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Two-Dimensional Signals and Systems

Two-Dimensional Signals and Systems. Fundamental of Digital Image Processing ANIL K.JAIN Chap.2. Notation and definitions. One-dimensional signal Continuous signal : Sampled signal : Two-dimensional signal Continuous signal : Sampled signal :

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Two-Dimensional Signals and Systems

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  1. Two-Dimensional Signals and Systems Fundamental of Digital Image Processing ANIL K.JAINChap.2

  2. Notation and definitions • One-dimensional signal • Continuous signal : • Sampled signal : • Two-dimensional signal • Continuous signal : • Sampled signal : • i, j, k, l, m, n, … are usually used to specify integer indices • Separable form :

  3. Notation and definitions • 2-D delta function • Dirac : • Property • Scaling : • Kronecker delta : • Property

  4. Some special signals(or functions)

  5. Linear and shift invariant systems • Linearity • Output of linear systems impulse response, unit sample response, point spread function(PSF) by superposition • Definition of impulse response

  6. Shift invariance definition of shift invariance Output of LSI(linear shift invariant) systems (2-D convolution) by superposition of linearity by definition of impulse response by shift invariance

  7. n n n n m m m m B C A A B C 2-D convolution rotate by 180 degree and shift by (m,n) (b) output at location (m,n) is the sum of product of quantities in the area of overlap (a) impulse response (ex) n m

  8. Stability • Definition : bounded input, bounded output • Stable LSI systems(necessary and sufficient condition)

  9. The Fourier transform • Definition • 1-D Fourier transform • 2-D Fourier transform

  10. Properties • Spatial frequencies : u,v (reciprocals of x and y) • f(t)  F(w) ; w = frequency • f(x,y)  F(u,v) ; u,v = • representing luminance change with respect to spatial distance • spatial frequencies that represent the luminance change with respect to spatial distance

  11. Uniqueness • and are unique with respect to one another • Separarability • Eigenfunction of a linear shift invariant system Performing the change of variables frequency response property of eignfunction

  12. Convolution theorem • Inner product preservation • Hankel transform : polar coordinate form of FT Setting h=f, Parseval energy conservation formula where

  13. 1-D case • 2-D case • is periodic : period = 1 • Sufficient condition for existence of Fourier transform of sequences(Table2.4)

  14. original 256x256 lena normalized spectrum (log-scale)

  15. The Z-transform(or Laurent series) • Definition • ROC(region of convergence) z-plane 1

  16. Example By definition For convergence of

  17. Optical and modulation transfer functions • Optical transfer function(OTF) • Normalized frequency response • Modulation transfer function(MTF) • Magnitude of the OTF

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