Projective Geometry Conversion Euclidean -> Homogenous...

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1 CS252A, Fall 2017 Computer Vision I Image Formation and Cameras Computer Vision I CSE 252A Lecture 4 CS252A, Fall 2017 Computer Vision I Announcements HW0 is due on today HW1 will be assigned tomorrow. Read Chapters 1 & 2 of Forsyth & Ponce Everyone is off waitlist Welcome extension students send email to Bolun Zhang [email protected] (Subset of?) Final exam can be use for CSE MS Comprehensive Exam [Pending] CS252A, Fall 2017 Computer Vision I Projective Geometry Axioms of Projective Plane 1. Every two distinct points define a line 2. Every two distinct lines define a point (intersect at a point) 3. There exists three points, A,B,C such that C does not lie on the line defined by A and B. Different than Euclidean (affine) geometry Projective plane is “bigger” than affine plane – includes “line at infinity” Projective Plane Affine Plane = + Line at Infinity CS252A, Fall 2017 Computer Vision I Conversion Euclidean -> Homogenous -> Euclidean In 2-D Euclidean -> Homogenous: (x, y) -> λ(x,y,1)= (λx, λy, λ) Homogenous -> Euclidean: (x, y, w) -> (x/w, y/w) In 3-D Euclidean -> Homogenous: (x, y, z) -> λ(x ,y, z, 1) Homogenous -> Euclidean: (x, y, z, w) -> (x/w, y/w, z/w) X Y (x,y) (x,y,1) 1 W CS252A, Fall 2017 Computer Vision I The equation of projection: Euclidean & Homogenous Coordinates Cartesian coordinates: ( x, y, z) ( f x z , f y z ) U V W = 1 0 0 0 0 1 0 0 0 0 1 f 0 X Y Z T Homogenous Coordinates and Camera matrix CS252A, Fall 2017 Computer Vision I Affine Camera Model Take perspective projection equation, and perform Taylor series expansion about some point (x 0 ,y 0 ,z 0 ). Drop terms that are higher order than linear. Resulting expression is affine camera model 0 0 0 z y x Appropriate in Neighborhood About (x 0 ,y 0 ,z 0 )

Transcript of Projective Geometry Conversion Euclidean -> Homogenous...

Page 1: Projective Geometry Conversion Euclidean -> Homogenous ...cseweb.ucsd.edu/classes/fa17/cse252A-a/lec4.pdfProjective Geometry • Axioms of Projective Plane 1. Every two distinct points

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CS252A, Fall 2017

Computer Vision I

Image Formation and Cameras

Computer Vision I CSE 252A Lecture 4

CS252A, Fall 2017

Computer Vision I

Announcements •  HW0 is due on today •  HW1 will be assigned tomorrow. •  Read Chapters 1 & 2 of Forsyth & Ponce •  Everyone is off waitlist •  Welcome extension students – send email to

Bolun Zhang [email protected] •  (Subset of?) Final exam can be use for CSE

MS Comprehensive Exam [Pending]

CS252A, Fall 2017

Computer Vision I

Projective Geometry •  Axioms of Projective Plane

1.  Every two distinct points define a line 2.  Every two distinct lines define a point (intersect

at a point) 3.  There exists three points, A,B,C such that C

does not lie on the line defined by A and B. •  Different than Euclidean (affine) geometry •  Projective plane is “bigger” than affine

plane – includes “line at infinity”

Projective Plane

Affine Plane = + Line at

Infinity CS252A, Fall 2017

Computer Vision I

Conversion Euclidean -> Homogenous -> Euclidean In 2-D •  Euclidean -> Homogenous:

(x, y) -> λ(x,y,1)= (λx, λy, λ)

•  Homogenous -> Euclidean: (x, y, w) -> (x/w, y/w)

In 3-D •  Euclidean -> Homogenous:

(x, y, z) -> λ(x ,y, z, 1) •  Homogenous -> Euclidean:

(x, y, z, w) -> (x/w, y/w, z/w)

X

Y (x,y)

(x,y,1)

1 W

CS252A, Fall 2017

Computer Vision I

The equation of projection: Euclidean & Homogenous Coordinates

Cartesian coordinates:

(x,y,z)→( f xz, f y

z)

UVW

⎜ ⎜ ⎜

⎟ ⎟ ⎟

=

1 0 0 00 1 0 00 0 1

f 0

⎜ ⎜ ⎜ ⎜

⎟ ⎟ ⎟ ⎟

XYZT

⎜ ⎜ ⎜ ⎜

⎟ ⎟ ⎟ ⎟

Homogenous Coordinates and Camera matrix

CS252A, Fall 2017

Computer Vision I

Affine Camera Model

•  Take perspective projection equation, and perform Taylor series expansion about some point (x0,y0,z0).

•  Drop terms that are higher order than linear. •  Resulting expression is affine camera model

⎥⎥⎥

⎢⎢⎢

0

0

0

zyx

Appropriate in Neighborhood About (x0,y0,z0)

Page 2: Projective Geometry Conversion Euclidean -> Homogenous ...cseweb.ucsd.edu/classes/fa17/cse252A-a/lec4.pdfProjective Geometry • Axioms of Projective Plane 1. Every two distinct points

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CS252A, Fall 2017

Computer Vision I

Simplified Camera Models Perspective Projection

Scaled Orthographic Projection

Affine Camera Model

Orthographic Projection

Approximation: Taylor Series Expansion about (X0, Y0, Z0)

Case of affine where X0= Y0 = 0

Case of scaled orthograhic where f/Z0 = 1

CS252A, Fall 2017

Computer Vision I

Projective Transformation Mapping from Plane to Plane

•  3 x 3 linear transformation of homogenous coordinates

•  Points map to points •  Lines map to lines

⎥⎥⎥

⎢⎢⎢

⎥⎥⎥

⎢⎢⎢

=

⎥⎥⎥

⎢⎢⎢

3

2

1

333231

232221

211211

3

2

1

xxx

aaaaaaaaa

uuu

X U

CS252A, Fall 2017

Computer Vision I

^

Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision”

Mapping from a Plane to a Plane under Perspective is given by a projective transform H

x’ = Hx H is a 3x3 matrix, x, x’ are 3x1 vectors of homogenous coordinates

ij

CS252A, Fall 2017

Computer Vision I

The equation of projection

Cartesian coordinates:

(x,y,z)→( f xz, f y

z)

UVW

⎜⎜⎜

⎟⎟⎟=

1 0 0 00 1 0 00 0 1

f 0

⎜⎜⎜⎜

⎟⎟⎟⎟

XYZT

⎜⎜⎜⎜

⎟⎟⎟⎟

Homogenous Coordinates and Camera matrix

CS252A, Fall 2017

Computer Vision I

^

x = xi + yj +Oπ

3D point coordinates on plane π

As a matrix equation

X = i j Oπ⎡⎣⎢

⎤⎦⎥

xy1

⎢⎢⎢

⎥⎥⎥

In homogenous coordinates

x 'y 'w '

⎢⎢⎢

⎥⎥⎥=

1 0 0 00 1 0 00 0 1

f 0

⎢⎢⎢⎢

⎥⎥⎥⎥

i0

j0

1

⎢⎢

⎥⎥

xy1

⎢⎢⎢

⎥⎥⎥= H

xy1

⎢⎢⎢

⎥⎥⎥

And taking perspective projection onto π’

ijX = i

0j0

1

⎢⎢

⎥⎥

xy1

⎢⎢⎢

⎥⎥⎥

x’ = Hx

4x3

CS252A, Fall 2017

Computer Vision I

Application of Homography: OCRs, scan,…

Homography: Can be estimated from four points.

P1

P3

P4

P2

(0,0) (0,1)

(1,1) (1,0)

Page 3: Projective Geometry Conversion Euclidean -> Homogenous ...cseweb.ucsd.edu/classes/fa17/cse252A-a/lec4.pdfProjective Geometry • Axioms of Projective Plane 1. Every two distinct points

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CS252A, Fall 2017

Computer Vision I Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision”

Planar Homography

x =H1X X’ =H2X

x’ = H2X = H2(H1-1

x) = (H2H1-1)x

Application: Two photos of a white board CS252A, Fall 2017

Computer Vision I Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision”

Planar Homography: Pure Rotation

x’ = H2X = H2(H1-1

x) = (H2H1-1)x

Application: Panoramas

CS252A, Fall 2017

Computer Vision I

Application: Panoramas and image stitching

All images are warped to central image CS252A, Fall 2017

Computer Vision I

Lenses

CS252A, Fall 2010 Computer Vision I

Beyond the pinhole Camera Getting more light – Bigger Aperture

CS252A, Fall 2010 Computer Vision I

Pinhole Camera Images with Variable Aperture

1mm

.35 mm

.07 mm

.6 mm

2 mm

.15 mm

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CS252A, Fall 2010 Computer Vision I

Limits for pinhole cameras

CS252A, Fall 2010 Computer Vision I

The reason for lenses

CS252A, Fall 2010 Computer Vision I

Thin Lens

O

•  Rotationally symmetric about optical axis. •  Spherical interfaces.

Optical axis

CS252A, Fall 2010 Computer Vision I

Thin Lens: Center

O

•  All rays that enter lens along line pointing at O emerge in same direction.

F

CS252A, Fall 2010 Computer Vision I

Thin Lens: Focus

O

Parallel lines pass through the Focal Point, F

F

CS252A, Fall 2010 Computer Vision I

Thin Lens: Image of Point

O

All rays passing through lens and starting at P converge upon P’

F

P

P’

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CS252A, Fall 2010 Computer Vision I

Thin Lens: Image of Point

O F

P

P’ z’

f

z

1z '−1z=1f

Where z, z’, and f are the z-coordinates of P, P, and F. i.e., z is negative.

j k

i

CS252A, Fall 2010 Computer Vision I

Thin Lens: Image Plane

O F

P

P’

Image Plane

Q’

Q

A price: Whereas the image of P is in focus, the image of Q isn’t.

CS252A, Fall 2010 Computer Vision I

Thin Lens: Aperture

O

P

P’

Image Plane •  Smaller Aperture -> Less Blur •  Pinhole -> No Blur

CS252A, Fall 2010 Computer Vision I

Field of View

O Field of View

Image Plane

f

– Telephoto lens: small field of view (large f) – Wide angle lens: large field of view(small f) – Zoom lens: variable f using compound lens

CS252A, Fall 2010 Computer Vision I

Deviations from the lens model Deviations from this ideal are aberrations Two types

2. chromatic

1. geometrical ❑  spherical aberration ❑  astigmatism ❑  distortion ❑  coma

Aberrations are reduced by combining lenses

Compound lenses CS252A, Fall 2010 Computer Vision I

Spherical aberration

Rays parallel to the axis do not converge Outer portions of the lens yield smaller focal lengths

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CS252A, Fall 2010 Computer Vision I

Astigmatism An optical system with astigmatism is one where rays that propagate in

two perpendicular planes have different foci. If an optical system with astigmatism is used to form an image of a cross, the vertical and horizontal lines will be in sharp focus at two different distances.

object

CS252A, Fall 2010 Computer Vision I

Distortion magnification/focal length different for different angles of inclination

Can be corrected! (if parameters are know)

pincushion (tele-photo)

barrel (wide-angle)

CS252A, Fall 2010 Computer Vision I

Chromatic aberration

sinθ1sinθ2

=n2n1=λ1λ2

ni is the index of refraction λi is the wavelength of light in the medium

Snells Law

CS252A, Fall 2010 Computer Vision I

Chromatic aberration

rays of different wavelengths focused in different planes cannot be removed completely sometimes achromatization is achieved for more than 2 wavelengths

Vignetting: Spatial Non-Uniformity

camera Iris

Litvinov & Schechner, radiometric nonidealities CS252A, Fall 2010 Computer Vision I

–  Only part of the light reaches the sensor –  Periphery of the image is dimmer

Vignetting