Where am I? Using Vanishing Points
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1 Whee am I? Using Vanishing Points
2 Whee was I (how high)? Taken fom my hotel oom (6 th floo) Taken fom beach
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8 Vanishing point Vanishing line fo hoizon Vanishing point What can vanishing line tell us about me? Hoizon Camea pitch angle (looking down) Camea oll angle (tilted towad ight) Kelle Hall
9 Whee am I w..t. Gound Plane? Camea X u Y v K t C W Z 1 1 How to compute?
10 What can a Vanishing Point tell us about? Camea v
11 What can a Vanishing Point tell us about? Camea Z v
12 What can a Vanishing Point tell us about? Camea Z C v K t W 1 v
13 Single Vanishing Point Camea K -1 v 3 Z C v K t W 1 v K 3 v
14 Single Vanishing Point Camea K -1 v 3 Z C v K t W 1 v K 3 Kv because 3 is a unit vecto Kv Z vanishing point tells us about the suface nomal of the gound plane v
15 Single Vanishing Point Camea K -1 v 3 Z C v K t W 1 v K 3 Kv because 3 is a unit vecto Kv Z vanishing point tells us about the suface nomal of the gound plane Rotation ambiguity v
16 Single Vanishing Point Camea Kv -1 Kv 3-1 K -1 v 3 Z Roll and pitch angle can be computed by the goun v
17 Single Vanishing Point Camea Kv -1 Kv 3-1 K -1 v 3 Z Roll and pitch angle can be computed by the goun Yaw: otation about z axis: R yaw cos -sin sin cos 1 v
18 Single Vanishing Point Camea Kv -1 Kv 3-1 K -1 v 3 Z Roll and pitch angle can be computed by the goun Yaw: otation about z axis: R yaw Pitch: otation about y Raxis: pitch cos -sin sin cos 1 cos sin 1 -sin cos v
19 Single Vanishing Point Camea Kv -1 Kv 3-1 K -1 v 3 Z Roll and pitch angle can be computed by the goun Yaw: otation about z axis: R yaw Pitch: otation about y Raxis: pitch cos -sin sin cos 1 cos sin 1 -sin cos Roll: otation about x axis: R oll 1 cos -sin sin cos v
20 Single Vanishing Point Camea Kv -1 Kv 3-1 K -1 v 3 Z Roll and pitch angle can be computed by the goun Yaw: otation about z axis: R yaw Pitch: otation about y Raxis: pitch cos -sin sin cos 1 cos sin 1 -sin cos Roll: otation about x axis: R oll 1 cos -sin sin cos v -sin T yaw pitch oll cos sin R R R? cos cos
21 Single Vanishing Point Camea Kv -1 Kv 3-1 K -1 v 3 Z Roll and pitch angle can be computed by the goun Yaw: otation about z axis: R yaw Pitch: otation about y Raxis: pitch cos -sin sin cos 1 cos sin 1 -sin cos Roll: otation about x axis: R oll 1 cos -sin sin cos v -sin T yaw pitch oll cos sin R R R cos cos
22 Single Vanishing Point Camea Kv -1 Kv 3-1 K -1 v 3 Z Roll and pitch angle can be computed by the goun -sin T yaw pitch oll cos sin R R R cos cos Pitch: Roll: tan tan v
23 Single Vanishing Point Camea Kv -1 Kv 3-1 K -1 v 3 Z Roll and pitch angle can be computed by the goun -sin T yaw pitch oll cos sin R R R cos cos Pitch: Roll: tan tan v
24 Single Vanishing Point (Execise) ComputeCameaUsingVanishingPoint.m v f = 1224; K = [f size(im,2)/2; f size(im,1)/2; 1]; m1 = [2563;25;1]; m2 = [2439;545;1]; m3 = [571;25;1]; m4 = [723;498;1]; l1 = GetLineFomTwoPoints(m1,m2); l2 = GetLineFomTwoPoints(m3,m4); v1 = GetPointFomTwoLines(l1,l2); v1 = v1/v1(3); 3 = inv(k)*v1/nom(inv(k)*v1) pitch = atan(-3(1)/nom(3(2:3))) oll = atan(3(2)/3(3)) 3 = pitch =.736 oll = 1.116
25 Single Vanishing Point (Execise) ComputeCameaUsingVanishingPoint.m f = 1224; K = [f size(im,2)/2; f size(im,1)/2; 1]; m1 = [2563;25;1]; m2 = [2439;545;1]; m3 = [571;25;1]; m4 = [723;498;1]; l1 = GetLineFomTwoPoints(m1,m2); l2 = GetLineFomTwoPoints(m3,m4); v1 = GetPointFomTwoLines(l1,l2); v1 = v1/v1(3); 3 = inv(k)*v1/nom(inv(k)*v1) pitch = atan(-3(1)/nom(3(2:3))) oll = atan(3(2)/3(3)) 3 = pitch =.736 oll = 1.116
26 Two Vanishing Points Camea v Y v X
27 Camea 3 Two Vanishing Points 1 2 X 1 v Y v X Y 1
28 Camea 3 Two Vanishing Points 1 2 X 1 C vx K t W X K 1 v Y v X Y 1
29 v Y Camea Two Vanishing Points 3 v X 1 2 X Y 1 1 C vx K t W X K 1 C vy K t W Y K 2
30 v Y Camea Two Vanishing Points 1 X C v X K t W X 3 K 1 2 v X Y 1 C vy K t W Y K v K v -1-1 X 1, -1 2 K vx -1 1 K K v Y Y : Othogonality constaint
31 Geometic Intepetation -1 KvX X
32 Geometic Intepetation -1 KvX X KvY Y 1
33 Geometic Intepetation -1 KvX X KvY Y 1
34 Geometic Intepetation K v K v -1-1 X Y -1 KvX X KvY Y 1
35 Geometic Intepetation (Tanslation Ambiguity)
36 Geometic Intepetation (Tanslation Ambiguity)
37 Geometic Intepetation (Tanslation Ambiguity)
38 Two Vanishing Points Camea ComputeCameaUsingTwoVanishingPoints.m f = 4; K = [f size(im,2)/2; f size(im,1)/2; 1]; v Y 2 v X 1 3 l11 = GetLineFomTwoPoints(m11,m12); l12 = GetLineFomTwoPoints(m13,m14); l21 = GetLineFomTwoPoints(m21,m22); l22 = GetLineFomTwoPoints(m23,m24); v1 = GetPointFomTwoLines(l11,l12); v2 = GetPointFomTwoLines(l21,l22); 1 = inv(k)*v1/nom(inv(k)*v1); 2 = inv(k)*v2/nom(inv(k)*v2); 3 = Vec2Skew(1)*2; R = Not othogonal mat det(r) =.577 R *R =
39 Two Vanishing Points f = 1224; K = [f size(im,2)/2; f size(im,1)/2; 1]; ComputeCameaUsingTwoVanishingPoints.m Change focal length l11 = GetLineFomTwoPoints(m11,m12); l12 = GetLineFomTwoPoints(m13,m14); l21 = GetLineFomTwoPoints(m21,m22); l22 = GetLineFomTwoPoints(m23,m24); v1 = GetPointFomTwoLines(l11,l12); v2 = GetPointFomTwoLines(l21,l22); 1 = inv(k)*v1/nom(inv(k)*v1); 2 = inv(k)*v2/nom(inv(k)*v2); 3 = Vec2Skew(1)*2; R = Othogonal matix! det(r) =.9948 R *R =
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