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3D object
Lens
CCD sensor
Slide by HyunSoo Park
3D Point Projection (Metric Space)3D point
mf
( , , )X Y Z
: Focal length in meter
Pinhole-fm
ccd X
u fZm
ccd Y
v fZm
ccd ccd X Y
u v f fZ Zm m( , ) ( , )
2D projection onto CCD plane
( , , )X Y Z
Projection plane
ccd ccdu v( , )
3D Point Projection (Metric Space)3D point
mf
( , , )X Y Z
: Focal length in meter
Pinhole=
center of projection-fm
ccd X
u fZm
ccd Y
v fZm
ccd ccd X Y
u v f fZ Zm m( , ) ( , )
2D projection onto CCD plane
( , , )X Y Z
Projection plane
( , )u vccd ccd
fm
ccd ccdu v( , )
http://www.joshuanava.biz/perspective/in-other-words-the-observer-simply-points-in-the-same-direction-as-the-lines-in-order-to-find-their-vanishing-point.html
recall we can paint a picture by
1) placing an image plane in front of us,2) draw a ray from our eye to a point in the image plane, to form a ray3) intersect that ray with the 3D world, and mark the intersection point,4) take the color of that 3D point, and paint it onto the 2D pixel
The ray itself, extended infinitely long, can be also thought of representative of a point in infinity of all 3D lines in direction, and we will see that would allow us to recover the orientation of the camera.
Locating Center of Projection
we will demonstrate how to find COP of your cell phone camera.
1) draw a set of radiating lines on a paper2) place the camera vertical to the paper
the radiating lines on paper should look like this, in the 3d world
Locating Center of Projection
1) using a live displaying mode, we can see these radiating lines as transformed in to the camera image plane. what does it look like?2) make sure we keep the camera vertical and move it forward and backward, until the imaged lines look all vertical 3) when we see this we can mark the front of the camera.
Locating Center of Projection
Locating Center of Projection
-- tracing the radiating lines to their intersection, we obtained a rough estimated of the physical center of projection of your camera
-- we now have an idea of where is the optical center of the camera, how about the focal length itself
Focal Length
Digital SLR cameras have focal length at order of 35mm, or 50 mm. Our cell phone, because it is very thin, has very short focal lengths, on order of a few mm.
Nikon’s 1200-1700mm f/5.6-8P lens.Professional photographer has lens with much longer focal lengths, on order of 1-2meters
Zooming, the process of changing focal length, can amplifying a scene. We can do the same with a cell phone, just we have to walk a lot to get those images.
Large Focal Length compresses depth
© 1995-2005 Michael Reichmann
400 mm 200 mm 100 mm 50 mm 28 mm 17 mm
zooming does not only create a sense of magnification, it can also create a sensation of moving,shown here, from 400mm, to 17mm, we can see the size of background objects is much reduced
3D Point Projection (Metric Space)3D point
mf
( , , )X Y Z
: Focal length in meterccd
Xu f
Zm
ccd Y
v fZm
ccd ccd X Y
u v f fZ Zm m( , ) ( , )
2D projection onto CCD plane
( , , )X Y Z
Projection plane
( , )u vccd ccd
fm
Center of Projection
3D Point Projection (Metric Space)3D point
mf
( , , )X Y Z
: Focal length in meterccd
Xu f
Zm
ccd Y
v fZm
ccd ccd X Y
u v f fZ Zm m( , ) ( , )
2D projection onto CCD plane
( , , )X Y Z
Projection plane
( , )u vccd ccd
fm
( , )u vimg img
img img ccd ccdu v u v( , ) ( , ) ?
MetricPixel
3D Point Projection (Pixel Space)
CCD sensor (mm) Image (pixel)
3D Point Projection (Pixel Space)
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
3D Point Projection (Pixel Space)
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
ccdw
ccdh
u vimg img( , )
3D Point Projection (Pixel Space)
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
ccdw
ccdh
u vimg img( , )
3D Point Projection (Pixel Space)
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
ccdw
ccdh
(0,0)
Projection of pinhole
u vimg img( , )
3D Point Projection (Pixel Space)
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
ccdw
ccdh
(0,0)
Projection of pinhole
imgw
imgh
u vimg img( , )
3D Point Projection (Pixel Space)
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
ccdw
ccdh
(0,0)
Projection of pinhole
(0,0) imgw
imgh
x yp p( , ): Image principal point
u vimg img( , )
3D Point Projection (Pixel Space)
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
Projection of pinhole
(0,0) imgw
imgh
x yp p( , ): Image principal point
xu pu
w wimgccd
ccd img
3D Point Projection (Pixel Space)
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
Projection of pinhole
(0,0) imgw
imgh
x yp p( , ): Image principal point
xu pu
w wimgccd
ccd img
yv pv
h himgccd
ccd img
3D Point Projection (Pixel Space)
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
Projection of pinhole
(0,0) imgw
imgh
x yp p( , ): Image principal point
xu pu
w wimgccd
ccd img
yv pv
h himgccd
ccd img
x
wu u p
wimg
img ccdccd
y
hv v p
himg
img ccdccd
3D Point Projection (Pixel Space)
O
( , )u vccd ccd
mf
Projection plane ( , , )X Y Z
Z
ccd ccd m m( , ) ( , )X Y
u v f fZ Z
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
(0,0) imgw
imghx yp p( , )
: Metric projection
3D Point Projection (Pixel Space)
O
( , )u vccd ccd
mf
Projection plane ( , , )X Y Z
Z
ccd ccd m m( , ) ( , )X Y
u v f fZ Z
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
(0,0) imgw
imghx yp p( , )
: Metric projection
x x
w w Xu u p f p
w w Zimg img
img ccd mccd ccd
y y
h h Yv v p f p
h h Zimg img
img ccd mccd ccd
Pixel projection
3D Point Projection (Pixel Space)
O
( , )u vccd ccd
mf
Projection plane ( , , )X Y Z
Z
ccd ccd m m( , ) ( , )X Y
u v f fZ Z
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
(0,0) imgw
imghx yp p( , )
: Metric projection
x x
w w Xu u p f p
w w Zimg img
img ccd mccd ccd
y y
h h Yv v p f p
h h Zimg img
img ccd mccd ccd
Pixel projection
3D Point Projection (Pixel Space)
O
Projection plane ( , , )X Y Z
Z
ccd ccd m m( , ) ( , )X Y
u v f fZ Z
x x
w w Xu u p f p
w w Zimg img
img ccd mccd ccd
y y
h h Yv v p f p
h h Zimg img
img ccd mccd ccd
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
(0,0) imgw
imghx yp p( , )
: Metric projection
Focal length in pixel
Pixel projection
img imgu v( , )
f
3D Point Projection (Pixel Space)
O
Projection plane ( , , )X Y Z
Z
ccd ccd m m( , ) ( , )X Y
u v f fZ Z
x x
w w Xu u p f p
w w Zimg img
img ccd mccd ccd
y y
h h Yv v p f p
h h Zimg img
img ccd mccd ccd
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
(0,0) imgw
imghx yp p( , )
: Metric projection
Focal length in pixel
xf
yf
imgm
ccd
x
wf f
wimg
mccd
y
hf f
hwhere
Pixel projection
img imgu v( , )
f
3D Point Projection (Pixel Space)
O
Projection plane ( , , )X Y Z
Z
ccd ccd m m( , ) ( , )X Y
u v f fZ Z
x x
w w Xu u p f p
w w Zimg img
img ccd mccd ccd
y y
h h Yv v p f p
h h Zimg img
img ccd mccd ccd
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
(0,0) imgw
imghx yp p( , )
: Metric projection
Focal length in pixel
f
f
img imgm m
ccd ccd
w h
f f fw h
where
Pixel projection
img img
ccd ccd
w h
w hif
img imgu v( , )
f
3D Point Projection (Pixel Space)
O
Projection plane ( , , )X Y Z
Z
img x
Xu f p
Z
img y
Yv f p
Z
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
(0,0) imgw
imghx yp p( , )
img imgu v( , )
f
3D Point Projection (Pixel Space)
O
Projection plane ( , , )X Y Z
Z
img x
Xu f p
Z
img y
Yv f p
Z
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
(0,0) imgw
imghx yp p( , )
img imgu v( , )
f
img xZu fX p Z
img yZv fY p Z
3D Point Projection (Pixel Space)
O
Projection plane ( , , )X Y Z
Z
img x
Xu f p
Z
img y
Yv f p
Z
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
(0,0) imgw
imghx yp p( , )
img imgu v( , )
f
img xZu fX p Z
img yZv fY p Z
img
img
11
x
y
u f p X
Z v f p Y
Z
3D Point Projection (Pixel Space)
O
Projection plane ( , , )X Y Z
Z
img x
Xu f p
Z
img y
Yv f p
Z
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
(0,0) imgw
imghx yp p( , )
img imgu v( , )
f
img xZu fX p Z
img yZv fY p Z
img
img
11
x
y
u f p X
v f p Y
Z
Metric spacePixel space
3D Point Projection (Pixel Space)
O
Projection plane ( , , )X Y Z
Z
img x
Xu f p
Z
img y
Yv f p
Z
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
(0,0) imgw
imghx yp p( , )
img imgu v( , )
f
img xZu fX p Z
img yZv fY p Z
img
img
11
x
y
u f p X
v f p Y
Z
Metric spacePixel space
+Camera intrinsic parameter: metric space to pixel space
3D Point Projection (Pixel Space)
O
Projection plane ( , , )X Y Z
Z
img x
Xu f p
Z
img y
Yv f p
Z
CCD sensor (mm) Image (pixel)
u vccd ccd( , )
u vimg img( , )
ccdw
ccdh
(0,0)
(0,0) imgw
imghx yp p( , )
img imgu v( , )
f
img xZu fX p Z
img yZv fY p Z
img
img
11
x
y
u f p X
v f p Y
Z
K
+Camera intrinsic parameter: metric space to pixel space
Metric spacePixel space
Camera Intrinsic Parameter
O
Projection plane
Z
img
img
11
x
y
u f p X
v f p Y
Z
K
Metric space
( , , )X Y Z
Camera Intrinsic Parameter
O
Projection plane
Z
img
img
11
x
y
u f p X
v f p Y
Z
K
Metric space
( , , )X Y Z
f
Camera Intrinsic Parameter
O
Projection plane
Z
img
img
11
x
y
u f p X
v f p Y
Z
K
Metric space
( , , )X Y Z
f
img imgu v( , )
Pixel space
2D Inverse Projection
O
Projection plane
img
img
11
x
y
u f p X
v f p Y
Z
K
Metric space
f
img imgu v( , )
Pixel space
u X
v Y
Z
K
img
-1img
1
2D Inverse Projection
O
Projection plane
img
img
11
x
y
u f p X
v f p Y
Z
K
Metric space
f
img imgu v( , )
Pixel space
u X
v Y
Z
K
img
-1img
1
u
vK
img
-1img
1
2D point == 3D ray
3D ray
2D Inverse Projection
O
Projection plane
img
img
11
x
y
u f p X
v f p Y
Z
K
Metric space
f
img imgu v( , )
Pixel space
u X
v Y
Z
K
img
-1img
1
u
vK
img
-1img
1
( , , )X Y Z
2D point == 3D ray
3D ray
2D Inverse Projection
O
Projection plane
img
img
11
x
y
u f p X
v f p Y
Z
K
Metric space
f
img imgu v( , )
Pixel space
u X
v Y
Z
K
img
-1img
1
u
vK
img
-1img
1
( , , )X Y Z
2D point == 3D ray
3D ray
The 3D point must lie in the 3D ray passing through the origin and 2D image point.
Exercise
324 m
1280 pix
960 pix
fm ?
What f to make the height of Eifel tower appear 960 pixel distance?
21.8 mm
ccd size
1500 m
Exercise
324 m
1280 pix
960 pix
fm ?
What f to make the height of Eifel tower appear 960 pixel distance?
m
hY Yy f f
Z h Zimg
imgccd
21.8 mm
ccd size
1500 m
Exercise
324 m
1280 pix
960 pix
fm ?
What f to make the height of Eifel tower appear 960 pixel distance?
m
hY Yy f f
Z h Zimg
imgccd
21.8 mm
ccd size
1280 324960 0.0757m
0.02181500 m mf f
1500 m
Focal Length
Exercise
324 m
Z ?
1280 pix
960 pix
What Z to make the height of Eifel tower appear 960 pixel distance?
m
hY Yy f f
Z h Zimg
imgccd
21.8 mm
ccd size
fm = 50 mm
Exercise
324 m
Z ?
1280 pix
960 pix
What Z to make the height of Eifel tower appear 960 pixel distance?
m
hY Yy f f
Z h Zimg
imgccd
21.8 mm
ccd size
ZZ
1280 324960 0.05 990.826m
0.0218
fm = 50 mm
Exercise
324 m
1500
What Zp to make the height of Eifel tower appear twice of the person?
fm = 50 mm
Z p
1.7 m
Exercise
324 m
1500
What Zp to make the height of Eifel tower appear twice of the person?
Y
h fZe
fm = 50 mm
Z p
h e
h p
1.7 m Y
h fZ
pp
p
h
h ep 2
s.t.
Exercise
324 m
1500
What Zp to make the height of Eifel tower appear twice of the person?
Y
h fZe
Y Y
f fZ Z
p
p 2
fm = 50 mm
Z p
h e
h p
1.7 m Y
h fZ
pp
p
h
h ep 2
s.t.
p
1.72 1500 15.74m
324 Z
Where Was I?
Z ?
1.7 m
Sensor size Sensor spec
img
ccd
m
hY Yy f f
Z h Z
Where Was I?
Z ?
1.7 m
Sensor size Sensor spec
img
ccd
m
hY Yy f f
Z h Z
Where Was I?
Circa 1984
Where Was I?
Circa 1984
250 pix
670 pix
Where Was I?
Circa 1984
324 m
fm = 50 mm
0.9 m
250 pix
670 pix
Where Was I?
Circa 1984
324 m
Z 1
0.9 m
img img1 1 11 1
1 ccd 1 ccd 1
1280 0.90.05 8.03m
0.0218 250 m m
h hY Y Yy f f Z f
Z h Z h y
250 pix
670 pix
fm = 50 mm
Where Was I?
Circa 1984
324 m
fm = 38 mm
Z 1
0.9 m
img img1 1 11 1
1 ccd 1 ccd 1
1280 0.90.05 8.03m
0.0218 250 m m
h hY Y Yy f f Z f
Z h Z h y
250 pix
670 pix
Where Was I?
Circa 1984
324 m
Z 1
0.9 m
img img1 1 11 1
1 ccd 1 ccd 1
1280 0.90.05 8.03m
0.0218 250 m m
h hY Y Yy f f Z f
Z h Z h y
250 pix
670 pix
fm = 50 mm
Where Was I?
Circa 1984
324 m
Z 1
0.9 m
Z 2
250 pix
670 pix
m m
h hY Y Yy f f Z f
Z h Z h yimg img2 2 2
2 22 ccd 2 ccd 2
1280 3240.05 1079m
0.0218 670
img img1 1 11 1
1 ccd 1 ccd 1
1280 0.90.05 8.03m
0.0218 250 m m
h hY Y Yy f f Z f
Z h Z h y
fm = 50 mm
Where Was I?
Circa 1984
250 pix
670 pix
m m
h hY Y Yy f f Z f
Z h Z h yimg img2 2 2
2 22 ccd 2 ccd 2
1280 3240.05 1079m
0.0218 670
200m
400m
600m
800m
1000m
800 m
Where Was I?
Focal Length
f d
Strong perspective
Focal Length
f d
Weak perspective
Focal Length
f
Orhographic
d
Focal Length
Dolly Zoom (Vertigo Effect)
Dolly Zoom (Vertigo Effect)
Dolly Zoom
Given focal length (fm=100mm),what Z100 to make the height of the person remain the same as fm=50mm?
f50 = 50 mm
h p H p
Z 50 =157.41m
Dolly Zoom
Given focal length (fm=100mm),what Z100 to make the height of the person remain the same as fm=50mm?
Y
h fZ50 50
50
h p
Y
h fZ100 100
100
h h100 50s.t.
H p
Z 50 =157.41m
f50 = 50 mm
Dolly Zoom
Given focal length (fm=100mm),what Z100 to make the height of the person remain the same as fm=50mm?
Y
h fZ50 50
50
f
Z Zf100
100 5050
f100 = 100 mm
Z 50 =157.41m
h p
Y
h fZ100 100
100
h h100 50s.t.
Z100
100157.41 314.8m
50
H p
Z 100 ?
Where am I with Dolly Zoom?
1 400pixh
2 120pixh
Where am I with Dolly Zoom?
1 400pixh
2 120pixh
How far I need to step back with zoom factor x2?
How will h2 change?
Where am I with Dolly Zoom?
A
B
C
camera
Image plane
Z
1 400pixh
2 120pixh
Top view
1 4mH
2 6mH
How will h2 change?
How far I need to step back with zoom factor x2?
Where am I with Dolly Zoom?
1 400pixh
2 120pixh
A
B
C
camera
Image plane
Z
f = 2md1d
Top view
How far I need to step back with zoom factor x2?
1 4mH
2 6mH
Unknowns: f, d1How will h2 change?
Where am I with Dolly Zoom?
1 400pixh
2 120pixh
A
B
C
camera
Image plane
Z
2f 1d
Top view
1 4mH
2 6mH
Unknowns: f, d1,
d
d
= 2md
How will h2 change?
How far I need to step back with zoom factor x2?
Where am I with Dolly Zoom?
1 400pixh
2 120pixh
A
B
C
camera
Image plane
Z
2f 1d
Top view
1 4mH
2 6mH
d
1
11 h f
H
d
Equations:
= 2md
Unknowns: f, d1, dHow will h2 change?
How far I need to step back with zoom factor x2?
Where am I with Dolly Zoom?
1 400pixh
2 120pixh
A
B
C
camera
Image plane
Z
2f 1d
Top view
1 4mH
2 6mH
d
1
11 h f
H
d
Equations:
1
11 2
+
H
h fd d
= 2md
Unknowns: f, d1, d How will h2 change?
How far I need to step back with zoom factor x2?
Where am I with Dolly Zoom?
1 400pixh
2 120pixh
A
B
C
camera
Image plane
Z
2f 1d
Top view
1 4mH
2 6mH
d
1
11 h f
H
d
Equations:1
11 2
+
H
h fd d 1 d d
= 2md
Unknowns: f, d1, d How will h2 change?
How far I need to step back with zoom factor x2?
Where am I with Dolly Zoom?
1 400pixh
2 120pixh
A
B
C
camera
Image plane
Z
2f 1d
Top view
How far I need to step back with zoom factor x2?
1 4mH
2 6mH
d
1
11 h f
H
d
Equations:1
11 2
+
H
h fd d
2
12 +
Hh f
d d
= 2md
Unknowns: f, d1, d How will h2 change?
1 d d
A
B
C
camera
Image plane
Z
2f 1d
Top view
1 4mH
2 6mH
d
1
11 h f
H
d
Equations:1
11 2
+
H
h fd d 1 d d
21
1
1 2
1= 2.5m
1
d
h HhH
d
= 2md
Unknowns: f, d1, d
2
12 +
Hh f
d d
1 400pixh
2 120pixh
2 120pixh 2 200pix h
2.5m d
A
B
C
camera
Image plane
Z
2f 1d
Top view
1 4mH
2 6mH
d
1
11 h f
H
d
Equations:1
11 2
+
H
h fd d 1 d d
21
1
1 2
1= 2.5m
1
d
h HhH
d
= 2md
Unknowns: f, d1, d
2
12 +
Hh f
d d
1 400pixh
2 120pixh
2 120pixh
2.5m d 250pixf
1 400pixh
2 120pixh
A
B
C
camera
Image plane
Z
2f 1d
Top view
How far I need to step back with zoom factor x2?
1 4mH
2 6mH
d
1
11 h f
H
d
Equations:1
11 2
+
H
h fd d
2
12 +
Hh f
d d
= 2md
Unknowns: f, d1, d How will h2 change?
2
12 2 200pix
+ + f
d d
Hh
d
250pixf1 2.5m d d
1 400pixh
2 120pixh
A
B
C
camera
Image plane
Z
2f 1d
Top view
How far I need to step back with zoom factor x2?
1 4mH
2 6mH
d
1
11 h f
H
d
Equations:1
11 2
+
H
h fd d
2
12 +
Hh f
d d2
12 2 429pix
+ + f
d d
Hh
d
= 2md
Unknowns: f, d1, d How will h2 change?
250pixf1 2.5m d d
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