Lecture 13: Camera Projection II - Penn State College of ...rtc12/CSE486/lecture13.pdfLecture 13:...

CSE486, Penn StateRobert Collins

Lecture 13: Camera Projection II

Reading: T&V Section 2.4


Recall: Imaging Geometry

VU

W

Object of Interestin World CoordinateSystem (U,V,W)


Imaging Geometry

Z

f

Camera Coordinate System (X,Y,Z).

• Z is optic axis• Image plane located f units

out along optic axis• f is called focal length

X

Y


Imaging Geometry

VU

W

Z

Forward Projection onto image plane.3D (X,Y,Z) projected to 2D (x,y)

y

xX

Y


Imaging Geometry

VU

W

Zy

Our image gets digitizedinto pixel coordinates (u,v)

xX

Y

u

v


Imaging Geometry

VU

W

Zy

World Coordinates

CameraCoordinates

Image (film)Coordinates

PixelCoordinates

u

v

xX

Y


Forward Projection

UVW

XYZ

xy

uv

WorldCoords

CameraCoords

FilmCoords

PixelCoords

We want a mathematical model to describehow 3D World points get projected into 2DPixel coordinates.

Our goal: describe this sequence of transformations by a big matrix equation!


Intrinsic Camera Parameters

UVW

XYZ

xy

uv

WorldCoords

CameraCoords

FilmCoords

PixelCoords

Affine Transformation


Intrinsic parameters

• Describes coordinate transformation between film coordinates (projected image) and pixel array

• Film cameras: scanning/digitization

• CCD cameras: grid of photosensors

still in T&V section 2.4


Intrinsic parameters (offsets)

film plane(projected image)

x

y

(0,0)

u (col)

v (row)

(0,0)

pixel array

yo

xo

xouZ

Xf yov

Z

Yf

ox and oy called image center or principle point


Intrinsic parameters

film plane

x

y

(0,0)

u (col)

v (row)

(0,0)

pixel array

yo

xo

sometimes one or more coordinate axes are flipped (e.g. T&V section 2.4)

xouZ

Xf yov

Z

Yf


analog

Intrinsic parameters (scales)

pixel array

sampling determines how many rows/cols in the image

C cols x R rows

film scanning resolution

CCD

resample


Effective Scales: sx and sy

O.Camps, PSU

xouZ

Xf yov

Z

Yf

1

sx

1

sy

Note, since we have different scale factors in x and y,we don’t necessarily have square pixels!

Aspect ratio is sy / sx


Perspective projection matrix

10

0

0

10

/

0

0

0

/

'

'

'

Z

Y

X

o

o

sf

sf

z

y

x

y

x

y

x

O.Camps, PSU

Adding the intrinsic parameters into theperspective projection matrix:

xouZ

Xf

1

sxyov

Z

Yf

1

sy

uz’

x’

vz’

y’

To verify:


Note:

Sometimes, the image and the camera coordinate systemsSometimes, the image and the camera coordinate systemshave opposite orientations: [the book does it this way] have opposite orientations: [the book does it this way]

10

0

0

10

/

0

0

0

/

'

'

'

Z

Y

X

o

o

sf

sf

z

y

x

y

x

y

x

yy

xx

sovZ

Yf

souZ

Xf

)(

)(


Note 2

10

0

0

1

0

0

0

0

0

0

'

'

'

Z

Y

X

f

f

z

y

x

100

a13a12a11

a23a22a21

w’

v’

u’

In general, I like to think of the conversion asa separate 2D affine transformation from filmcoords (x,y) to pixel coordinates (u,v):

u = Mint PC = Maff Mproj PC

MprojMaff


Huh?

Did he just say it was “a fine” transformation?

No, it was “affine” transformation, a type of2D to 2D mapping defined by 6 parameters.

More on this in a moment...


Summary : Forward Projection

UVW

XYZ

xy

uv

WorldCoords

CameraCoords

FilmCoords

PixelCoords

Mext Mproj Maff

MUVW

uv

34

24

14

33

23

13

31

22

12

31

21

11

m

m

m

m

m

m

m

m

m

m

m

m

MintUVW

XYZ

uv

Mext


Summary: Projection Equation

World to cameraPerspectiveprojection

Film planeto pixels

MextMprojMaff

Mint

M


Lecture 13/14: Intro to Image Mappings


from R.Szeliski

Image MappingsOverview


Geometric Image Mappings

image transformed imageGeometric

transformation

(x,y)

(x’,y’)

x’ = f(x, y, {parameters})y’ = g(x, y, {parameters})


Linear Transformations

image transformed imageGeometric

transformation

(x,y)

(x’,y’)

x’y’1

xy1

(Can be written as matrices)

M(params)=


Translation

0 1

1

0tx

ty

x

y

x’

y’

equations matrix form

transform


Scale

0 1

1

0x

y

x’

y’


transform

0 S

S

0


Rotation

0 1

1

0x

y

x’

y’


transform

)


Euclidean (Rigid)

0 1

1

0x

y

x’

y’


transform )

tx

ty


Partitioned Matrices

http://planetmath.org/encyclopedia/PartitionedMatrix.html


Partitioned Matrices

2x1 2x2 2x12x1

1x1 1x2 1x11x1

equation form

matrix form


Another Example (from last time)

1

W

V

U

1

Z

Y

X tx

1000

r13r12r11

r23r22r21

r33r32r31

ty

tz

PC = R PW + T

PC

1=

R T

0 1

PW

1

3x1

1x1

3x3 3x1

1x3 1x1 1x1

3x1


Similarity (scaled Euclidean)

0 1

1

0x

y

x’

y’


transform )

tx

ty

S


Affine

0 1

1

0x

y

x’

y’


transform


Projective

0 1

1

0x

y

x’

y’


transform

Note!


Summary of 2D Transformations



Euclidean



Similarity



Affine



Projective



from R.Szeliski

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