Session 5

------ is a mathematical framework that is used in computer vision to analyze and manipulate images and their properties, such as perspective, scale, and rotation. It involves transforming the 3D world into a 2D image by mapping the points in the world onto the image plane using a series of mathematical operations. This allows for accurate analysis and manipulation of images in a way that is independent of the specific camera or viewpoint used to capture them.

• Faysals Geometry

• Euclidean Geometry

• Spatial Geometry

• Projective Geometry

Select the applications of projective geometry in computer vision

• Reconstruction: Projective geometry is used for 3D reconstruction from multiple 2D images.

• Perspective correction: Projective geometry allows for perspective correction, which is essential for accurate image analysis and manipulation.

• Object recognition and tracking: Projective geometry is used for object recognition and tracking in computer vision applications.

• Camera calibration: Projective geometry is used for camera calibration, which is essential for accurate image analysis and manipulation.

Which of the following has these advantages?

• The central projection process (pinhole model-perspective transformation) is projective: it is not Euclidean (does not preserve distances) nor affine (does not preserve parallelism)

• It provides a linear model (if there are no distortions) for the process of image acquisition

• It allows for the representation of entities in the infinite (vanishing point)

• Homogenization of elements. Duality between points and lines lying on a plane.

• Transformation between projective planes (between scene and image or between two images)

• Correlation between points and lines when operating on multiple projective planes(e.g. two images)

• Uradumasius

  Geometry

• Euclidean Geometry

• Projective Geometry

• Affine Geometry

Which of the following geometries has these transformations / properties:

• Invariance of lengths and areas.

• Circles can be defined.

• Rotation and translation transformations.

• Cartesian coordinates.

• Homogeneous Geometry

• Euclidean Geometry

• Affine Geometry

• Projective Geometry

Which of the following geometries has these transformations / properties?

• Parallelism invariance.

• Distance ratios invariance

• Parallelograms, parabolas, ellipses and hyperbolas can be defined

• Oblique cartesian coordinates

• Homogeneous Geometry

• Euclidean Geometry

• Affine Geometry

• Projective Geometry

Which of the following geometries has these transformations / properties?

• Cross-ratio invariance

• Conic sections, quadric surfaces

• Projective or homogeneous coordinates

• Homogeneous Geometry

• Euclidean Geometry

• Affine Geometry

• Projective Geometry

The (2D) image plane (2D) captures the scene’s (3D) information.Some information is lost as all points belonging to the line betweenthe optical center and each 3D point are projected as a single point inthe image.This process is modelled using a ------

• Euclidean Transformation

• Affine Transformation

• Perspective Transformation

• Projective Transformation

In Euclidean Geometry there are three types of conic sections: Hyperbola, Ellipse and Parabola

• True

• False

What is the difference between projective planes Pn, P2 and P3?

Answer explanation

The projective plane Pn is a mathematical space used to model the relationship between 2D images and 3D space, where P2 specifically refers to the projective plane of dimension 2 used to represent the relationship between 2D images and 3D space, and P3 refers to the projective space of dimension 3 used to represent the relationship between 3D space and 2D images captured by a camera.

Projective plane P2 is used for tasks such as perspective correction, object recognition, and camera calibration.

• True

• False