a projective drawing system - university of waterloo 3d illusion elodie fourquet a projective...

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

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

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.

Perspective Scene

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

2D freehand sketch

Traditional Art 2D Drawings

Renaissance

Still fundamental part of art and design education

Its basic elements

Vanishing pointsProjective grids

Provide 3D illusion

Renaissance

Its basic elements

Provide 3D illusion

Renaissance

Its basic elements

Provide 3D illusion

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.

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.

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.

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.

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.

Computer Vision Projection Applications

Geometric Scene Reconstruction

Shape from Motion ProblemComputing 3-D motion from two images

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)

Projective Plane

(a, b, c) = λ(a, b, c), λ 6= 0

(3,−5, 2) = (−9, 15,−6)

Projective Plane

(a, b, c) = λ(a, b, c), λ 6= 0

(3,−5, 2) = (−9, 15,−6)

Projective Plane

Or points on the unit sphere.

Projective Plane

Or points on the unit sphere.

Or unit vectors.

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

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 ?

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

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.

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

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.

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 ).

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.

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.

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.

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.

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.

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.

In Summary

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).

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 !!

Proof....

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.

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.

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.

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.

Main Contribution

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