2D DSP Basics: 2D Systems

Transcription

1 - Digital Image Processig ad Compressio D DSP Basics: D Systems

2 D Systems T[ ] y = T [ ] Liearity Additivity: If T y = T [ ] The + T y = y + y Homogeeity: If The T y = T [ ] a T y = ay = at [ ] Liearity superpositio priciple holds i a i i T a i y i where y i = T [ i ] i

3 D Systems Shift ivariace Shift i the iput results i a correspodig shift i the output Ep. If T y The -s -s T y -s -s

4 D Systems Why LSI Systems? Most frequetly used Easy to desig ad aalyze Need to simplify Mathematical tractable ad rich theory BUT Superpositio ot always useful for images e.g. occlusio

5 D Systems Impulse respose What is it? δ T h Geeral system Not SI δ -s -s T h ss Shift Ivariat SI system δ -s -s T h -s -s

6 Liear ad Shift Ivariat LSI Systems Impulse respose h uiquely characterizes system for iput ca compute output usig h sice: = δ = = = y = T = T δ = = [ ] [ ] = h = = = ** h D Covolutio

7 D Systems h y = h D filter LSI Ep. Let be a oisy image choose impulse respose h of a low-pass filter to remove high frequecy oise; y : ehaced or restored image LSI System is characterized by: impulse respose h i time/space domai frequecy respose Hω ω i frequecy domai where H jω jω ω ω = DTFT{ h } = h e e = =

8 D Systems Two mai possible implemetatios of LSI systems: Time/space domai implemetatio: D covolutio Use actual iput image values ad impulse respose h to get y h y = h Frequecy trasform domai implemetatio Use DTFT Xω ω of ad DTFT Hω ω of h to get y Xω ω Hω ω Yω ω = Xω ω Hω ω y = DTFT - {Yω ω }

9 D Systems Uique oe-to-oe relatio mappig betwee time/space domai ad frequecy domai: DTFT h Hω ωω LSI systems fully defied by h or Hω ω

10 D DTFT D DTFT Two-Dimesioal Discrete- Time Fourier Trasform D DTFT: ω ω X { } = = = = j j e e DTFT X ω ω ω ω Note: D Covolutio Property ω ω ω ω H X h

11 D DTFT D DTFT periodic with period π i ω ad ω : X ω ω = X ω + π ω = X ω ω + π ω π π π ω π Iverse D DTFT evaluated over oe period: 4 π π j ω + ω = X ω ω e dωdω 4π π π

12 D Systems Usually filters are implemeted i practice i the time/space domai sice: DTFT is cotiuous i the variables ω ad ω very dese grid required to represet cotiuous domai If is fiite-etet of size N N DTFT Xω ω ca be sampled i ω ad ω resultig i the N N DFT XK K : πk N XK K = Xω ω with ω = ad ω = πk N A sparse fiite grid ca be used whe of fiite size as for images DFT but size of eeded grid depeds o iput image size DTFT ad DFT are importat for trasform domai based methods such as trasform codig ad fast covolutios overlap-add ad overlap-save covolutio methods.

13 D Covolutio Sum y Notatio: y = h h = = = Eample L h M L M

14 D Covolutio Sum h- - correspods to a 80 o rotated versio of h reflect flip aroud ais ad the ais or vice-versa h- - h - - is a 80 o rotated ad shifted versio of h Sample of h- - atorigi ow is at

15 D Covolutio Sum Covolutio: Flip slide to multiply ad sum Repeat for other desired values of y L +M - L +M - h- - h - - : M M h : L L Covolutio Result: y : L + M - L + M -

16 D Covolutio Sum Summary of D Covolutio steps Graphical Method: Reflect h about both ad aes Traslate so that sample h00 lies at the poit Multiply the sequeces ad h - - poit-by-poit Sum o-zero samples of obtaied product sequece to compute the output sample y Vary ad repeat from Step Direct evaluatio of the D covolutio sum ca be sometimes easier tha usig the graphical method. Covolutio sum is directly used or is used i combiatio with the graphical method whe both or oe of the sigals caot be easily represeted graphically.

17 D Covolutio Sum Covolutio with a D impulse: δ - - =?

18 D Covolutio Sum Covolutio with a D Impulse: δ - - = - - D Covolutio Eample: All oes All oes

19 D Covolutio Sum Properties of D covolutio: h = h commutative h g = h g associative h g y h g y commutative h + g = h + g distributive g h y h g h + g y h + g y

20 Separable Systems A D LSI system is separable if its impulse respose is a separable sequece h = h h If LSI system is separable we ca implemet it more efficietly y = N N = 0 = 0 h If h is NN we eed N multiplicatios ad N - adds for each computed output sample y for a geeral D LSI system.

21 Separable Systems Separable Systems If h separable: = = = 0 0 N N h h y N N = = = 0 0 g h h D covolutio i directio = = 0 N g h D covolutio i directio The 0 h g N = = = N h g y The =0 h g y

22 Separable Systems D Covolutio Algorithm:. To get g covolve each colum of with h. To get y covolve each row of the result from Step i.e. of g with h Algorithm ca be summarized as follows: D covolutio o rows the D covolutio o colums or Vice-Versa Note: Sigal does ot eed to be separable oly the system

23 Separable Systems Eample Cosider lowpass filterig of images Lowpass filter with rectagular support: ω π Hω ω H ω H ω π π ω = π π ω π π π Istead of a D filter we eed to desig oly D filters Hω ω = H ω H ω

24 Separable systems Eample Cosider lowpass filterig of images No-separable lowpass filter with circular support: ω π Hω ω π π ω π