1. Computer Vision – 2D Signal & System Hanyang University Jong-Il Park

2. Department of Computer Science and Engineering, Hanyang University 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. Department of Computer Science and Engineering, Hanyang University 2-D delta function  Dirac :  Property  Scaling :  Kronecker delta :  Property Delta function

4. Department of Computer Science and Engineering, Hanyang University Some special signals(or functions) Special signals

5. Department of Computer Science and Engineering, Hanyang University Linear and shift invariant systems Linearity Definition of impulse response Output of linear systems by superposition impulse response, unit sample response, point spread function(PSF)

6. Department of Computer Science and Engineering, Hanyang University Shift invariance Output of LSI(linear shift invariant) systems definition of shift invariance by superposition of linearity by definition of impulse response by shift invariance (2-D convolution) Shift invariance

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

8. Department of Computer Science and Engineering, Hanyang University Stability  Definition : bounded input, bounded output  Stable LSI systems(necessary and sufficient condition) Stability

9. Department of Computer Science and Engineering, Hanyang University The Fourier transform Definition  1-D Fourier transform  2-D Fourier transform

10. Department of Computer Science and Engineering, Hanyang University Properties  f(t)  F(  ) ;  = angular frequency  f(x,y)  F(u,v) ; u,v = spatial frequencies that represent the luminance change with respect to spatial distance Frequency domain

11. Department of Computer Science and Engineering, Hanyang University  Uniqueness  and are unique with respect to one another  Separability  Convolution theorem Properties of Fourier transform

12. Department of Computer Science and Engineering, Hanyang University  Inner product preservation  Hankel transform : polar coordinate form of FT Setting h=f, Parseval energy conservation formula where

13. Department of Computer Science and Engineering, Hanyang University 1-D case Fourier series

14. Department of Computer Science and Engineering, Hanyang University 2D Fourier series 2-D case  is periodic : period = 1  Sufficient condition for existence of

15. Department of Computer Science and Engineering, Hanyang University original 256x256 lena normalized spectrum (log-scale) Eg. 2D Fourier transform

16. Department of Computer Science and Engineering, Hanyang University

17. Department of Computer Science and Engineering, Hanyang University Optical and modulation transfer functions  Optical transfer function(OTF)  Normalized frequency response  Modulation transfer function(MTF)  Magnitude of the OTF OTF & MTF

18. Department of Computer Science and Engineering, Hanyang University Readings Gonzalez and Woods, Digital Image Processing, 3 rd ed.  Sect. 2.4. Sampling & Quantization  Sect. 3.4. Fundamentals of Spatial Filtering