Equation of Perspective Projection. Projective Geometry. Conversion Euclidean -> Homogenous -> Euclidean. Image Formation and Cameras

Equation of Perspective Projection Image Formation and Cameras Guest Professor: Dr. Ana Murillo Computer Vision I CSE 252A Lecture 4 Cartesian coordi...
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Equation of Perspective Projection Image Formation and Cameras Guest Professor: Dr. Ana Murillo Computer Vision I CSE 252A Lecture 4

Cartesian coordinates: •  We have, by similar triangles, that (x, y, z) -> (f’ x/z, f’ y/z, f’) •  Establishing an image plane coordinate system at C’ aligned with i and j, we get (x, y,z) →( f ' x , f ' y ) z

CS252A, Fall 2012

Computer Vision I

z

CS252A, Fall 2012

Computer Vision I



Projective Geometry • 

Projective geometry provides an elegant means for handling these different situations in a unified way and homogenous coordinates are a way to represent entities (points & lines) in projective spaces.

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

CS252A, Fall 2012

Computer Vision I

=

Affine Plane

+

Line at Infinity

CS252A, Fall 2012

Conversion

Computer Vision I

The equation of projection: Euclidean & Homogenous Coordinates

Euclidean -> Homogenous -> Euclidean In 2-D •  Euclidean -> Homogenous: (x, y) -> k (x,y,1)

• 

Homogenous -> Euclidean:

(x,y,1)

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

In 3-D •  Euclidean -> Homogenous: (x, y, z) -> k (x,y,z,1)

• 

Homogenous -> Euclidean:

1

Y

Homogenous Coordinates and Camera matrix

(x,y)

Cartesian coordinates: x y (x, y,z) →( f , f ) z z

(x, y, z, w) -> (x/w, y/w, z/w) CS252A, Fall 2012

⎛ U ⎞ ⎛1 0 ⎜ ⎟ ⎜ ⎜ V ⎟ = ⎜0 1 ⎜ ⎟ ⎜⎜ ⎝W ⎠ ⎝0 0

X

Computer Vision I

CS252A, Fall 2012



0 0 1

f

⎞⎛ X ⎞ 0⎟⎜ ⎟ Y 0⎟⎜ ⎟ ⎟⎜ Z ⎟ 0⎟⎜ ⎟ ⎠⎝ T ⎠ Computer Vision I



1

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

Projective transformation

x’ = Hx

•  Also called a homography •  This is a mapping from 2-D to 2-D in homogenous coordinates •  3 x 3 linear transformation of homogenous coordinates

H is a 3x3 matrix, x is a 3x1 vector of homogenous coordinates

•  Points map to points •  Lines map to lines Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision” CS252A, Fall 2012

Computer Vision I

CS252A, Fall 2012

Application: Panoramas

Computer Vision I

Planar Homography: Pure Rotation

Coordinates between pairs of images are related by projective transformations

Transforms

x’ = H2X = H2(H1-1 x) = (H2H1-1)x Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision” CS252A, Fall 2012

Computer Vision I

Planar Homography

x =H1X

CS252A, Fall 2012

Computer Vision I

More applications: OCRs, scan,…

x’ =H2X

x’ = H2X = H2(H1-1 x) = (H2H1-1)x Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision” CS252A, Fall 2012

Computer Vision I

CS252A, Fall 2012

Computer Vision I

2

Augmented reality

Vanishing Point •  In the projective space, parallel lines meet at a point at infinity.

Region of interest

Corresponding points

•  The vanishing point is the perspective projection of that point at infinity, resulting from multiplication by the camera matrix.

H matrix?

CS252A, Fall 2012

Computer Vision I

CS252A, Fall 2012

Some applications …

Computer Vision I

Simplified Camera Models Perspective Projection Approximation Affine Camera Model Scaled Orthographic Projection

CS252A, Fall 2012

Figure from “Handling Urban Location Recognition as a 2D Homothetic Problem” G. Baatz, K. Koser1, D. Chen, R. Grzeszczuk, M. Pollefeys ECCV 2010.

Affine Camera Model

Computer Vision I

Orthographic Projection

CS252A, Fall 2012

Appropriate in Neighborhood About (x0,y0,z0)

Particular case

Computer Vision I

•  Perspective

•  Assume that f=1, and perform a Taylor series expansion about (x0, y0, z0)

•  Take perspective projection equation, and perform Taylor series expansion about some point P= (x0,y0,z0). •  Drop terms that are higher order than linear. •  Resulting expression is affine camera model CS252A, Fall 2012

Computer Vision I

⎡u ⎤ 1 ⎡ x 0 ⎤ 1 ⎡ x 0 ⎤ 1 ⎡1 ⎤ ⎢ ⎥ = ⎢ ⎥ − 2 ⎢ ⎥( z − z0 ) + ⎢ ⎥( x − x 0 ) z0 ⎣0 ⎦ ⎣v ⎦ z0 ⎣ y 0 ⎦ z0 ⎣ y 0 ⎦ 2 1 ⎡0 ⎤ 1 2 ⎡ x 0 ⎤ + ⎢ ⎥( y − y 0 ) + 3 ⎢ ⎥( z − z0 ) + z0 ⎣1 ⎦ 2 z0 ⎣ y 0 ⎦

•  Dropping higher order terms and regrouping. € CS252A, Fall 2012

⎡ x ⎤ ⎡u ⎤ 1 ⎡ x 0 ⎤ ⎡1/z0 0 −x 0 /z02 ⎤ ⎢ ⎥ y = Ap + b ⎢ ⎥ ≈ ⎢ ⎥ + ⎢ 2 ⎥ ⎢ ⎥ ⎣v ⎦ z0 ⎣ y 0 ⎦ ⎣0 1/z0 −y 0 /z0 ⎦ ⎢⎣ z ⎥⎦

Computer Vision I



3

Scaled orthographic projection

Affine camera model in Euclidean Coordinates

Starting with Affine Camera Model, take Taylor series about (xo, y0, z0) = (0, 0, z0) – a point on the optical axis Rewrite affine camera model in terms of Homogenous Coordinates (0, 0, z0)

⎡u ⎤ 1 ⎡ x ⎤ ⎢ ⎥ = ⎢ ⎥ ⎣v ⎦ z0 ⎣ y ⎦ –  That is the z coordinate is dropped, and the image a scaling of the x and y coordinates, where the scale is 1/z0, the depth of the point of the expansion. CS252A, Fall 2012

Computer Vision I

CS252A, Fall 2012

Computer Vision I



The projection matrix for scaled orthographic projection ⎛ U ⎞ ⎛1/z0 0 ⎜ ⎟ ⎜ ⎜ V ⎟ = ⎜ 0 1/z0 ⎜ ⎟ ⎜ 0 ⎝W ⎠ ⎝ 0

⎛ X ⎞ 0 0⎞⎜ ⎟ ⎟⎜ Y ⎟ 0 0⎟ ⎟⎜ Z ⎟ 0 1⎠⎜ ⎟ ⎝ 1 ⎠

For all cameras?

•  Parallel lines project to parallel lines € •  Ratios of distances are preserved under orthographic projection

CS252A, Fall 2012

Computer Vision I

CS252A, Fall 2012

Other camera models

Computer Vision I

Some Alternative “Cameras”

•  Generalized camera – maps points lying on rays and maps them to points on the image plane. Omnicam (hemispherical)

CS252A, Fall 2012

Light Probe (spherical)

Computer Vision I

CS252A, Fall 2012

Computer Vision I

4

Beyond the pinhole Camera

CS252A, Fall 2012

Beyond the pinhole Camera Getting more light – Bigger Aperture

Computer Vision I

CS252A, Fall 2012

Pinhole Camera Images with Variable Aperture 2 mm

1mm

.6 mm

.35 mm

.15 mm

.07 mm

CS252A, Fall 2012

Computer Vision I

Limits for pinhole cameras

CS252A, Fall 2012

The reason for lenses

Computer Vision I

Computer Vision I

Lenses

We need light, but big pinholes cause blur.

CS252A, Fall 2012

Computer Vision I

CS252A, Fall 2012

Computer Vision I

5

Thin Lens

O

Thin Lens: Center

Optical axis F

•  Rotationally symmetric about optical axis. •  Spherical interfaces.

CS252A, Fall 2012

O

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

Computer Vision I

CS252A, Fall 2012

Thin Lens: Focus

Computer Vision I

Thin Lens: Image of Point P F

O

F

O

P’ –  All rays passing through lens and starting at P converge upon P’ –  So light gather capability of lens is given the area of the lens and all the rays focus on P’ instead of become blurred like a pinhole

Parallel lines pass through the focus, F CS252A, Fall 2012

Computer Vision I

CS252A, Fall 2012

Computer Vision I

Thin Lens: Image of Point

Thin Lens: Image Plane Q’

F P’

Z’

f

P

P

O

F

Z

P’ Image Plane

Relation between depth of Point (Z) and the depth where it focuses (Z’)

CS252A, Fall 2012

Computer Vision I

O Q

A price: Whereas the image of P is in focus, the image of Q isn’t. CS252A, Fall 2012

Computer Vision I

6

Thin Lens: Aperture

Field of View Image Plane

P

O P’ Image Plane

f

O

Field of View

•  Smaller Aperture -> Less Blur •  Pinhole -> No Blur – Field of view is a function of f and size of image plane.

CS252A, Fall 2012

Computer Vision I

CS252A, Fall 2012

Computer Vision I

Spherical aberration

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

Rays parallel to the axis do not converge

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

Outer portions of the lens yield smaller focal lengths

2. chromatic Aberrations are reduced by combining lenses

CS252A, Fall 2012

Compound lenses

Computer Vision I

CS252A, Fall 2012

Astigmatism

An optical system with astigmatism is one where rays that propagate in two perpendicular planes have different focus. 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.

Computer Vision I

Distortion magnification/focal length different for different angles of inclination

object pincushion (tele-photo)

barrel (wide-angle)

Can be corrected! (if parameters are know) CS252A, Fall 2012

Computer Vision I

CS252A, Fall 2012

Computer Vision I

7

Chromatic aberration

Chromatic aberration

(great for prisms, bad for lenses)

rays of different wavelengths focused in different planes

cannot be removed completely

CS252A, Fall 2012

Computer Vision I

CS252A, Fall 2012

Vignetting

Computer Vision I

Human eye

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

CS252A, Fall 2012

Computer Vision I

CS252A, Fall 2012

Computer Vision I

8

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