EE 290-T Lecture #3 Notes Scribe: Nikhil Naikal ([email protected]) Topics: 1. Euclidean Geometry 2. Projective Geometry 3. Projective Space 4. 3D to 2D projection 1. Euclidean Geometry • Euclidean geometry describes the world well. • It allows to measure lengths and angles. • Length, angles, parallelism, orthogonality, and all other properties that are related via a linear/Euclidean transform are preserved. • Euclidean coordinates of a point in a plane are given by a 2-tuple ~ ሾݑ,ݒሿ். • Ex: Consider the transformation that rotates 2 points, ܲଵ,ܲଶ, in a plane counter-clockwise θo with respect to the origin as shown in Fig. 1. The transformation can be represented by the linear equations, ܲଵᇱ= ܴ.ܲଵ=ቂcosθ sinߠ−sinߠ cosߠቃ.ܲଵ 1.1 ܲଶᇱ= ܴ.ܲଶ =ቂcosθ sinߠ−sinߠ cosߠቃ.ܲଶ 1.2 Since the transformation is Euclidean, the length between the two points, and the angle subtended at the origin, before and after the transformation remains the same. Q – Why do we need Projective geometry? A – Because 3D objects are projected on to a 2D plane on capturing an image. O P1 P2 P1’ P2’ θ llll llll Fig. 1. – Rotating a point about the origin is a Euclidean transformation.2. Projective Geometry • Describes projection to lower dimensions well. For instance, parallel lines in 3D space are no longer parallel in a 2D image projection, and appear to meet. Such properties are captured well by projective geometry. • The horizon has the same projection. • Since parallelism between lines is not preserved, distances or angles are not preserved either. • Projective geometry describes a larger class of transformations. It is an extension of Euclidean geometry and deals with the perspective projection of a camera. • Projective coordinates of a point in a plane are homogenized and represented by a 3-tuple: ሾݑ,ݒ,1ሿ். • Rule: Scaling the projective coordinates by a non-zero factor does not change the Euclidean point it represents as it is homogenized. i.e., ሾݑ,ݒ,1ሿ்≡ሾߣݑ,ߣݒ,ߣሿ். 3. Projective Space The Euclidean coordinates of a point in a plane can be represented by a 2-tuple: ሾݑ,ݒሿ். It’s projective coordinates are obtained by appending a 1 to the vector: ሾݑ,ݒ,1ሿ். By representing the point by this 3-tuple in projective coordinates, a one-to-one mapping is established between the 2D point in Euclidean coordinates and the corresponding point in projective coordinates. Thus, scaling the point by a non-zero zero factor does not change the Euclidean point it represents as it is homogenized. i.e., ሾݑ,ݒ,1ሿ்≡ሾߣݑ,ߣݒ,ߣሿ். Thus, projective coordinates represent naturally the operations performed by cameras. Definition: The space of (݊ + 1)-tuples of coordinates, with the rule that proportional (or scaled) (݊ + 1)-tuples represent the same point, is called a projective space of dimension ݊, and is denoted ۾௡. In general, given coordinates in ܀௡, the corresponding projective coordinates are obtained as, ሾݔଵ,ݔଶ,…,ݔ௡ሿ்܀೙→۾೙ሱۛۛۛሮሾݔଵ,ݔଶ,…,ݔ௡,1ሿ். 3.1 To transform a point from projective coordinates back to Euclidean coordinates, we just need to divide by the last coordinate and the drop the last coordinate, ሾݔଵ,ݔଶ,…,ݔ௡,ݔ௡ାଵሿ்۾೙→܀೙ሱۛۛۛሮ൤ݔଵݔ௡ାଵ,ݔଶݔ௡ାଵ,…,ݔ௡ݔ௡ାଵ൨். 3.2 Points with last coordinate ݔ௡ାଵ≠ 0 are usual points with representations in ܀௡, but points of the form ሾݔଵ,ݔଶ,…,ݔ௡,0ሿ், do not have an equivalent representation in Euclidean coordinates. If we consider them as the limit of ሾݔଵ,ݔଶ,…,ݔ௡,ߣሿ், when ߣ → ∞, (i.e. the limit of ሾݔଵ/ߣ,ݔଶ/ߣ,…,ݔ௡/ߣ,1ሿ்) then they represent the limit of a point in ܀௡ going to infinity in the direction ሾݔଵ,ݔଶ,…,ݔ௡ሿ். Such points are called points at infinity. Thus projective space contains more points than the Euclidean space of same dimensions, and is a union of the usual space ܀௡ and the set of points at infinity. i.e., ۾௡= ܀௡∪ሼሾݔଵ,ݔଶ,…,ݔ௡,0ሿ்ሽ. 3.3 As a result of this formalism, points at infinity are represented without exceptions in projective space.4. 3D to 2D Projection Fig.2. represents the pinhole model of projection of a point in 3D onto the 2D image plane, R. Point C is the optical center, and does not belong to R. The projection, m, of a point in 3D space, M, is the intersection of the optical ray, (C, M), with the image plane. The optical axis is the line through C perpendicular to the image plane, pierces the image plane at the principal point, c. The camera coordinate system is established by the orthonormal vectors, x, y and z, centered at the optical center, C. Here the image coordinate system is aligned with the camera coordinate system. The distance between C and R is the focal length, and is chosen as unity without loss of generality. Thus the relationship between the camera-coordinates of point M=ሾܺ,ܻ,ܼሿ், and the corresponding image-coordinates, m =ሾݑ,ݒሿ் is, ݑ = ܼܺ, ݒ =ܻܼ 4.1 Once the projection has been captured by the image, the true 3D depth of the point M, can no longer be inferred from a single image due to the inherent nature of 3D to 2D projection. Thus any other point, M’=ሾߣܺ,ߣܻ,ߣܼሿ், that lies on the optical ray (C, M), also has the same 2D-projection, m. This depth ambiguity cannot be inferred from a single image of the point using geometry alone, and the only information available from the single image projection is the vector along which the 3D point lies in space. C x y z u v M=ሾܺ,ܻ,ܼሿ் m Image plane Focal Point Point in 3D space Projection onto 2D plane Fig.2. – Perspective projection of a 3D point onto a 2D image plane M’=ሾߣܺ,ߣܻ,ߣܼሿ் Scaled point in