Introduction to Computer Vision IntroductionIntroduction
Transcript of Introduction to Computer Vision IntroductionIntroduction
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Introduction to
Computer Vision IntroductionIntroduction
Image Formation
CMPSCI 591A/691ACMPSCI 570/670
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Introduction to
Computer Vision Lecture OutlineLecture Outline
■ Light and Optics● Pinhole camera model● Perspective projection● Thin lens model● Fundamental equation● Distortion: spherical & chromatic aberration, radial distortion● Reflection and Illumination: color, Lambertian and specular
surfaces, Phong, BDRF
■ Sensing Light■ Conversion to Digital Images■ Sampling Theorem■ Other Sensors: frequency, type, ….
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Introduction to
Computer Vision Abstract ImageAbstract Image
■ An image can be represented by an image functionwhose general form is f(x,y).
■ f(x,y) is a vector-valued function whose argumentrepresents a pixel location.
■ The value of f(x,y) can have different interpretations indifferent kinds of images.
Examples
Intensity Image - f(x,y) = intensity of the scene
Range Image - f(x,y) = depth of the scene from imaging system
Color Image - f(x,y) = {fr(x,y), fg(x,y), fb(x,y)}
Video - f(x,y,t) = temporal image sequence
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Introduction to
Computer Vision Basic RadiometryBasic Radiometry
■ Radiometry is the part of image formation concerned withthe relation among the amounts of light energy emittedfrom light sources, reflected from surfaces, and registeredby sensors.
Surface
Optics
CCD Array
P
Light Source
L(P,d)
in
p
e
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Introduction to
Computer Vision Light and MatterLight and Matter
■ The interaction between light and matter can takemany forms:● Reflection● Refraction● Diffraction● Absorption● Scattering
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Introduction to
Computer Vision Lecture AssumptionsLecture Assumptions
■ Typical imaging scenario:● visible light
● ideal lenses
● standard sensor (e.g. TV camera)
● opaque objects
■ Goal
To create 'digital' images which can beprocessed to recover some of thecharacteristics of the 3D world whichwas imaged.
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Introduction to
Computer Vision StepsSteps
World Optics Sensor
Signal Digitizer
Digital Representation
World realityOptics focus {light} from world on sensorSensor converts {light} to {electrical energy}Signal representation of incident light as continuous electrical energyDigitizer converts continuous signal to discrete signalDigital Rep. final representation of reality in computer memory
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Introduction to
Computer Vision Factors in Image FormationFactors in Image Formation
■ Geometry● concerned with the relationship between points in the
three-dimensional world and their images
■ Radiometry● concerned with the relationship between the amount of
light radiating from a surface and the amount incident atits image
■ Photometry● concerned with ways of measuring the intensity of light
■ Digitization● concerned with ways of converting continuous signals
(in both space and time) to digital approximations
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Introduction to
Computer Vision Image FormationImage Formation
Light (Energy) Source
Surface
Pinhole Lens
Imaging Plane
World Optics Sensor Signal
B&W Film
Color Film
TV Camera
Silver Density
Silver densityin three colorlayers
Electrical
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Introduction to
Computer Vision GeometryGeometry
■ Geometry describes the projection of:
two-dimensional(2D) image plane.
three-dimensional(3D) world
■ Typical Assumptions
● Light travels in a straight line
■ Optical Axis: the perpendicular from the image plane through thepinhole (also called the central projection ray)
■ Each point in the image corresponds to a particular direction defined bya ray from that point through the pinhole.
■ Various kinds of projections:
● - perspective - oblique
● - orthographic - isometric
● - spherical
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Introduction to
Computer Vision Basic OpticsBasic Optics
■ Two models are commonly used:● Pin-hole camera● Optical system composed of lenses
■ Pin-hole is the basis for most graphics and vision● Derived from physical construction of early cameras● Mathematics is very straightforward
■ Thin lens model is first of the lens models● Mathematical model for a physical lens● Lens gathers light over area and focuses on image plane.
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Introduction to
Computer Vision Pinhole Camera ModelPinhole Camera Model
■ World projected to 2D Image● Image inverted
● Size reduced
● Image is dim
● No direct depth information
■ f called the focal length of the lens
■ Known as perspective projection
Pinhole lens
Optical Axis
f
Image Plane
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Introduction to
Computer Vision Pinhole camera imagePinhole camera image
Photo by Robert Kosara, [email protected]
http://www.kosara.net/gallery/pinholeamsterdam/pic01.html
Amsterdam
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Introduction to
Computer Vision Equivalent GeometryEquivalent Geometry
■ Consider case with object on the optical axis:
fz
■ More convenient with upright image:
- fz
Projection plane z = 0
■ Equivalent mathematically
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Introduction to
Computer Vision
f
IMAGEPLANE
OPTICAXIS
LENS
i o
1 1 1f i o
= + ‘THIN LENS LAW’
Thin Lens ModelThin Lens Model
■ Rays entering parallel on one side converge at focal point.■ Rays diverging from the focal point become parallel.
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Introduction to
Computer Vision Coordinate SystemCoordinate System
■ Simplified Case:● Origin of world and image coordinate systems coincide
● Y-axis aligned with y-axis
● X-axis aligned with x-axis
● Z-axis along the central projection ray
WorldCoordinateSystem
Image Coordinate System
Z
X
Y
Y
ZX
(0,0,0)
y
x
P(X,Y,Z)p(x,y)
(0,0)
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Introduction to
Computer Vision Perspective ProjectionPerspective Projection
■ Compute the image coordinates of p in terms of theworld coordinates of P.
■ Look at projections in x-z and y-z planes
x
y
Z
P(X,Y,Z)p(x, y)
Z = 0
Z=-f
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Introduction to
Computer Vision X-Z ProjectionX-Z Projection
■ By similar triangles:
Z- f
X
x
=x
f
X
Z+f
=xfX
Z+f
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Introduction to
Computer Vision Y-Z ProjectionY-Z Projection
■ By similar triangles: =y
f
Y
Z+f
=yfY
Z+f
- f
Z
Yy
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Introduction to
Computer Vision Perspective EquationsPerspective Equations
■ Given point P(X,Y,Z) in the 3D world
■ The two equations:
■ transform world coordinates (X,Y,Z)
into image coordinates (x,y)
=yfY
Z+f=x
fX
Z+f
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Introduction to
Computer Vision Reverse ProjectionReverse Projection
■ Given a center of projection and image coordinates of apoint, it is not possible to recover the 3D depth of the pointfrom a single image.
In general, at least two images of the same point takenfrom two different locations are required to recover depth.
All points on this linehave image coordi-nates (x,y).
p(x,y)
P(X,Y,Z) can be any-where along this line
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Introduction to
Computer Vision Stereo GeometryStereo Geometry
■ Depth obtained by triangulation
■ Correspondence problem: pl and pr must correspondto the left and right projections of P, respectively.
Object point
CentralProjection
Rays
Vergence Angle
pl
pr
P(X,Y,Z)
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Introduction to
Computer Vision RadiometryRadiometry
■ Image: two-dimensional array of 'brightness' values.
■ Geometry: where in an image a point will project.
■ Radiometry: what the brightness of the point will be.
■ Brightness: informal notion used to describe bothscene and image brightness.
■ Image brightness: related to energy flux incident onthe image plane:
IRRADIANCE
■ Scene brightness: brightness related to energy fluxemitted (radiated) from a surface.
RADIANCE
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Introduction to
Computer Vision LightLight
■ Electromagnetic energy
■ Wave model
■ Light sources typically radiate over a frequency spectrum■ Φ watts radiated into 4π radians
r
Φ watts
dω
R = Radiant Intensity = dω
dΦ Watts/unit solid angle (steradian)
(of source)
Φ = ∫ dΦsphere
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Introduction to
Computer Vision IrradianceIrradiance
■ Light falling on a surface from all directions.
■ How much?
dA
■ Irradiance: power per unit area falling on a surface.
dΦ
Irradiance E =dΑ
dΦwatts/m2