image transform - university of minnesota · 2018-01-28 · euclidean transform se(3) rotation...
TRANSCRIPT
![Page 1: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/1.jpg)
Image Transform
Panorama Image (Keller+Lind Hall)
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Image Warping (Coordinate Transform)
2 1( ) = ( )I v I u
I1 I2
u
v
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Image Warping (Coordinate Transform)
2 1( ) = ( )I v I u
I1 I2
uv
: Pixel transport
![Page 4: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/4.jpg)
Image Warping (Coordinate Transform)
2 1( ) = ( )I v I u
I1 I2
uv
: Pixel transport
![Page 5: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/5.jpg)
Cf) Image Filtering (Pixel Transform)
gI I2 1= ( )
I1 I2
: Pixel transform
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Uniform Scaling
I1
u
I2
v
2 1( ) = ( )I v I u
![Page 7: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/7.jpg)
Uniform Scaling
I1
u
I2
x x x
y y y
1 1 1
v s u
v s u?
v
2 1( ) = ( )I v I u
![Page 8: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/8.jpg)
Uniform Scaling
I1
u
I2
s sx y=
v
x x x
y y y
1 1 1
v s u
v s u
2 1( ) = ( )I v I u
![Page 9: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/9.jpg)
Aspect Ratio Change
I1I2
u v
s sx y
x x x
y y y
1 1 1
v s u
v s u
2 1( ) = ( )I v I u
![Page 10: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/10.jpg)
Translation
I1
uv
I2
t x
t y
x x x
y y y
1
1
1 1 1
v t u
v t u?
2 1( ) = ( )I v I u
![Page 11: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/11.jpg)
Translation
I1
uv
I2
t x
t y
x x x
y y y
1
1
1 1 1
v t u
v t u
2 1( ) = ( )I v I u
![Page 12: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/12.jpg)
Rotation
I1
u
v I2
x x
y y
cos sin
sin cos
11 1
v u
v u
θ θ
θ θ
θ
?
2 1( ) = ( )I v I u
![Page 13: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/13.jpg)
Rotation
I1
u
v I2
θ
x x
y y
cos sin
sin cos
11 1
v u
v u
θ θ
θ θ
2 1( ) = ( )I v I u
![Page 14: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/14.jpg)
x x x
y y y
cos sin
sin cos
1 1 1
v t u
v t u
θ θ
θ θ
Euclidean Transform SE(3)
Rotation around the center of image
I1 I2
?
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Euclidean Transform SE(3)
x x x
y y y
cos sin
sin cos
1 1 1
v t u
v t u
θ θ
θ θ
Rotation around the center of image
I1 I2
![Page 16: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/16.jpg)
Euclidean Transform SE(3)
Rotation around the center of image
I1 I2Invariant properties
• Length • Angle • Area
Degree of freedom
3 (2 translation+1 rotation)
x x x
y y y
cos sin
sin cos
1 1 1
v t u
v t u
θ θ
θ θ
![Page 17: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/17.jpg)
Euclidean Transform SE(3)
Rotation around the center of image
I1 I2Invariant properties
• Length • Angle • Area
Degree of freedom
3 (2 translation+1 rotation)
x x x
y y y
cos sin
sin cos
1 1 1
v t u
v t u
θ θ
θ θ1 1 1
v R t u
0
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Euclidean Transform SE(3)
I2
Rotate about the image center
?
vu
t
1 1 1
v R t u
0+v Ru t
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Euclidean Transform SE(3)
Rotate about the image center
vu
? t
1 1 1
v R t u
0+v Ru t
![Page 20: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/20.jpg)
Euclidean Transform SE(3)
I2
Rotate about the image center
p
vu
p
1 1 1
v R t u
0+v Ru t
![Page 21: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/21.jpg)
Euclidean Transform SE(3)
I2
Rotate about the image center
= -u u pp
vu
p = -v v p
v Ru
1 1 1
v R t u
0+v Ru t
![Page 22: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/22.jpg)
Euclidean Transform SE(3)
I2
Rotate about the image center
= -u u pp
vu
p = -v v p
v Ru
1 1 1
v R t u
0+v Ru t
v -p R u-p
v Ru-Rp+p
![Page 23: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/23.jpg)
Euclidean Transform SE(3)
I2
Rotate about the image center
= -u u pp
vu
p = -v v p
v Ru
1 1 1
v R t u
0+v Ru t
v -p R u-p
v Ru-Rp+p
t -Rp+p
![Page 24: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/24.jpg)
Similarity Transform
I1 I2
x x x x
y y y y
cos sin
sin cos1
11 1 1 1
R t
0
v t u u
v t u u
θ θ
θ θ?
![Page 25: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/25.jpg)
Similarity Transform
I1 I2
x x x x
y y y y
cos sin
sin cos1
11 1 1 1
R t
0
v t u u
v t u u
θ θ
θ θ
![Page 26: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/26.jpg)
Similarity Transform
I1 I2Invariant properties
• Length ratio • Angle
x x x x
y y y y
cos sin
sin cos1
11 1 1 1
R t
0
v t u u
v t u u
θ θ
θ θ
Degree of freedom
4 (2 translation+1 rotation+1 scale)
![Page 27: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/27.jpg)
Affine Transform
I1 I2
x x
y y1
1 1
R t
0
v u
v u
Euclidean transform
![Page 28: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/28.jpg)
Affine Transform
I1 I2
x x
y y1
1 1
R t
0
v u
v ux 11 12 13 x x
y 21 22 23 y y1
0 0 11 1 1
A t
0
v a a a u u
v a a a u u
Euclidean transform
![Page 29: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/29.jpg)
Affine Transform
I1 I2
x x
y y1
1 1
R t
0
v u
v ux 11 12 13 x x
y 21 22 23 y y1
0 0 11 1 1
A t
0
v a a a u u
v a a a u u
Euclidean transform
Invariant properties
• Parallelism • Ratio of area • Ratio of length
Degree of freedom
6
![Page 30: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/30.jpg)
Perspective Transform (Homography)
I1 I2
x 11 12 13 x x
y 21 22 23 y y
31 32 11 1 1
H
v h h h u u
v h h h u u
h h
![Page 31: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/31.jpg)
Perspective Transform (Homography)
I1 I2
x 11 12 13 x x
y 21 22 23 y y
31 32 11 1 1
H
v h h h u u
v h h h u u
h h
: General form of plane to plane linear mapping
![Page 32: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/32.jpg)
Ground plane
Camera plane
u
Perspective Transform (Homography)
u K R t Xu
Ground plane Camera plane
X
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Ground plane
Camera plane
u
Perspective Transform (Homography)
u K R t X
0
1
X =
X
Y
uGround plane Camera plane
![Page 34: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/34.jpg)
Ground plane
Camera plane
u
Perspective Transform (Homography)
u K R t X
0
1
X =
X
Y
uGround plane Camera plane
1 2 30
1
u K r r r t
X
Y
![Page 35: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/35.jpg)
Ground plane
Camera plane
u
Perspective Transform (Homography)
u K R t X
0
1
X =
X
Y
uGround plane Camera plane
1 2 30
1
u K r r r t
X
Y
1 2
1
u K r r t
X
Y
![Page 36: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/36.jpg)
Ground plane
Camera plane
u
Perspective Transform (Homography)
u K R t X
0
1
X =
X
Y
uGround plane Camera plane
1 2 30
1
u K r r r t
X
Y
1 2
1
u K r r t
X
Y
x
y
11
H
u X
u Y
![Page 37: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/37.jpg)
Perspective Transform (Homography)
I1 I2Invariant properties
• Cross ratio
• Concurrency
• Colinearity
x 11 12 13 x x
y 21 22 23 y y
31 32 11 1 1
H
v h h h u u
v h h h u u
h h
Degree of freedom
8 (9 variables 1 scale)
![Page 38: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/38.jpg)
Hierarchy of Transformations
Euclidean (3 dof) Similarity (4 dof) Affine (6 dof) Projective (8 dof)
• Length • Angle • Area
• Length ratio • Angle
• Parallelism • Ratio of area • Ratio of length
• Cross ratio • Concurrency • Colinearity
x
y
cos sin
sin cos
1
t
t
θ θ
θ θ
x
y
cos sin
sin cos
1
t
t
θ θ
θ θ
11 12 13
21 22 23
0 0 1
a a a
a a a11 12 13
21 22 23
31 32 1
h h h
h h h
h h
![Page 39: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/39.jpg)
u1
u2
u3
u4
Fun with Homography
v1 v2
v3v4
H
The image can be rectified as if it is seen from top view.
![Page 40: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/40.jpg)
Fun with Homography u = [u1'; u2'; u3'; u4']; v = [v1'; v2'; v3'; v4']; % Need at least non-colinear four points H = ComputeHomography(v, u); im_warped = ImageWarping(im, H);
u1
u2
u3
u4
RectificationViaHomography.m
![Page 41: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/41.jpg)
Fun with Homography u = [u1'; u2'; u3'; u4']; v = [v1'; v2'; v3'; v4']; % Need at least non-colinear four points H = ComputeHomography(v, u); im_warped = ImageWarping(im, H);
u1
u2
u3
u4
RectificationViaHomography.m
u_x = H(1,1)*v_x + H(1,2)*v_y + H(1,3); u_y = H(2,1)*v_x + H(2,2)*v_y + H(2,3); u_z = H(3,1)*v_x + H(3,2)*v_y + H(3,3); u_x = u_x./u_z; u_y = u_y./u_z; im_warped(:,:,1) = reshape(interp2(im(:,:,1), u_x(:), u_y(:)), [h, w]); im_warped(:,:,2) = reshape(interp2(im(:,:,2), u_x(:), u_y(:)), [h, w]); im_warped(:,:,3) = reshape(interp2(im(:,:,3), u_x(:), u_y(:)), [h, w]); im_warped = uint8(im_warped);
ImageWarping.m
x x
y y
1 1
H
v u
v u
u_x = H(1,1)*v_x + H(1,2)*v_y + H(1,3); u_y = H(2,1)*v_x + H(2,2)*v_y + H(2,3);
Cf) ImageWarpingEuclidean.m
![Page 42: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/42.jpg)
Fun with Homography
H v1
v2
v3v4
u1
u2
u3
u4
![Page 43: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/43.jpg)
Fun with Homography
u1
u2
u3
u4
![Page 44: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/44.jpg)
Fun with Homography
![Page 45: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/45.jpg)
Fun with Homography
![Page 46: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/46.jpg)
Image Warping
![Page 47: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/47.jpg)
Image Inpainting
![Page 48: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/48.jpg)
Virtual Advertisement
![Page 49: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/49.jpg)
Image Transform via Plane
Keller entrance left Keller entrance right
![Page 50: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/50.jpg)
Plane
Image Transform via 3D Plane
X
YX =
0
1
u
u K R t X
![Page 51: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/51.jpg)
Plane
Image Transform via 3D Plane
X
YX =
0
1
1 2
1 2
=
1
=
1
u K r r t
v K r r t
X
Y
X
Y
u
u K R t X
![Page 52: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/52.jpg)
Plane
Image Transform via 3D Plane
X
YX =
0
1
1 2
1 2
=
1
=
1
u K r r t
v K r r t
X
Y
X
Y
u
u K R t X
v
![Page 53: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/53.jpg)
Plane
Image Transform via 3D Plane
X
YX =
0
1
1 2
1 2
=
1
=
1
u K r r t
v K r r t
X
Y
X
Y
u
u K R t X
v
How are two image coordinates (u,v) related?
![Page 54: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/54.jpg)
Plane
Image Transform via 3D Plane
X
YX =
0
1
1 2
1 2
=
1
=
1
u K r r t
v K r r t
X
Y
X
Y
u
u K R t X
v
How are two image coordinates (u,v) related?
-1-1 -1 -1
1 2 1 2
-1 -11 2 1 2
= =
1
=
=
r r t K u r r t K v
v K r r t r r t K u
v Hu
X
Y
![Page 55: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/55.jpg)
Image Transform via 3D Plane
Keller entrance left Keller entrance right
![Page 56: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/56.jpg)
Image Transform via 3D Plane
Keller entrance left Right image to left
![Page 57: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/57.jpg)
Image Transform via 3D Plane
![Page 58: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/58.jpg)
Image Transform via 3D Plane
Left image to right Right image to left
![Page 59: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/59.jpg)
Image Transform via 3D Plane
![Page 60: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/60.jpg)
360 Panorama
https://www.youtube.com/watch?v=H6SsB3JYqQg
![Page 61: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/61.jpg)
Camera
Image Transform by Pure 3D Rotation
X
![Page 62: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/62.jpg)
Camera
Image Transform by Pure 3D Rotation
X1 = u KX
![Page 63: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/63.jpg)
Camera
Image Transform by Pure 3D Rotation
X
2 = v KRX
1 = u KX
![Page 64: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/64.jpg)
Camera
Image Transform by Pure 3D Rotation
X
2 = v KRX
1 = u KX
-1 T -11 2= = X K u R K v
![Page 65: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/65.jpg)
Camera
Image Transform by Pure 3D Rotation
X
2 = v KRX
1 = u KX
-1 T -11 2= = X K u R K v
-1=v KRK u
![Page 66: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/66.jpg)
Camera
Image Transform by Pure 3D Rotation
X
2 = v KRX
1 = u KX
-1 T -11 2= = X K u R K v
-1=v KRK u
-1=H KRK
![Page 67: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/67.jpg)
Camera
Image Transform by Pure 3D Rotation
X
2 = v KRX
1 = u KX
-1 T -11 2= = X K u R K v
-1=v KRK u
-1=H KRK-1=R K HK
![Page 68: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/68.jpg)
Lind Hall Left Lind Hall Right
u Hv=
![Page 69: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/69.jpg)
Euclidean Transform (Translation)
![Page 70: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/70.jpg)
Homography
![Page 71: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/71.jpg)
Homography
Why misalignment?
![Page 72: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/72.jpg)
Image Panorama (Cylindrical Projection)
![Page 73: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/73.jpg)
Image Panorama (Cylindrical Projection)
![Page 74: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/74.jpg)
Image Panorama (Cylindrical Projection)
h
![Page 75: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/75.jpg)
Image Panorama (Cylindrical Projection)
-1 TX = K R u
u
1R 2R 11R3R 4R 5R 6R 7R 8R 9R 10R
-1=R K HK
![Page 76: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/76.jpg)
Image Panorama (Cylindrical Projection)
u
-1=R K HK
h
H
cos
-2
sin
u =
f
Hh
f
-1 TX = K R u
![Page 77: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/77.jpg)
Image Panorama (Cylindrical Projection)
u
-1=R K HK
h
cos
-2
sin
u =
f
Hh
f
f
H
-1 TX = K R u
![Page 78: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/78.jpg)
Image Panorama (Cylindrical Projection)
u
-1=R K HK
h
cos
-2
sin
u =
f
Hh
f
f
H
h-1 TX = K R u
![Page 79: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/79.jpg)
Image Panorama (Cylindrical Projection)
u
-1=R K HK
h
cos
-2
sin
u =
f
Hh
f
f
H
h
-1 T= K R u
-1 TX = K R u
![Page 80: Image Transform - University of Minnesota · 2018-01-28 · Euclidean Transform SE(3) Rotation around the center of image I 1 I 2 Invariant properties • Length • Angle • Area](https://reader033.vdocuments.net/reader033/viewer/2022060405/5f0f3dc17e708231d4432f43/html5/thumbnails/80.jpg)
Image Panorama (Cylindrical Projection)
h