Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
A Projective Drawing SystemOsama Tolba, Julie Dorsey & Leonard McMillan, ’01
Elodie Fourquet
June 16, 2004
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
1 Paper Motivation2D Drawings under-estimated in Computer Graphics2D Drawings in Art vs. in Computer Program
2 Paper IssuesAimsNeed Computer Vision Techniques
3 Projective GeometryDifferent Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
4 Projective Geometry in Paper
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
2D Drawings under-estimated in Computer Graphics2D Drawings in Art vs. in Computer Program
Perspective Scene
Often solved with 3D computer graphics.- 3D models and ray-tracing.
Rarely composed as traditional illustration.- artist or architecture sketches.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
2D Drawings under-estimated in Computer Graphics2D Drawings in Art vs. in Computer Program
Perspective Scene
Often solved with 3D computer graphics.- 3D models and ray-tracing.
Rarely composed as traditional illustration.- artist or architecture sketches.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
2D Drawings under-estimated in Computer Graphics2D Drawings in Art vs. in Computer Program
Perspective Scene
Often solved with 3D computer graphics.- 3D models and ray-tracing.
Rarely composed as traditional illustration.- artist or architecture sketches.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
2D Drawings under-estimated in Computer Graphics2D Drawings in Art vs. in Computer Program
3D Modelling vs. 2D Drawing
3D scene description
Dynamic, walkthrough
Tedious to construct
Perspective algorithmembedded projection
2D freehand sketch
Static, single fix viewpoint
Digital notepad, scanning
User 2D fix perspective,no possible re-projection
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
2D Drawings under-estimated in Computer Graphics2D Drawings in Art vs. in Computer Program
3D Modelling vs. 2D Drawing
3D scene description
Dynamic, walkthrough
Tedious to construct
Perspective algorithmembedded projection
2D freehand sketch
Static, single fix viewpoint
Digital notepad, scanning
User 2D fix perspective,no possible re-projection
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
2D Drawings under-estimated in Computer Graphics2D Drawings in Art vs. in Computer Program
3D Modelling vs. 2D Drawing
3D scene description
Dynamic, walkthrough
Tedious to construct
Perspective algorithmembedded projection
2D freehand sketch
Static, single fix viewpoint
Digital notepad, scanning
User 2D fix perspective,no possible re-projection
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
2D Drawings under-estimated in Computer Graphics2D Drawings in Art vs. in Computer Program
Traditional Art 2D Drawings
Renaissance
Still fundamental part of art and design education
Its basic elements
Vanishing pointsProjective grids
Provide 3D illusion
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
2D Drawings under-estimated in Computer Graphics2D Drawings in Art vs. in Computer Program
Traditional Art 2D Drawings
Renaissance
Still fundamental part of art and design education
Its basic elements
Vanishing pointsProjective grids
Provide 3D illusion
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
2D Drawings under-estimated in Computer Graphics2D Drawings in Art vs. in Computer Program
Traditional Art 2D Drawings
Renaissance
Still fundamental part of art and design education
Its basic elements
Vanishing pointsProjective grids
Provide 3D illusion
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
2D Drawings under-estimated in Computer Graphics2D Drawings in Art vs. in Computer Program
Traditional 2D Drawings Computer Program
DeceivingUniversal Euclidean representation of points (2 coordinates)What about powerful representation of 2D points, such as setof projective 2D points (3 coordinates) ??Euclidean points are a subset of 2D projective points.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
AimsNeed Computer Vision Techniques
Paper Goals
Give to 2D drawing 3D-like capabilities.
Combine multiple 2D drawings together in a 3D scene.
Intuitive navigation of a virtual camera with zooming androtation. No translation since no depth information.
Primitives transfomations in 3D with apparent translation androtation.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
AimsNeed Computer Vision Techniques
Paper Goals
Give to 2D drawing 3D-like capabilities.
Combine multiple 2D drawings together in a 3D scene.
Intuitive navigation of a virtual camera with zooming androtation. No translation since no depth information.
Primitives transfomations in 3D with apparent translation androtation.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
AimsNeed Computer Vision Techniques
The Problem
Give to the 2D stroke some 3D definition when integrated inthe 3D-like drawing system.Each stroke = list of projective points (back-projecting imagepoints to lie on surface unit sphere).Projective representation to generate novel re-projection ofthe drawing. Re-projection can be interpreted as rotating andzooming a camera about a single fix point in 3D space.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
AimsNeed Computer Vision Techniques
The Problem
Give to the 2D stroke some 3D definition when integrated inthe 3D-like drawing system.Each stroke = list of projective points (back-projecting imagepoints to lie on surface unit sphere).Projective representation to generate novel re-projection ofthe drawing. Re-projection can be interpreted as rotating andzooming a camera about a single fix point in 3D space.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
AimsNeed Computer Vision Techniques
Computer Vision Projection Applications
Geometric Scene Reconstruction
Shape from Motion ProblemComputing 3-D motion from two images
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Projective Plane
<P2 = {rA},A ∈ <3r 6= 0, set of lines passing by origin.
(a, b, c) = λ(a, b, c), λ 6= 0
(3,−5, 2) = (−9, 15,−6)
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Projective Plane
<P2 = {rA},A ∈ <3r 6= 0, set of lines passing by origin.
(a, b, c) = λ(a, b, c), λ 6= 0
(3,−5, 2) = (−9, 15,−6)
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Projective Plane
<P2 = {rA},A ∈ <3r 6= 0, set of lines passing by origin.
(a, b, c) = λ(a, b, c), λ 6= 0
(3,−5, 2) = (−9, 15,−6)
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Projective Plane
<P2 = {rA},A ∈ <3r 6= 0, set of lines passing by origin.
Or points on the unit sphere.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Projective Plane
<P2 = {rA},A ∈ <3r 6= 0, set of lines passing by origin.
Or points on the unit sphere.
Or unit vectors.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Point Definition
(m1,m2,m3), homogeneous coordinates.
λ(m1,m2,m3) = (m1,m2,m3) represents same point.
To keep reasonable magnitude, homogeneous coordinates arekept as “unit vector”.
N-vector of point (a, b) on image plane : ~m = ±N
abf
.
Where Z=f = image plane, & N[~u] = normalization of ~u.
If m3 6= 0 identified with image plane point(f ∗m1/m3, f ∗m2/m3), inhomogeneous coordinates.
If m3 = 0, point = (m1,m2, 0) is located at infinity and iscalled an ideal point. Why ?
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Example
P = (X ,Y ,Z ) projected on image plane Z = f (origino = (0, 0, f ) and x− and y−axes are parallel to X− andY−axes).Intersection ray from V → P and Z = f .p = (x , y) of P = (X ,Y ,Z ) are :
x = f XZ y = f Y
Z
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
In Summary
Scene coordinates can be identified with homogeneouscoordinates on the image plane.The N-vector ~m of a point P can be interpreted as the unitvector starting from the viewpoint V and pointing toward P.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Line Definition
(n1, n2, n3), homogeneous coordinates.
λ(n1, n2, n3) = (n1, n2, n3) represents same line.
To keep reasonable magnitude, homogeneous coordinates arekept as “unit vector”.
N-vector of line Ax + By + C = 0 on image plane :
~m = ±N
AB
C/f
Z=f = image plane & N[~u] = normalization of vector ~u.
If n1 or n2 6= 0 line appears on image plane asn1x + n2y + n3f = 0.
If n1 = n2 = 0, line interpreted to be the ideal line at infinity.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Example
Plane AX + BY + CZ = 0 through viewpoint V.~n = (A,B,C ) surface normal.Plane intersects image plane Z = f along the lineAx + By + Cf = 0, whose homogeneous coordinates are(A,B,C ).
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
In Summary
The N-vector ~n of a line l can be interpreted as the unitvector normal to the plane passing through viewpoint V andintersecting image plane along l.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Vanishing Point
Theorem 1
A line in the scene extending along unit vector ~m has, whenprojected, a vanishig point of N-vector ~m.
Corollary 1
Projection of parallel lines in the scene intersect at a commonvanishing point.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Vanishing Point
Theorem 1
A line in the scene extending along unit vector ~m has, whenprojected, a vanishig point of N-vector ~m.
Corollary 1
Projection of parallel lines in the scene intersect at a commonvanishing point.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Vanishing Line
Theorem 1’
A planar surface in the scene whose unit surface normal is ~n has,when projected, a vanishig line of N-vector ~n.
Corollary 1’
Projection of planar surfaces mutually parallel in the scene define acommon vanishing line.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Vanishing Line
Theorem 1’
A planar surface in the scene whose unit surface normal is ~n has,when projected, a vanishig line of N-vector ~n.
Corollary 1’
Projection of planar surfaces mutually parallel in the scene define acommon vanishing line.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
In Summary
If a vanishig point is detected on the image plane, its N-vectorindicates the 3D orientation of the line.
If a vanishig line is detected on the image plane, its N-vectorindicates the surface normal of the planar surface.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
In Summary
If a vanishig point is detected on the image plane, its N-vectorindicates the 3D orientation of the line.
If a vanishig line is detected on the image plane, its N-vectorindicates the surface normal of the planar surface.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
In Summary
If a vanishig point is detected on the image plane, its N-vectorindicates the 3D orientation of the line.
If a vanishig line is detected on the image plane, its N-vectorindicates the surface normal of the planar surface.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Collineation Definition
A one-to-one mapping thats maps
the set of all points to a set of all pointsand the set of all lines to the set of all lines is a collineation if
1 collinear points are mapped to collinear points
2 concurrent lines are mapped to concurrent lines
3 the incidence is preserve (= if a point is on a line, the mappedpoint is on the mapped line).
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Collineation Formula
Linear mapping of N-vectors in the form~m′ = ±N[~AT ~m], ~n′ = ±N[~A−1~n]
where ~A : 3-dimensional singular matrix~m and ~n : N-vectors of original point and line ~m′ and ~n′ :N-vectors of point and line after mapping
Proof....
Involve invariance of cross-ratio and projective coordinatesdefined in tems of cross ratio under collineations !!
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Collineation Formula
Linear mapping of N-vectors in the form~m′ = ±N[~AT ~m], ~n′ = ±N[~A−1~n]
where ~A : 3-dimensional singular matrix~m and ~n : N-vectors of original point and line ~m′ and ~n′ :N-vectors of point and line after mapping
Proof....
Involve invariance of cross-ratio and projective coordinatesdefined in tems of cross ratio under collineations !!
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Collineation Formula
Linear mapping of N-vectors in the form~m′ = ±N[~AT ~m], ~n′ = ±N[~A−1~n]
where ~A : 3-dimensional singular matrix~m and ~n : N-vectors of original point and line ~m′ and ~n′ :N-vectors of point and line after mapping
Proof....
Involve invariance of cross-ratio and projective coordinatesdefined in tems of cross ratio under collineations !!
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Collineation Matrix ~A
Equation on blackboard... maybe ??
Set of all colineations of 2D projective space forms a group oftransformations = group of 2D projective transformations.
Matrix ~A, 8 degrees of freedom (scale indeterminacy).
So need 4 points (or lines) in general postion over two imagesto define unique collineation.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Different Projective Plane RepresentationPoints and LinesVanishing Points and LinesCollineations : a special 1-to-1 mapping
Collineation Matrix ~A
Equation on blackboard... maybe ??
Set of all colineations of 2D projective space forms a group oftransformations = group of 2D projective transformations.
Matrix ~A, 8 degrees of freedom (scale indeterminacy).
So need 4 points (or lines) in general postion over two imagesto define unique collineation.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Where Projective Geometry Appears
2D image projected to unit sphere, for further re-projection ondifferent camera zooming and rotation.
User aligns 2D image to integrate perspective of others 2Ddrawings already included in unit sphere.
Semi-automatic alignment ?User selects 2D image vanishing points (correspond to 2orthogonal directions from view) to compute focal distance f.
Perspective shape manipulation with apparent translation androtation. Image points are transformed using collineationmatrix ~A inferred by user input points, surface normal, andmotion trajectory selected by user from vanishing point.Finding information for the 4 points mapping needed.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Where Projective Geometry Appears
2D image projected to unit sphere, for further re-projection ondifferent camera zooming and rotation.
User aligns 2D image to integrate perspective of others 2Ddrawings already included in unit sphere.
Semi-automatic alignment ?User selects 2D image vanishing points (correspond to 2orthogonal directions from view) to compute focal distance f.
Perspective shape manipulation with apparent translation androtation. Image points are transformed using collineationmatrix ~A inferred by user input points, surface normal, andmotion trajectory selected by user from vanishing point.Finding information for the 4 points mapping needed.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Where Projective Geometry Appears
2D image projected to unit sphere, for further re-projection ondifferent camera zooming and rotation.
User aligns 2D image to integrate perspective of others 2Ddrawings already included in unit sphere.
Semi-automatic alignment ?User selects 2D image vanishing points (correspond to 2orthogonal directions from view) to compute focal distance f.
Perspective shape manipulation with apparent translation androtation. Image points are transformed using collineationmatrix ~A inferred by user input points, surface normal, andmotion trajectory selected by user from vanishing point.Finding information for the 4 points mapping needed.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Where Projective Geometry Appears
2D image projected to unit sphere, for further re-projection ondifferent camera zooming and rotation.
User aligns 2D image to integrate perspective of others 2Ddrawings already included in unit sphere.
Semi-automatic alignment ?User selects 2D image vanishing points (correspond to 2orthogonal directions from view) to compute focal distance f.
Perspective shape manipulation with apparent translation androtation. Image points are transformed using collineationmatrix ~A inferred by user input points, surface normal, andmotion trajectory selected by user from vanishing point.Finding information for the 4 points mapping needed.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Where Projective Geometry Appears
2D image projected to unit sphere, for further re-projection ondifferent camera zooming and rotation.
User aligns 2D image to integrate perspective of others 2Ddrawings already included in unit sphere.
Semi-automatic alignment ?User selects 2D image vanishing points (correspond to 2orthogonal directions from view) to compute focal distance f.
Perspective shape manipulation with apparent translation androtation. Image points are transformed using collineationmatrix ~A inferred by user input points, surface normal, andmotion trajectory selected by user from vanishing point.Finding information for the 4 points mapping needed.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Main Contribution
Apply to computer graphics the projective geometry that isused extensively in vision.
Limited-dynamic 2D drawing navigation
1 by projecting onto unit sphere2 by defining a fix point for the camera.
Provide a drawing system that includes features to helpperspective drawing, vanishing points, grids, semi-automaticalignment.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Main Contribution
Apply to computer graphics the projective geometry that isused extensively in vision.
Limited-dynamic 2D drawing navigation
1 by projecting onto unit sphere2 by defining a fix point for the camera.
Provide a drawing system that includes features to helpperspective drawing, vanishing points, grids, semi-automaticalignment.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Main Contribution
Apply to computer graphics the projective geometry that isused extensively in vision.
Limited-dynamic 2D drawing navigation
1 by projecting onto unit sphere2 by defining a fix point for the camera.
Provide a drawing system that includes features to helpperspective drawing, vanishing points, grids, semi-automaticalignment.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Main Contribution
Apply to computer graphics the projective geometry that isused extensively in vision.
Limited-dynamic 2D drawing navigation
1 by projecting onto unit sphere2 by defining a fix point for the camera.
Provide a drawing system that includes features to helpperspective drawing, vanishing points, grids, semi-automaticalignment.
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Videos
Elodie Fourquet A Projective Drawing System
Paper MotivationPaper Issues
Projective GeometryProjective Geometry in Paper
Questions
Elodie Fourquet A Projective Drawing System