CS计算机代考程序代写 2/4/2021

2/4/2021
CSE 473/573
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Introduction to Computer Vision and Image Processing
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Questions From Last Class?
• I will try to come on a few minutes early to answer questions each class. Simply Ask!
• Syllabus and Slides are online
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Creating an Image… Lets Drill Down…
Digital Camera
Image Processing Image Processing
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SCENE
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IMAGE FORMATION
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What type of things can effect the image that gets formed?
• LightingConditions • SceneGeometry
• Surfac‘-eProperties • CameraProperties
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Reflections
• For a rough surface, light is reflected in all directions from any local point
• A light source is providing a lot of rays in parallel
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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?
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Slide source: Seitz
Answer is No….
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Part 1: Geometry of Image Formation
Mapping between image and world coordinates • Pinhole camera model (any photographers?)
• Projective geometry ‘-
‐ Vanishing points and lines • Projection matrix
• SLIDE CREDITS : Hays, Darrell
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Pinhole camera
Idea 2: add a barrier to block off most of the rays • This reduces blurring
• The opening known as the aperture
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Slide source: Seitz
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Pinhole camera
f
c
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f = focal length
c = center of the camera
Figure from Forsyth
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You can build one
• Cardboard box
• Cut two holes in one end • One on left
• One on right
• Put a white piece of paper
at the other end
• Cover the right hole with a piece of aluminum foil
• Poke a hole it with a pin
• Point the box at a light and look in the other hole
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Camera obscura: the pre-camera
• Known during classical period in China and Greece (e.g. Mo-Ti, China, 470BC to 390BC)
Illustration of Camera Obscura
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Freestanding camera obscura at UNC Chapel Hill
Photo by Seth Ilys
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Camera Obscura used for Tracing
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Lens Based Camera Obscura, 1568
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First Photograph
Oldest surviving photograph • Took 8 hours on pewter plate
Joseph Niepce, 1826
Photograph of the first photograph
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Stored at UT Austin
Niepce later teamed up with Daguerre, who eventually created Daguerrotypes
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Geometry of Image Formation
Mapping between image and world coordinates • Pinhole camera model (any photographers?)
• Projective geometry ‘-
‐ Vanishing points and lines • Projection matrix
• SLIDE CREDITS : Hays, Darrell
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Vanishing points and lines
Parallel lines in the world intersect in the image at a “vanishing point”
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Vanishing points and lines
Vanishing Line
Vanishing Point
Vanishing Point oo
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Vanishing points and lines
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Vertical vanishing point
(at infinity)
Vanishing point
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Vanishing point
Slide from Efros, Photo from Criminisi
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Exercise
• Can you identify other pairs of parallel lines in the image?
• Can you “estimate” the vanishing point of these parallel lines?
• Do they meet on the vanishing line that is defined by the two vanishing
points in the previous slide?
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• What properties must the green lines in the previous slide have, so they will not meet along at the vanishing line?
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What do we notice about perspective?
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Perspective effects
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Projection properties
Many-to-one: any points along same ray map to same point in image
Points  ? • points
Lines  ? • lines
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Distances and angles are / are not ? preserved • are not
Degenerate cases:
– Line through focal point projects to a point.
– Plane through focal point projects to line
– Plane perpendicular to image plane projects to part of the image.
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Length and area are not preserved
A’ C’
B’
Figure by David Forsyth
Here:
• C and B are approximately equal
• A is shorter then C
Here:
• A and C are approximately equal • A andCareshorterthenB‘-
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Perspective and art
• Use of correct perspective projection indicated in 1st century B.C. frescoes
• Skill resurfaces in Renaissance: artists develop systematic methods to determine perspective
projection (around 1480-1515)
Raphael
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Durer, 1525
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K. Grauman
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Geometry of Image Formation
Mapping between image and world coordinates • Pinhole camera model (any photographers?)
• Projective geometry ‘-
‐ Vanishing points and lines • Projection matrix
• SLIDE CREDITS : Hays, Darrell
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What measurements do we lose?
• Projection from 3D world to 2D Image
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What do we lose?
• Angles
• Distances
• and therefore Area
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Observations
• If we want to reason about images, we have to be able to “convert” between 3D space 2D Images
• In Euclidean Space, Parallel Lines can not meet
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• Homogeneous coordinates are a way of representing N-dimensional coordinates with N+1 numbers.
• 2D, or 3D, or ND it does not matter
• In Projective Space, Parallel Lines can meet
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Homogeneous coordinates
Converting to homogeneous coordinates ‘-
homogeneous image homogeneous scene coordinates coordinates
Converting from homogeneous coordinates
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Exercise: Parallel Lines Never Meet
• Euclidean Space linear systems
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• There is no Solution because C = D means they are the same line
• But what if x and y are points in projective (homogeneous) space? Is there a solution?
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Perspective projection & calibration
• Perspective equations so far in terms of camera’s reference frame….
• Camera’s intrinsic (Camera) and‘-extrinsic (World) parameters needed to calibrate geometry.
Camera frame
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K. Grauman
Idealized (Pinhole) Assumptions
• Everything is a unit length
• Focal Length is 1 unit, pixels are 1 unit
• The camera and the world coordinate systems are aligned with the image plane. ‘-
• No Rotation or Translation
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Projection matrix
Slide Credit: Savarese
jw
kw Ow
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WORLD COORDINATE SYSTEM
R,t
X
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x iw
CAMERA COORDINATE SYSTEM
𝐱􏰉𝐊𝐑 𝐭 𝐗
x: Image Coordinates: (u,v,1) K: Intrinsic Matrix (3×3)
R: Rotation (3×3)
t: Translation (3×1)
X: World Coordinates: (X,Y,Z,1)
Intrinsic parameters: from world coordinates to pixel coordinates
p = (u,v)
Perspective projection
Forsyth&Ponce
focal length = f
P = (x,y,z)
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𝑢 􏰉 𝑓 𝑥𝑧 𝑣 􏰉 𝑓 𝑦𝑧
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W. Freeman
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Show Geometrically that we can co.mpute (u,v)𝑥 𝑋􏰉𝑦
𝑧
.f.
3 5y
‘-z
-2
.v ?
Center
u
𝑥􏰉 𝑢𝑣
Camera (0, 0, 0)
IfX=2,Y=3, Z = 5, and f = 2
What are u and v?
𝑣 􏰥𝑓
􏰉 𝑦 𝑧
𝑢􏰉􏰥𝑥∗𝑓𝑧 𝑢􏰉􏰥2∗25 𝑣 􏰉 􏰥𝑦 ∗ 𝑓 𝑣 􏰉 􏰥3 ∗ 25
𝑧 39
Intrinsic parameters
𝑢 􏰉 𝛼 𝑥𝑧 𝑣 􏰉 𝛼 𝑦𝑧
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Relax pixels of unit length:
The “pixels” are in some arbitrary spatial units
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W. Freeman
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Intrinsic parameters
𝑢 􏰉 𝛼 𝑥𝑧 𝑣 􏰉 𝛽 𝑦𝑧
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Relax pixels aspect: Pixels may not be square
𝛽 𝛼
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W. Freeman
Intrinsic parameters
Relax image center is (0,0):
Origin of our camera pixel coordinates is offset
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𝑢 􏰉 𝛼 𝑥𝑧 􏰜 𝑢 􏰃 𝑣 􏰉 𝛽 𝑦𝑧 􏰜 𝑣􏰃
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W. Freeman
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Intrinsic parameters
2/4/2021
􏰭 ‘-𝑣𝑣􏰭 𝑣􏰭sin􏰁𝜃􏰈􏰉𝑣 􏰭
𝑢 􏰉𝑢􏰥cos􏰁𝜃􏰈𝑣 􏰉𝑢􏰥cot􏰁𝜃􏰈𝑣
𝑢􏰉𝛼 𝑥􏰥𝛼cot􏰁𝜃􏰈𝑦􏰜𝑢 𝑧 𝑧 􏰃
𝜃 𝑢􏰭 𝑢
Relaxtheimageplaneis orthogonal to the camera axis:
May be skew between camera pixel axes
𝑣 􏰉
𝛽 𝑦 􏰜 𝑣􏰃 sin􏰁 𝜃􏰈 𝑧
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W. Freeman
Intrinsic parameters, homogeneous coordinates
𝑢 􏰉 𝛼 𝑥𝑧 􏰥 𝛼 c o t 􏰁 𝜃 􏰈 𝑦𝑧 􏰜 𝑢 􏰃
𝑣 􏰉
𝛽 𝑦 􏰜 𝑣􏰃 sin􏰁𝜃􏰈 𝑧
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Using homogenous coordinates,
we can write this as:
or:

x  
  cot() u0 u
0   y
  
v0 v 0
 
  sin() 0 z   
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 0 0 1 
 1  
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W. Freeman
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Projection matrix
Extrinsic Assumptions • No rotation
• Camera at (0,0,0)
Intrinsic Assumptions • Unit aspect ratio
• Optical center at (0,0) • No skew
X
x
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K
𝐱􏰉𝐊𝐈 𝟎 𝐗
u f 0 0 0x wv0 f 0 0y
1 0 0 1 0z   1
Slide Credit: Savarese

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Projection matrix
Extrinsic Assumptions • No rotation
• Camera at (0,0,0)
Intrinsic Assumptions • Unit aspect ratio
• Optical center at (0,0) •Noskew
X
x
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𝐱􏰉𝐊𝐈 𝟎 𝐗
u f 0 0 0x wv0 f 0 0y
1 0 0 1 0z   1
Slide Credit: Savarese

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2/4/2021
Remove assumption: known optical center
Intrinsic Assumptions • Unit aspect ratio
• No skew
𝐱􏰉𝐊𝐈 𝟎 𝐗
Extrinsic Assumptions • No rotation
• Camera at (0,0,0)
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u 
0x 0y
f 0 u0 0 f v0 001
wv
1   
0z 1

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Remove assumption: square pixels
Intrinsic Assumptions • No skew
𝐱􏰉𝐊𝐈 𝟎 𝐗
Extrinsic Assumptions • No rotation
• Camera at (0,0,0)
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u  0 u 0x   0 y
wv  0  v0 0z
1 0 0 1 0   1

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2/4/2021
Remove assumption: non-skewed pixels
Intrinsic Assumptions
𝐱􏰉𝐊𝐈 𝟎 𝐗
Note: different books use different notation for parameters
Extrinsic Assumptions • No rotation
• Camera at (0,0,0)
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u  s u 0x   0 y
wv  0  v0 0z
1 0 0 1 0   1

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Remove assumption: non-skewed pixels
Intrinsic Assumptions
𝐱􏰉𝐊𝐈 𝟎 𝐗
Note: different books use different notation for parameters
Extrinsic Assumptions • No rotation
• Camera at (0,0,0)
‘-
u  s u 0x   0 y
wv  0  v0 0z
1 0 0 1 0   1

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