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Computer Vision (CS708) Lecture02 - Image formation and OpenCV
Dr. Abdul Basit Assistant Professor - CSIT University of Balochistan
September 29, 2016
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
1 / 54
Contents
The contents of this lecture are taken form Computer Vision: Algorithms and Applications (Richard Szeliski ) Introduction to computer vision CS Learning OpenCV Computer Vision in C++ with OpenCV library Adrian Kaebler & Gary Bradski
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
2 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
3 / 54
Computer vision overview Overview of vision, examples of the state of the art, preview of projects
Figure: Computer vision overview. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
4 / 54
Computer vision algorithms
Figure: A pictorial summary of the computer vision algorithms. Reprinted form Szeliski (2011). Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
5 / 54
Computer vision algorithms (Cond)
Figure: A pictorial summary of the computer vision algorithms. Reprinted form Szeliski (2011). Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
6 / 54
Contents overview
Figure: Relationship between images, geometry, and photometry, as well as a taxonomy of the topics covered in this book. Szeliski (2011). Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
7 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
8 / 54
A camera
What do we need to make a camera from scratch?
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
9 / 54
Image formation This lecture
Mapping between image and world coordinates Pinhole camera model Projective geometry Vanishing points and lines Projection matrix
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
10 / 54
Image formation Camera and world geometry
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
11 / 54
Camera and world geometry
How tall is the woman? How high the camera is? What is the camera rotation? What is the focal length of the camera? Which ball is closer?
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
12 / 54
Image formation Let’s design a camera Idea 1: put a piece of film in front of an object Do we get a reasonable image?
Figure: Image formation understanding.
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
13 / 54
Image formation Idea 2: add a barrier to block off most of the rays This reduce blurring The opening known as the aperture
Figure: Image formation understanding.
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
14 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
14 / 54
Image formation Pinhole camera
Figure: A pinhole camera. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
15 / 54
Image formation Dimensionality Reduction Machine (3D to 2D)
Figure: A pinhole camera. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
16 / 54
Projective geometry What is lost?
What is lost? Length Which one is closer?
Dr. Abdul Basit (CSIT - UoB)
Computer Figure: What weVision lost after projection.
September 29, 2016
17 / 54
Projective geometry Length cannot be preserved?
Figure: Lenght is lost. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
18 / 54
Projective geometry Length cannot be preserved?
What is lost? Length Angles
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
19 / 54
Projective geometry What is preserved?
What is preserved? Straight lines are still straight
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
20 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
21 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
21 / 54
2D Modeling Transformation 2D Translation
Translation is a simple straight line movement of the object
Figure: 2D translation.
old coordinates are (x, y) and the new coordinates are (x’, y’) 0
x = x + Tx 0
y = y + Ty Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
22 / 54
2D Modeling Transformation 2D Translation
Consider the following equation 0
x =x +3 0
y =y +1 This can be rewritten with matrices 0 x x 3 = + 0 1 y y
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
23 / 54
2D Modeling Transformation 2D scaling
Scaling alters the size of the object. Sx and Sy are the scaling factors: 0
0
Old coordinates are (x, y ) and the new coordinates are (x , y )
0
x = x ∗ Sx 0
y = y ∗ Sy
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
24 / 54
2D Modeling Transformation 2D scaling
Figure: 2D scaling.
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
25 / 54
2D Modeling Transformation 2D Rotation
Example of a 2D rotation through an angle w where the coordinates x, y go into x’, y’. Note that w is positive for a counterclockwise rotation and that that rotation is about the origin (0, 0).
Figure: 2D rotation. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
26 / 54
2D Modeling Transformation Formula for rotation 0
0
Old coordinates are (x, y ) and the new coordinates are (x , y ) θ = initial angle, α = angle of rotation.
x = r cos θ y = r sin θ
0
x = r cos(θ + α) = r cos θ cos α − r sin θ sin α 0
y = r sin(θ + α) = r sin θ cos α + r cos θ sin α hence, 0
x = x cos α − y sin α 0
y = y cos α + x sin α Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
27 / 54
2D Modeling Transformation Martix representation of transformations
0
0
Scaling: x = x ∗ Sx , y = y ∗ Sy , which can be written as:
x
0
y
0
T x S = × x y 0
0 Sy
Rotation: 0
x = x cos α − y sin α 0
y = y cos α + x sin α which can be written as: 0 x
Dr. Abdul Basit (CSIT - UoB)
y
0
T x cos α sin α = × y − sin α cos α
Computer Vision
September 29, 2016
28 / 54
2D Modeling Transformation Martix representation of transformations
So if we wanted to scale and then rotate the object we could do the following:
Dr. Abdul Basit (CSIT - UoB)
x
00
x
0
y
00
y
0
T x S = × x y 0
0 Sy
0 T cos α sin α x = 0 × − sin α cos α y
Computer Vision
September 29, 2016
29 / 54
2D Modeling Transformation Matrix representation of transformations
But this is the same as:
x
00
Dr. Abdul Basit (CSIT - UoB)
y
00
T x Sx 0 cos α sin α = × × y 0 Sy − sin α cos α T x Sx cos α Sx sin α = × y −Sy sin α Sy cos α
Computer Vision
September 29, 2016
30 / 54
2D Transformations Matrix representation of transformations
What about translation? 0
x = x ∗ Tx 0
y = y ∗ Ty There is no way to express this as a multiplicative transformation with a 2D matrix. Therefore we go into hyperspace (N + 1 dimensions) , i.e., we embed 2D space into 3D space. We express all coordinates as (xh, yh, z). This is called a ”Homogeneous” coordinate system.
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
31 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
31 / 54
2D transformations Homogeneous coordinates
Translation (represented by T = (Tx , Ty )): T x 1 0 0 0 1 x y 1 = y × 0 1 Tx Ty
0 0 1
which gives 0
x = x + 1 ∗ Tx 0
y = y + 1 ∗ Ty 1=1
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
32 / 54
2D transformations Homogeneous coordinates
Scaling (represented by S = (Sx , Sy )) T x Sx 0 0 x y 1 = y × 0 1 0x
0 Sy 0
0 0 1
which gives 0
x = x ∗ Sx 0
y = y ∗ Sy
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
33 / 54
2D transformations Homogeneous coordinates
Rotation (represented by R(q): T x cos q sin q 0 0 0 x y 1 = y × − sin q cos q 0 1 0 0 1 which gives 0
x = x cos q − y sin q 0
y = y cos q + x sin q
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
34 / 54
Homogeneous coordinates Projection world to image coordinates
Figure: Projection from world to image coordinates. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
35 / 54
Homogeneous coordinates
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
36 / 54
Homogeneous coordinates Scaling
Invariant to scaling
Point in Cartesian is ray in Homogeneous
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
37 / 54
Homogeneous coordinates Projection Matrix
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
38 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
38 / 54
Projection Matrix
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
39 / 54
Projection Matrix Remove assumption: known optical center
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
40 / 54
Projection Matrix Remove assumption: square pixels
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
41 / 54
Homogeneous coordinates Oriented and translated camera
Figure: Oriented and translated camera. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
42 / 54
Projection Matrix Allow camera translation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
43 / 54
Projection Matrix 3D rotation of points
Rotation around the coordinate axes, counter-clockwise:
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
44 / 54
Projection Matrix Allow camera rotation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
45 / 54
Projection Degrees of freedom
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
46 / 54
Projection Field of View (Zoom, focal length)
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
47 / 54
Projection Matrix Beyond Pinholes: Radial Distortion
Figure: Radial distortion. Reprinted from
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
48 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
48 / 54
Projective geometry Vanishing points and lines
Parallel lines in the world intersect in the image at a “vanishing point”
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
49 / 54
Projective geometry Vanishing points and lines
Figure: Vanishing points and lines. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
50 / 54
Projective geometry Vanishing points and lines
Figure: Vanishing points and lines. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
51 / 54
Things to remember
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
52 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
53 / 54
The End
59 / 59
Dr. Abdul Basit Assistant Professor - CSIT University of Balochistan
September 29, 2016
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
1 / 54
Contents
The contents of this lecture are taken form Computer Vision: Algorithms and Applications (Richard Szeliski ) Introduction to computer vision CS Learning OpenCV Computer Vision in C++ with OpenCV library Adrian Kaebler & Gary Bradski
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
2 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
3 / 54
Computer vision overview Overview of vision, examples of the state of the art, preview of projects
Figure: Computer vision overview. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
4 / 54
Computer vision algorithms
Figure: A pictorial summary of the computer vision algorithms. Reprinted form Szeliski (2011). Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
5 / 54
Computer vision algorithms (Cond)
Figure: A pictorial summary of the computer vision algorithms. Reprinted form Szeliski (2011). Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
6 / 54
Contents overview
Figure: Relationship between images, geometry, and photometry, as well as a taxonomy of the topics covered in this book. Szeliski (2011). Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
7 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
8 / 54
A camera
What do we need to make a camera from scratch?
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
9 / 54
Image formation This lecture
Mapping between image and world coordinates Pinhole camera model Projective geometry Vanishing points and lines Projection matrix
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
10 / 54
Image formation Camera and world geometry
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
11 / 54
Camera and world geometry
How tall is the woman? How high the camera is? What is the camera rotation? What is the focal length of the camera? Which ball is closer?
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
12 / 54
Image formation Let’s design a camera Idea 1: put a piece of film in front of an object Do we get a reasonable image?
Figure: Image formation understanding.
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
13 / 54
Image formation Idea 2: add a barrier to block off most of the rays This reduce blurring The opening known as the aperture
Figure: Image formation understanding.
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
14 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
14 / 54
Image formation Pinhole camera
Figure: A pinhole camera. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
15 / 54
Image formation Dimensionality Reduction Machine (3D to 2D)
Figure: A pinhole camera. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
16 / 54
Projective geometry What is lost?
What is lost? Length Which one is closer?
Dr. Abdul Basit (CSIT - UoB)
Computer Figure: What weVision lost after projection.
September 29, 2016
17 / 54
Projective geometry Length cannot be preserved?
Figure: Lenght is lost. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
18 / 54
Projective geometry Length cannot be preserved?
What is lost? Length Angles
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
19 / 54
Projective geometry What is preserved?
What is preserved? Straight lines are still straight
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
20 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
21 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
21 / 54
2D Modeling Transformation 2D Translation
Translation is a simple straight line movement of the object
Figure: 2D translation.
old coordinates are (x, y) and the new coordinates are (x’, y’) 0
x = x + Tx 0
y = y + Ty Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
22 / 54
2D Modeling Transformation 2D Translation
Consider the following equation 0
x =x +3 0
y =y +1 This can be rewritten with matrices 0 x x 3 = + 0 1 y y
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
23 / 54
2D Modeling Transformation 2D scaling
Scaling alters the size of the object. Sx and Sy are the scaling factors: 0
0
Old coordinates are (x, y ) and the new coordinates are (x , y )
0
x = x ∗ Sx 0
y = y ∗ Sy
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
24 / 54
2D Modeling Transformation 2D scaling
Figure: 2D scaling.
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
25 / 54
2D Modeling Transformation 2D Rotation
Example of a 2D rotation through an angle w where the coordinates x, y go into x’, y’. Note that w is positive for a counterclockwise rotation and that that rotation is about the origin (0, 0).
Figure: 2D rotation. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
26 / 54
2D Modeling Transformation Formula for rotation 0
0
Old coordinates are (x, y ) and the new coordinates are (x , y ) θ = initial angle, α = angle of rotation.
x = r cos θ y = r sin θ
0
x = r cos(θ + α) = r cos θ cos α − r sin θ sin α 0
y = r sin(θ + α) = r sin θ cos α + r cos θ sin α hence, 0
x = x cos α − y sin α 0
y = y cos α + x sin α Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
27 / 54
2D Modeling Transformation Martix representation of transformations
0
0
Scaling: x = x ∗ Sx , y = y ∗ Sy , which can be written as:
x
0
y
0
T x S = × x y 0
0 Sy
Rotation: 0
x = x cos α − y sin α 0
y = y cos α + x sin α which can be written as: 0 x
Dr. Abdul Basit (CSIT - UoB)
y
0
T x cos α sin α = × y − sin α cos α
Computer Vision
September 29, 2016
28 / 54
2D Modeling Transformation Martix representation of transformations
So if we wanted to scale and then rotate the object we could do the following:
Dr. Abdul Basit (CSIT - UoB)
x
00
x
0
y
00
y
0
T x S = × x y 0
0 Sy
0 T cos α sin α x = 0 × − sin α cos α y
Computer Vision
September 29, 2016
29 / 54
2D Modeling Transformation Matrix representation of transformations
But this is the same as:
x
00
Dr. Abdul Basit (CSIT - UoB)
y
00
T x Sx 0 cos α sin α = × × y 0 Sy − sin α cos α T x Sx cos α Sx sin α = × y −Sy sin α Sy cos α
Computer Vision
September 29, 2016
30 / 54
2D Transformations Matrix representation of transformations
What about translation? 0
x = x ∗ Tx 0
y = y ∗ Ty There is no way to express this as a multiplicative transformation with a 2D matrix. Therefore we go into hyperspace (N + 1 dimensions) , i.e., we embed 2D space into 3D space. We express all coordinates as (xh, yh, z). This is called a ”Homogeneous” coordinate system.
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
31 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
31 / 54
2D transformations Homogeneous coordinates
Translation (represented by T = (Tx , Ty )): T x 1 0 0 0 1 x y 1 = y × 0 1 Tx Ty
0 0 1
which gives 0
x = x + 1 ∗ Tx 0
y = y + 1 ∗ Ty 1=1
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
32 / 54
2D transformations Homogeneous coordinates
Scaling (represented by S = (Sx , Sy )) T x Sx 0 0 x y 1 = y × 0 1 0x
0 Sy 0
0 0 1
which gives 0
x = x ∗ Sx 0
y = y ∗ Sy
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
33 / 54
2D transformations Homogeneous coordinates
Rotation (represented by R(q): T x cos q sin q 0 0 0 x y 1 = y × − sin q cos q 0 1 0 0 1 which gives 0
x = x cos q − y sin q 0
y = y cos q + x sin q
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
34 / 54
Homogeneous coordinates Projection world to image coordinates
Figure: Projection from world to image coordinates. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
35 / 54
Homogeneous coordinates
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
36 / 54
Homogeneous coordinates Scaling
Invariant to scaling
Point in Cartesian is ray in Homogeneous
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
37 / 54
Homogeneous coordinates Projection Matrix
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
38 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
38 / 54
Projection Matrix
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
39 / 54
Projection Matrix Remove assumption: known optical center
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
40 / 54
Projection Matrix Remove assumption: square pixels
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
41 / 54
Homogeneous coordinates Oriented and translated camera
Figure: Oriented and translated camera. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
42 / 54
Projection Matrix Allow camera translation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
43 / 54
Projection Matrix 3D rotation of points
Rotation around the coordinate axes, counter-clockwise:
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
44 / 54
Projection Matrix Allow camera rotation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
45 / 54
Projection Degrees of freedom
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
46 / 54
Projection Field of View (Zoom, focal length)
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
47 / 54
Projection Matrix Beyond Pinholes: Radial Distortion
Figure: Radial distortion. Reprinted from
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
48 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
48 / 54
Projective geometry Vanishing points and lines
Parallel lines in the world intersect in the image at a “vanishing point”
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
49 / 54
Projective geometry Vanishing points and lines
Figure: Vanishing points and lines. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
50 / 54
Projective geometry Vanishing points and lines
Figure: Vanishing points and lines. Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
51 / 54
Things to remember
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
52 / 54
Contents
1
Computer vision techniques
2
Image formation Pinhole camera
3
Projective geometry 2D geometry Homogeneous coordinates 3D projection Vanishing points and lines
4
OpenCV Installation
Dr. Abdul Basit (CSIT - UoB)
Computer Vision
September 29, 2016
53 / 54
The End
59 / 59
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