1. ENEE631 Digital Image Processing (Spring'04) Signal Processing: From 1-D to 2-D (m-D) Spring ’04 Instructor: Min Wu ECE Department, Univ. of Maryland, College Park   www.ajconline.umd.edu (select ENEE631 S’04)   minwu@eng.umd.edu Based on ENEE631 Spring’04 Section 4

2. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [2] Overview Last Time: Point processing (zero-memory operations) –Gamma and gamma correction –Contrast stretching –Histogram and histogram equalization –Coping with contour artifacts: dithering Today –Cont’d on dithering and halftoning –A systematic examination of 2-D signals and systems Assignment-1: Due Friday 2/20 5pm (See course webpage) Jasmines Lab reserved for our class Friday 10am – 1pm Office Hour: change to Monday 4:30-6:30pm UMCP ENEE631 Slides (created by M.Wu © 2004)

3. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [3] Signals and Systems: 1-D to 2-D UMCP ENEE631 Slides (created by M.Wu © 2004)

4. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [4] 1-D and 2-D Sig. Proc: Similarity and Differences Many signal processing concepts can be extended from 1-D to 2-D to multi-dimension Major differences –The amount of data involved becomes several magnitude higher u Audio: CD quality 44.1K samples/second u Video: DVD quality 720*480 at 30 frames/sec => 10.4 M samples/sec –Less complete mathematic foundations for multi-dimension SP u E.g. A 1-D polynomial can be factored as a product of first-order polynomials (as we see and use in ZT, filter design, etc) u A general 2-D polynomial cannot always be factored as a product of lower-order polynomials –Notion of causality: u Causal processing a 2-D signal: from top to bottom & left to right u Causality often matters more for temporal signal than spatial signal

5. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [5] 1-D and 2-D Sig. Proc: Similarity and Differences Many signal processing concepts can be extended from 1-D to 2-D to multi-dimension Major differences –The amount of data involved becomes several magnitude higher u Audio: CD quality 44.1K samples/second u Video: DVD quality 720*480 at 30 frames/sec => 10.4 M samples/sec –Less complete mathematic foundations for multi-dimension SP u E.g. A 1-D polynomial can be factored as a product of first-order polynomials (as we see and use in ZT, filter design, etc) u A general 2-D polynomial cannot always be factored as a product of lower-order polynomials –Notion of causality: u Causal processing a 2-D signal: from top to bottom & left to right u Causality often matters more for temporal signal than spatial signal

6. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [6] 2-D Signals Continuously indexed vs. discretely indexed (sampled) 2-D Impulse (unit sample function) Any 2-D discrete function can be represented as linear combination of impulses

7. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [7] 2-D Signals Continuously indexed vs. discretely indexed (sampled) 2-D Impulse (unit sample function) Any 2-D discrete function can be represented as linear combination of impulses

8. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [8] 2-D Signals (cont’d) 2-D step function Extensions: line impulse and 1-sided step function

9. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [9] Periodicity x[n 1,n 2 ] is periodic with a period T 1 -by- T 2 if x[n 1,n 2 ] = x[n 1 +T 1, n 2 ] = x[n 1, n 2 +T 2 ] for  (n 1, n 2 ) Example: cos[  n 1 /2 +  n 2 ] is periodic with a period 4-by-2

10. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [10] Separability x[n 1,n 2 ] is called a separable signal if it can be expressed as x[n 1,n 2 ] = f[n 1 ]  g[n 2 ] –E.g. the impulse signal is separable:  [n 1,n 2 ] =  [n 1 ]   [n 2 ] Separable signals form a special class of multi- dimensional signals –Consider indices range: 0  n 1  N 1  1, 0  n 2  N 2  1 –A general 2-D signal x[n 1,n 2 ] has N 1  N 2 degrees of freedom –A separable signal has only N 1 + N 2  1 degrees of freedom

11. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [11] Separability x[n 1,n 2 ] is called a separable signal if it can be expressed as x[n 1,n 2 ] = f[n 1 ]  g[n 2 ] –E.g. the impulse signal is separable:  [n 1,n 2 ] =  [n 1 ]   [n 2 ] Separable signals form a special class of multi- dimensional signals –Consider indices range: 0  n 1  N 1  1, 0  n 2  N 2  1 –A general 2-D signal x[n 1,n 2 ] has N 1  N 2 degrees of freedom –A separable signal has only N 1 + N 2  1 degrees of freedom

12. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [12] 2-D System A 2-D system often refers to a system that maps a 2-D input signals to a 2-D output signal –Such as system can be represented by y[n 1,n 2 ] = H ( x[n 1,n 2 ] ) Linearity of a system H( ) : for all a, b, x1[ ], x2[ ] H (a  x 1 [n 1,n 2 ] + b  x 2 [n 1,n 2 ]) = a  H (x 1 [n 1,n 2 ]) + b  H (x 2 [n 1,n 2 ]) Impulse response h(m,n; m’,n’) = H (  [m-m’, n-n’] ) is the output at location (m,n) in response to a unit impulse at (m’,n’) => Point Spread Function (PSF): impulse response for system with positive inputs & outputs (such as intensity of light in imaging system) A linear sys can be characterized by its impulse response

13. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [13] 2-D System A 2-D system often refers to a system that maps a 2-D input signals to a 2-D output signal –Such as system can be represented by y[n 1,n 2 ] = H ( x[n 1,n 2 ] ) Linearity of a system H( ) : for all a, b, x1[ ], x2[ ] H (a  x 1 [n 1,n 2 ] + b  x 2 [n 1,n 2 ]) = a  H (x 1 [n 1,n 2 ]) + b  H (x 2 [n 1,n 2 ]) Impulse response h(m,n; m’,n’) = H (  [m-m’, n-n’] ) is the output at location (m,n) in response to a unit impulse at (m’,n’) => Point Spread Function (PSF): impulse response for system with positive inputs & outputs (such as intensity of light in imaging system) A linear sys can be characterized by its impulse response

14. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [14] Shift Invariance Shift invariance: If H (x[m, n]) = y[m,n], then H ( x[m - m 0, n - n 0 ] ) = y[m - m 0, n - n 0 ] Impulse response for Linear Shift-Invariant (LSI) System –A function of the two displacement index variable only: h(m,n; m’,n’) = h[ m-m’, n-n’] –i.e. the shape of the impluse response does not change as the input impulse move in the (m,n) plane I/O relation for a LSI system: –Equal to the convolution of the input with the impulse response

15. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [15] Shift Invariance Shift invariance: If H (x[m, n]) = y[m,n], then H ( x[m - m 0, n - n 0 ] ) = y[m - m 0, n - n 0 ] Impulse response for Linear Shift-Invariant (LSI) System –A function of the two displacement index variable only: h(m,n; m’,n’) = h[ m-m’, n-n’] –i.e. the shape of the impluse response does not change as the input impulse move in the (m,n) plane I/O relation for a LSI system: –Equal to the convolution of the input with the impulse response

16. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [16] 2-D Convolution 1.Rotate the impulse response array h( ,  ) around the original by 180 degree 2.Shift by (m, n) and overlay on the input array x(m’,n’) 3.Sum up the element-wise product of the above two arrays 4.The result is the output value at location (m, n) From Jain’s book Example 2.1

17. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [17] Summary of Today’s Lecture Dithering / Halftoning 2-D Signals and Systems 2-D Fourier Transform Next time –Image enhancement via spatial filtering Readings –Jain’s book 4.12, 2.1—2.6 Gonzalez’s book 4.1—4.2, 2.6, 5.5 UMCP ENEE631 Slides (created by M.Wu © 2004)

18. ENEE631 Digital Image Processing (Spring'04) Lec4 – Multi-dim Sig/Sys [18] Readings and Assignment Readings –Jain’s book 4.12, 2.1—2.6 Gonzalez’s book 4.1—4.2 Assignment-1 –See course webpage for handout and image files needed UMCP ENEE631 Slides (created by M.Wu © 2001,2004)