge inside the image frame)
What are Vanishing Lines?
Vanishing Lines: Connecting two vanishing points. Vertical parallel lines give a vertical vanishing point
What is the Horizon?
The horizon is a special vanishing line when the set of parallel planes are parallel to the ground reference.
- Changes with the height of the observer
How to calculate vanishing points and lines?
- A point in image is defined by its (u,v) coordinates
- Two points determine a line
- Intersection of two parallel lines will give us a vanishing point
- Two vanishing points will give us the corresponding vanishing line
- Two clicks on each parallel line will solve the problem
What is the further understanding of Vanishing Lines?
- A vanishing line is the result of projecting a set of parallel planes (not necessarily parallel to the image plane) into a 2D image.
- These planes intersect the image plane along a line, which is known as the vanishing line.
What can we find from the vanishing point and lines?
- Camera Orientation:
○ The horizon line (vanishing line of ground plane) gives the camera’s tilt and viewpoint. - Relative Geometry:
○ You can determine whether lines or planes are parallel in 3D space.
○ Can deduce relationships like collinearity, concurrency, and orthogonality. - Metric Information:
○ With just a single image, vanishing points and lines allow estimation of real-world measurements (e.g., object heights). - Scene Reconstruction:
○ Helps with rectification, 3D reconstruction, and understanding perspective layout. - Single View Metrology:
○ Used for height estimation, angle recovery, or distance computation when certain references are known.
What are the understanding of the points and lines in Homogenous coordinates?
REFER TO NOTES
What is the Human height measurement example?
Demonstrates how vanishing geometry can be used for single-view measurement
- The basic idea is if you know the height of one reference object in an image (person or building) you can estimate the height of other objects or people using projective geometry
What are the steps for calculating height?
- Calculate vertical vanishing point
- Calculate the vanishing line of the reference plane
- Compute the metric factor
- Repeat
How do you ensure the measurements are precise when calculating height?
- Radial distortion needs to be removed first
- Robust detection of parallel lines
- Vanishing point detection based on multiple parallel lines
- Heights are not always vertical. Ideally, the vertical lines should meet at the vertical vanishing point.
What are the projective invariants
Projective transformations preserve:
* Collinearity: Points that lie on the same line in 3D will still lie on a line after projection.
* Concurrency: Lines that meet at a point in 3D will still appear to meet at a point in the image.
* Cross-ratio: The most important projective invariant; a value computed from four collinear points, which remains constant under projection.
What is Cross-ratio?
The cross-ratio is the ratio of two ratios of lengths
REFER TO SLIDES FOR FORMULA
What are the properties of Cross-ratio?
-It can always be calculated from an image
- The cross-ratio will always be exactly the same as in the real world
- If a measuremeant is unknown in the real world is can be calculated used the cross-ratio
What are the 6 possible cross-ratios?
REFER TO NOTES
What is Homography in projective geography?
The mapping between twp planes in a projective space
Calculated using a homogrpahy matrix: K[r1, r2, t]
What is Affine homography?
A more appropriate model if the image region in which the homography is computed is small or the image has been acquired with a large focal length.
- Affine homography is a special type of homography whose last row is fixed to h12 = h23 = 0 , h33 = 1
What is Image Rectification?
Transform a perspective image of a distorted plane into a frontal-parallel view (like a top-down orthographic view).
When is it possible to perform image rectification?
To perform rectification, you need:
* The vanishing line of the distorted plane.
* At least 4 known points on that plane.
* Two known reference distances or angles (for metric rectification).
What is the process of image rectification?
Process:
1. Identify the plane to rectify (e.g., a wall or floor).
2. Mark 4 points on this plane (not near the vanishing line).
3. Compute homography matrix H using these points.
4. Apply H to transform the image and flatten the plane.
How does homography play a role in rectification?
For a plane in 3D with Z=0, the mapping reduces to a 2D homography: x’ == 𝐻x
Once 𝐻 is computed, you can remap image pixels to achieve rectified geometry.
What are the Geometric Invarients?
- Euclidean Transformation
- Similarity Transformation
- Affine Transformation
- Perspective Transformation
REFER TO NOTES FOR MATRIX AND PROCESS
What does Similarity Transformation preserve?
Preserves:
* Angles
* Ratios of lengths
* Ratios of areas
* Parallelism
* Concurrency
* Collinearity
What does Affine Transformation preserve?
Preserves:
* Ratios of lengths
* Ratios of areas
* Parallelism
* Concurrency
* Collinearity
* Cross-ratio
What does Perspective Transformation preserve?
Preserves:
* Concurrency
* Collinearity
* Cross-ratio
