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13D GeometryOur Plan• Projection from 2D to 3D• Representing 3D pose• Projective Invariants.– Affine invariants (scaled orthographic projection of planar objects).– Projective invariants (planar, perspective).– Lack of invariants for 3D objects.2The equation of projection(Forsyth & Ponce)The equation of projection• Cartesian coordinates:– We have, by similar triangles, that (x, y, z) -> (f x/z, f y/z, -f)– Ignore the third coordinate, and get(x,y,z) → (fxz, fyz)3Weak perspective (scaled orthographic projection)• Issue– perspective effects, but not over the scale of individual objects– collect points into a group at about the same depth, then divide each point by the depth of its group(Forsyth & Ponce)The Equation of Weak Perspective),(),,( yxszyx→• s is constant for all points.4Projection• We’ll talk about a fixed camera, and moving object.• Key point: =111...212121nnnzzzyyyxxxPPoints =yxtssstsssS3,22,21,23,12,11,1Some matrix =nnvvvuuuI2121...The imageSPI=Then:RotationPrrrrrrrrr 3,32,31,33,22,21,23,12,11,1Represents a 3D rotation of the points in P.5First, look at 2D rotation (easier)  −nnyyyxxx2121...cossinsincosθθθθ −=θθθθcossinsincosRMatrix R acts on points by rotating them.• Also, RRT= Identity. RTis also a rotation matrix, in the opposite direction to R.Why does multiplying points by R rotate them?• Think of the rows of R as a new coordinate system. Taking inner products of each points with these expresses that point in that coordinate system. • This means rows of R must be orthonormal vectors (orthogonal unit vectors).• Think of what happens to the points (1,0) and (0,1). They go to (cos theta, -sin theta), and (sin theta, cos theta). They remain orthonormal, and rotate clockwise by theta.• Any other point, (a,b) can be thought of as a(1,0) + b(0,1). R(a(1,0)+b(0,1) = Ra(1,0) + Ra(0,1) = aR(1,0) + bR(0,1). So it’s in the same position relative to the rotated coordinates that it was in before rotation relative to the x, y coordinates. That is, it’s rotated.6Simple 3D Rotation  −nnnzzzyyyxxx212121...1000cossin0sincosθθθθRotation about z axis.Rotates x,y coordinates. Leaves z coordinates fixed.Full 3D Rotation − − −=ααααββββθθθθcossin0sincos0001cos0sin010sin0cos1000cossin0sincosR• Any rotation can be expressed as combination of three rotations about three axes. =100010001TRR• Rows (and columns) of R are orthonormal vectors.• R has determinant 1 (not -1).7• Intuitively, it makes sense that 3D rotations can be expressed as 3 separate rotations about fixed axes. Rotations have 3 degrees of freedom; two describe an axis of rotation, and one the amount.• Rotations preserve the length of a vector, and the angle between two vectors. Therefore, (1,0,0), (0,1,0), (0,0,1) must be orthonormal after rotation. After rotation, they are the three columns of R. So these columns must be orthonormal vectors for R to be a rotation. Similarly, if they are orthonormal vectors (with determinant 1) R will have the effect of rotating (1,0,0), (0,1,0), (0,0,1). Same reasoning as 2D tells us all other points rotate too. • Note if R has determinant -1, then R is a rotation plus a reflection.Full 3D Motion/ProjectionPrrrrrrrrrtttszyx   10000001000100010100013,32,31,33,22,21,23,12,11,1ScaleProjection3D Translation3D Rotation),,(),,(0),,(),,(where3,22,21,23,12,11,13,22,21,23,12,11,13,22,21,23,12,11,1ssssssssssssPstsssstsssyx==• ≡We can just write stxas txand sty as ty.8Definitions and Invariants• A definition of a class means that given a list of properties:– For all props, all objects have that prop.– No other objects have all properties• Invariant is an image property that:– For some objects, property is true for all images.– For all other objects, property is false for all images.Whiteboard• Definitions are composed of invariant properties.Invariants, a brief history• Invariance has long history in perception.Each movement we make by which we alter the appearance of objects should be thought of as an experiment desgned to test whether we have understood correctly the invariant relations of the phenomena before us, that is, their existence in definite spatial relations.– Helmholtz, 1878If invariants of the energy flux at the receptors of an organism exist, and if these invariants correspond to the permanent properties of the environment … then I think thee is new support for … a new theory of perception in psychology.– Gibson, 1967• In math, Erlanger program conceives geometry as study of invariant properties under a group of transformations.9Invariants on a line  =  = 111...111000000......211,21,1213,22,21,23,12,11,12121nyxnyxnnxxxtstsxxxtssstsssvvvuuu• WLOG, line is y=0,z=0.Then, we can show that ||p3-p2||/||p2-p1|| is invariant to this transformation.whiteboard||(s1x3+tx,s2x3+ty)-(s1x2+tx,s2x2+ty)||/||(s1x2+tx,s2x2+ty)-(s1x1+tx,s2x1+ty)||= ||(s1(x3-x2),s2(x3-x2))||/||(s1(x2-x1),s2(x2-x1))||=sqrt(s1^2+s2^2)(x3-x2)/sqrt(s1^2+s2^2)(x2-x1)=(x3-x2)/(x2-x1)10Planar Invariants  =  = 111...111000......21212,21,22,11,121213,22,21,23,12,11,12121nnyxnnyxnnyyyxxxtsstssyyyxxxtssstsssvvvuuuA tp1p2p3p4p4 = p1 + a(p2-p1) + b(p3-p1)A(p4)+ t = A(p1+a(p2-p1) + b(p3-p1)) + t= A(p1)+t + a(A(p2)+t – A(p1)-t) + b(A(p3)+t – A(p1)-t)p4 is linear combination of p1,p2,p3. Transformed p4 is same linear combination of transformed p1, p2, p3.We call (a,b) affine coordinates of p4.11Perspective Projection• Problem: perspective is non-linear.• Solution: Homogenous coordinates.– Represent points in plane as (x,y,w)– (x,y,w), (kx, ky, kw), (x/w, y/w, 1) represent same point.– If we think of these as points in 3D, they lie on a line through origin. Set


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UMD CMSC 828 - 3D Geometry

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