Pinhole cameras

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Pinhole cameras

• Abstract camera model - box with a small hole in
  it

• Pinhole cameras work in practice

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Distant objects are smaller
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Parallel lines meet
Common to draw film plane in front of the focal
point. Moving the film plane merely scales the
image.
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Vanishing points

• each set of parallel lines (direction) meets at
  a different point
• The vanishing point for this direction
• Sets of parallel lines on the same plane lead to
  collinear vanishing points.
• The line is called the horizon for that plane

• Good ways to spot faked images
• scale and perspective dont work
• vanishing points behave badly
• supermarket tabloids are a great source.

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Slide credit David Jacobs
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Properties of Projection

• Points project to points
• Lines project to lines
• Vanishing points for parallel lines
• Parallel lines parallel to image plane donot
  converge
• Closer objects appear bigger
• Angles are not preserved
• Degenerate cases
• Line through focal point projects to a point.
• Plane through focal point projects to line

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Pinhole Camera Terminology
Image plane
Optical axis
Principal point/ image center
Focal length
Camera center/ pinhole
Camera point
Image point
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The equation of projection
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The equation of projection

• Cartesian coordinates
• We have, by similar triangles, that
  (x, y, z) -gt (f x/z, f y/z, -f)
• Ignore the third coordinate, and get

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The camera matrix

• Turn previous expression into HCs
• HCs for 3D point are (X,Y,Z,T)
• HCs for point in image are (U,V,W)

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Weak perspective

• Issue
• perspective effects, but not over the scale of
  individual objects
• collect points into a group at about the same
  depth, then divide each point by the depth of its
  group
• Adv easy
• Disadv wrong

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The Equation of Weak Perspective

• s is constant for all points.
• Parallel lines no longer converge, they remain
  parallel.

Slide credit David Jacobs
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Pros and Cons of These Models

• Weak perspective has simpler math.
• Accurate when object is small and distant.
• Most useful for recognition.
• Pinhole perspective much more accurate for
  scenes.
• Used in structure from motion.
• When accuracy really matters, we must model the
  real camera
• Use perspective projection with other calibration
  parameters (e.g., radial lens distortion)

Slide credit David Jacobs
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Orthographic projection
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The projection matrix for orthographic projection
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Affine cameras
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Camera parameters

• Issue
• camera may not be at the origin, looking down the
  z-axis
• extrinsic parameters
• one unit in camera coordinates may not be the
  same as one unit in world coordinates
• intrinsic parameters - focal length, principal
  point, aspect ratio, angle between axes, etc.

Note the matrix dimensions
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Camera calibration

• Issues
• what are intrinsic parameters of the camera?
• what is the camera matrix? (intrinsicextrinsic)
• General strategy
• view calibration object
• identify image points
• obtain camera matrix by minimizing error
• obtain intrinsic parameters from camera matrix

• Error minimization
• Linear least squares
• easy problem numerically
• solution can be rather bad
• Minimize image distance
• more difficult numerical problem
• solution usually rather good,
• start with linear least squares
• Numerical scaling is an issue

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Human Eye

• The eye has an iris like a camera
• Focusing is done by changing shape of lens
• Retina contains cones (mostly used) and rods (for
  low light)
• The fovea is small region of high resolution
  containing mostly cones
• Optic nerve 1 million flexible fibres

http//www.cas.vanderbilt.edu/bsci111b/eye/human-e
ye.jpg
Slide credit David Jacobs
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The Image Formation Pipeline
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Outline

• Vector, matrix basics
• 2-D point transformations
• Translation, scaling, rotation, shear
• Homogeneous coordinates and transformations
• Homography, affine transformation

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Notes on Notation

• Vectors, points x, v (assume column vectors)
• Matrices R, T
• Scalars x, a
• Axes, objects X, Y, O
• Coordinate systems W, C
• Number systems R, Z
• Specials
• Transpose operator xT
  (as opposed to x0)
• Identity matrix Id
• Matrices/vectors of zeroes, ones 0, 1

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Block Notation for Matrices

• Often convenient to write matrices in terms of
  parts
• Smaller matrices for blocks
• Row, column vectors for ranges of entries on
  rows, columns, respectively
• E.g. If A is 3 x 3 and
  

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2-D Transformations

• Types
• Scaling
• Rotation
• Shear
• Translation
• Mathematical representation

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2-D Scaling
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2-D Scaling
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2-D Scaling
sx
1
Horizontal shift proportional to horizontal
position
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2-D Scaling
sy
1
Vertical shift proportional to vertical position
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2-D Scaling
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Matrix form of 2-D Scaling
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2-D Scaling
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2-D Rotation
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2-D Rotation
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2-D Rotation
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Matrix form of 2-D Rotation
(this is a counterclockwise rotation reverse
signs of sines to get a clockwise one)
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Matrix form of 2-D Rotation
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2-D Shear (Horizontal)
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2-D Shear (Horizontal)
Horizontal displacement proportional to vertical
position
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2-D Shear (Horizontal)
(Shear factor h is positive for the figure above)
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2-D Shear (Horizontal)
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2-D Shear (Vertical)
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2-D Translation
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2-D Translation
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2-D Translation
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2-D Translation
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2-D Translation
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2-D Translation
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Representing Transformations

• Note that weve defined translation as a vector
  addition but rotation, scaling, etc. as matrix
  multiplications
• Its inconvenient to have two different
  operations (addition and multiplication) for
  different forms of transformation
• It would be desirable for all transformations to
  be expressed in a common, linear form (since
  matrix multiplications are linear
  transformations)

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Example Trick of additional coordinate makes
this possible

• Old way
• New way

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Translation Matrix

• We can write the formula using this expanded
  transformation matrix as
• From now on, assume points are in this expanded
  form unless otherwise noted

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Homogeneous Coordinates

• Expanded form is called homogeneous coordinates
  or projective space
• Change to projective space by adding a scale
  factor (usually but not always 1)

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Homogeneous Coordinates Projective Space

• Equivalence is defined up to scale (non-zero
  for finite points)
• Think of projective points in P2 as rays in R3,
  where z coordinate is scale factor
• All Euclidean points along ray are same in this
  sense

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Leaving Projective Space

• Can go back to non-homogeneous representation by
  dividing by scale factor and dropping extra
  coordinate
• This is the same as saying Where does the ray
  intersect the plane defined by z 1?
• Analogy to perspective projection, where f1
  (image plane) and lambda is z of any point in the
  ray. For different lambdas along the line,
  projected point is the same, thus Equivalence
  class

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Homogeneous Coordinates Rotations, etc.

• A 2-D rotation, scaling, shear or other
  transformation normally expressed by a 2 x 2
  matrix R is written in homogeneous coordinates
  with the following 3 x 3 matrix
• The non-commutativity of matrix multiplication
  explains why different transformation orders give
  different resultsi.e., RT ? TR

In homogeneous form
In homogeneous form
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Example Transformations Dont Commute
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Example Transformations Dont Commute
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Example Transformations Dont Commute
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Example Transformations Dont Commute
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2-D Transformations

• Full-generality 3 x 3 homogeneous transformation
  is called a homography

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2-D Transformations

• Full-generality 3 x 3 homogeneous transformation
  is called a homography
• Translation components

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2-D Transformations

• Full-generality 3 x 3 homogeneous
• transformation is called a homography
• Scale/rotation components

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2-D Transformations

• Full-generality 3 x 3 homogeneous transformation
  is called a homography
• Shear/rotation components

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2-D Transformations

• Full-generality 3 x 3 homogeneous transformation
  is called a homography
• Homogeneous scaling factor

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2-D Transformations

• Full-generality 3 x 3 homogeneous transformation
  is called a homography
• When these are zero (as they have been so far), H
  is an affine transformation