数字视频技术课件PPT

© Yao Wang, 1/2003
Frequency Domain Analysis
Spatial Frequency
• Spatial frequency measures how fast the image intensity changes in the image plane • Spatial frequency can be completely characterized by the variation frequencies in two orthogonal directions (e.g horizontal and vertical)
• Temporal frequency measures temporal variation (cycles/s) • In a video, the temporal frequency is spatial position dependent, as every point may change differently • Temporal frequency is caused by camera or object motion
© Yao Wang, 1/2003
Frequency Domain Analysis
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Temporal Frequency caused by Linear Motion
© Yao Wang, 1/2003
Frequency Domain Analysis
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Relation between Motion, Spatial and Temporal Frequency
Frequency Domain Analysis
4
Continuous Space Systems
• General system over K-D continuous space
φ (x) = T (ψ (x)), x ∈ R k
• Linear and Space-Shift Invariant (LSI) System
Fourier Analysis of Video Signals & Frequency Response of the HVS
Yao Wang Polytechnic University, Brooklyn, NY11201 yao@vision.poly.edu
Based on: Y. Wang, J. Ostermann, and Y.-Q. Zhang, Video Processing and Communications, Prentice Hall, 2002.
© Yao Wang, 1/2003
Frequency Domain Analysis
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Illustration of Spatial Frequency
Angular Frequency
Problem with previous defined spatial frequency: Perceived speed of change depends on the viewing distance.
© Yao Wang, 1/2003
Frequency Domain Analysis
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Discrete Space Signals
• K-D Space Signals
ψ (n), n = [n1, n2 ,..., nK ] ∈ Z K
• Convolution
ψ (n) * h(n) =
m∈Z K
Rk
• Inverse transform
ψ (x) = ∫ Ψc (f ) exp( j 2πf T x)df
Rk
• Convolution theorem
ψ (x) * h(x) ⇔ Ψc (f ) H c (f ) ψ (x)h(x) ⇔ Ψc (f ) * H c (f )
© Yao Wang, 1/2003
© Yao Wang, 1/2003 Frequency Domain Analysis 14
Illustration of the Relation
© Yao Wang, 1/2003
Frequency Domain Analysis
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Outline
• Fourier transform over multidimensional space
⇔ Ψ ( f x , f y , ft ) = Ψ0 ( f x , f y )δ ( ft + vx f x + v y f y ) Relation between motion, spatial, and temporal frequency : f t = − (v x f x + v y f y )
© Yao Wang, 1/2003
© Yao Wang, 1/2003
Frequency Domain Analysis
Continuous Space Signals
• K-D Space Signals
ψ (x), x = [ x1 , x2 ,..., xK ] ∈ R k
• Convolution
ψ (x) * h(x) = ∫ψ (x − y )h(y )dy
∑ψ (n − m)h(m)
• Example function
– Delta function
© Yao Wang, 1/2003
Frequency Domain Analysis
6
Discrete Space Fourier Transform (DSFT)
• Forward transform Ψd (f ) = ∑ψ (n) exp(− j 2πf T n)
•
Frequency response of the HVS
– – – – Temporal frequency response and flicker Spatial frequency response Spatio-temporal response Smooth pursuit eye movement
•
Frequency response of the HVS
– – – – Spatial frequency response Temporal frequency response and flicker Spatio-temporal response Smooth pursuit eye movement
Consider an object moving with speed (v x , v y ). Assume the image pattern at t = 0 is ψ 0 ( x, y ), the image pattern at time t is
ψ ( x, y , t ) = ψ 0 ( x − v x t , y − v y t )
The temporal frequency of the image of a moving object depends on motion as well as the spatial frequency of the object. Example: A plane with vertical bar pattern, moving vertically, causes no temporal change; But moving horizontally, it causes fastest temporal change
– Continuous space FT (CSFT) – Discrete space FT (DSFT) – Sampled space FT (SSFT)
•
Frequency domain characterization of video signals
– Spatial frequency – Temporal frequency – Temporal frequency caused by motion
– fx: cycles/horizontal unit distance – fy : cycles/vertical unit distance
• It can also be specified by magnitude and angle of change
fm = f x2 + f y2 ,θ = arctan( f y / f x )
α1φ1 (x) + α 2φ2 (x) = T (α1ψ 1 (x) + α 2ψ 2 (x)) T (ψ (x + x0 )) = φ (x + x 0 )
• LSI systems can be completely described by its impulse response
h(x) = T (δ (x) ) φ (x) = ψ (x) * h(x) ⇔ Φ c (f ) = Ψc (f ) H c (f )
n∈R K
Ψd (f ) is periodic in each dimension with period of 1 Fundamental period : I K = {f , f k ∈ (−1 / 2,1 / 2)}
• Inverse transform ψ (n) = ∫ Ψd (f ) exp( j 2πf T n)df
Rk
• Example function
– Delta function
© Yao Wang, 1/2003
Frequency Domain Analysis
3
Continuous Space Fourier Transform (CSFT)
• Forward transform
Ψc (f ) = ∫ψ ( x) exp(− j 2πf T x)dx
Outline
• Fourier transform over multidimensional space
– Continuous space FT (CSFT) – Discrete space FT (DSFT) – Sampled space FT (SSFT)
•
Frequency domain characterization of video signals
– Continuous space FT (CSFT) – Discrete space FT (DSFT) – Sampled space FT (SSFT)
•
Frequency domain characterization of video signals
– Spatial frequency – Temporal frequency – Temporal frequency caused by motion
f ∈I K
• Convolution theorem
ψ (n) * h(n) ⇔ Ψd (f ) H d (f ) ψ (n)h(n) ⇔ Ψd (f ) * H d (f )
© Yao Wang, 1/2003 Frequency Domain Analysis 7
Outline
• Fourier transform over multidimensional space
θ = 2 arctan(h / 2d )(radian) ≈ h/d (radian) =
fθ =
© Yao Wang, ቤተ መጻሕፍቲ ባይዱ/2003
θ
fs
=
π d
180 h
180 h (degree) π d
f s (cycle/degree)
Frequency Domain Analysis 11
Temporal Frequency
– Spatial frequency – Temporal frequency – Temporal frequency caused by motion
•
Frequency response of the HVS
– – – – Spatial frequency response Temporal frequency response and flicker Spatio-temporal response Smooth pursuit eye movement