EECS 556 – Image Processing– W 09
2D CONTINUOUS‐SPACE SIGNALS/SYSTEMS
GSI: Gaurav Pandey
[email protected]
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System properties in terms of PSF
orthogonal representation of signals
2D Fourier series
2D Fourier transform: Properties of 2D FT ;New properties
OTF/MTF
2D‐FT Examples
rms bandwidth, rms time duration
time‐bandwidth product, gaussian example
Systems for 2D signals
Image processing
systems
S maps one function to another function
i.e., and image to another image
Linear Systems for 2D signals
• How to decompose a signal?
• Using the Dirac impulse and the sifting property:
Linear Systems for 2D signals
• input to a system S is an impulse centered at (x’, y’),
What’s the output? (point spread function) PSF
Impulse Response (point spread function) PSF
• g(x,y) expressed in terms of f(x,y) and PSF computed for all input coordinates (S fully characterized by its PSFs))
• Properties of S are also characterized in terms of conditions on PSFs
Properties of Linear systems
• Shift invariance: A linear system is shift‐
invariant iff
PSF does depend solely on the difference between input and output coordinates
Impulse Response (point spread function) PSF
Shift invariant
Separability
• Process x and y (rows and columns) of image independently
• An LSI imaging system is separable if and only if its PSF h(x, y) is separable: h(x, y) = h1(x) h2(y) for some 1D functions h1 and h2
two sets of 1D convolution operations
Resolution
• PSF
• Image is formed by two closely spaces impulse‐
like signals; • These two impulse can be no longer resolved
Example – 1D pinhole camera
f
f = focal length
c = center of the camera
Example – 1D pinhole camera
• What’s the PSF of this image capture system?
Source magnification
factor
Impulse at x’ Æ
Example – 1D pinhole camera
• What’s the output of the image capture system?
Superposition integral
Source magnification
factor
2D pinhole system
source magnification factor
Is this shift invariant? No! PSF does depend solely on the difference between input and output coordinates
2D pinhole system
Convolution Æ system is shift invariant
magnified, scaled and mirrored
version of the input image
Representation of Signals by Orthogonal Bases
• useful to represent signals by a linear combination of “simpler” signals
• convenient if we choose a set of signals that are orthogonal
Orthogonal signals
• Orthogonal vectors
= 0
u and v are N‐dim vectors
• n‐dimensional complex vectors
y’ denotes the Hermitian transpose of a vector (or matrix).
Orthogonal signals
Related to the concept of dot product is norm and distance
• norm of a vector: • distance: Orthogonal signals
• 2D continuous‐space images, which are functions defined on R2, rather than simple vectors in Rn or Cn
• Inner product, norm and distance are:
domain B in R2
Orthogonal signals
• 2D images are orthogonal if:
• A set of 2D signals is called orthogonal iff
Signal energy Kronecker delta function
• If Ek = 1 for all k, then we call the signals orthonormal
Orthogonal signals
• Once a orthonormal basis is defined:
Orthogonal signals ‐ example
• Harmonic sinusoid:
• Are these signals orthogonal in ?
Generalized Fourier Series
Goal: find orthogonal signals on the set B such that if g(x, y) has finite energy over B than:
Orthogonal series representation
Orthogonal basis
ck are called Fourier coefficients wrt set Generalized Fourier Series
Goal: find orthogonal signals on the set B such that if g(x, y) has finite energy over B than:
Generalized Fourier Series
Goal: find orthogonal signals on the set B such that if g(x, y) has finite energy over B than:
Orthogonal basis is complete if this representation is possible for every g(x,y) with finite energy over B
Generalized Fourier Series
In practice:
What’s ck?
is the coefficient that minimizes the approximation error:
for any N! (true because we use L2‐ for L1 doesn’t work)
Completeness revisited
• By choosing ck as above:
• Energy approximation Æ 0 iif
• This is a necessary and sufficient condition for a set of orthogonal functions to be complete
Parseval Theorem
• Relationship between energy of the signal and energy of basis signals
Examples of Orthogonal bases: Harr basis
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A set of 1D signals
Orthogonal
Complete on the interval (0,1) Example of Orthogonal bases: Harr basis
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Any finite‐energy signal can be approximated using harr basis as N is large enough (completeness)
Example of Orthogonal bases: Harmonic complex exponentials
Example of Orthogonal bases: Harmonic complex exponentials
Harmonic complex exponentials are important because they are eigenfunctions of LSI systems
Example of Orthogonal bases: Harmonic complex exponentials
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Fourier transform of h(x,y)
Example of Orthogonal bases: Harmonic complex exponentials
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Scaled version of the input signal
• Exponentials are eigenfunctions of LSI systems
• If we decompose g into complex exponentials, output of LSI is easily obtained by superposition
Fourier series
• If g(x, y) is periodic with period (TX, TY):
•Complete and orthogonal
•Integral on any rectangle of width Tx and Ty
Fourier series – properties
• Convergence: because of completeness
Fourier series – properties
• If g is separable:
and gX(x) has period TX and gY(y) has period TY
product of the 1D Fourier series coefficients
• Parseval’s theorem:
Power in the space domain equals the sum of the powers in each frequency component
Notice fourier basis is orthonormal (E=1)
Examples
• 2D “bed of nails”
Examples
• 2D “bed of nails”
Examples
2D Fourier Transform
• Inverse Fourier Transform 2D Fourier Transform
• Why it is important?
– Convolution in the space domain is a multiplication in the fourier domain
– Complex exponentials are eigenvalues of LSI systems
2D Fourier Transform
• Relation to the 1D case:
Properties
• Linearity
• Convolution
Properties
• Cross‐correlation:
complex conjugate product of the spectra of two signals
Properties
• Autocorrelation:
Power spectral density
(energy density spectrum)
Properties
• Shift
• Derivative
• DC value
• Scaling ?
Properties
• Duality
• Parseval
• If g is Hermitian symmetric, G is real
Properties
Unique to 2D Fourier transforms
• Rotation
• Separability
• Polar separability and circular symmetry
Some Fourier transforms
We need to introduce the concept of generalized Fourier transforms
I.e. transforms of functions that are not absolutely integrable
• Constant
• Impulse
• Complex exp
• Sinusoid
Courtesy of John M. Brayer Courtesy of John M. Brayer Some Fourier transforms
• Periodic Signals
Some Fourier transforms
Some Fourier transforms
Courtesy of John M. Brayer Courtesy of John M. Brayer Courtesy of John M. Brayer Next lecture
‐ Sampling
‐ Interpolation; ‐ 2D‐DSFT / 2D‐FT