Lect. 7 Discrete Fourier Transforms Discrete Fourier Transform: discrete representation of the Fourier Integral

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1 Biomedicl signl nd imge pocessing (Couse ) Lect. 7 Discete Fouie Tnsfoms Discete Fouie Tnsfom: discete epesenttion of the Fouie Integl ( f) ( x) ( i2 πfx) dx ( x) sin c[ π ( x x) / x] ; ( f ) f sin c[ π ( f f )/ f ] exp ; DFT: exp i2π IDFT: exp i2π Bsic popeties: cyclicity, symmeties, shift nd conolution theoems: ( ) mod ; ( ) mod ; { ± } { ± s} ; { ± } { ± } ; ( p) exp( i2πp/ ); nb( n) + mod mod β 2-D DFT: M ls +, s, l exp i2π. M l M, l M ls +, s exp i2π M s M 2-D cyclicity nd symmeties. Shifted Discete Fouie Tnsfoms: If smpling positions in signl nd Fouie domin e shifted by u nd with espect to oigin of signl nd its spectum co-odinte systems, espectiely, discete epesenttion of the Fouie integl is: exp( i2π ( + u)( + ) / ) exp i 2πu / gies ise to SDFT() nd ISDFT(): Remoing ielent constnt ( ) Impotnt specil cses of SDFT e: Discete cosine tnsfom: DCT: SDFT ( ) exp exp ( i2π / ) exp( i2π ( + u) / ) ( i2πu / ) exp( i2π ( + ) / ) ( + / 2) cos ; { } / 2, 2 : π 2 2 IDCT: ( ) ( + ) + / 2 ISDFT / 2, 2 : 2 cos π 2 Discete Sine Tnsfom: DST: ({ }) ( )( ) + + SDFT, 2 : sin π ; { 2 } 2 + ( )( ) IDST: + + sin π 2 + Enegy compction popeties of DFT, DCT nd DST: they edistibute signl enegy into smll fction of tnsfom coefficients. DCT nd DST emoe signl discontinuities t boundies nd hence compct enegy moe efficiently then DFT. DFT, DCT nd DST nd Khunen-Loew Tnsfom Poblems. Descibe distinctions between integl Fouie Tnsfom nd DFT nd bsic popeties of DFT. 2. Explin Shifted Discete Fouie Tnsfom, DCT nd DST nd thei eltion to DFT Home wo: Inestigte nd explin DFT spect of sinusoidl signls of diffeent fequencies

2 - een numbe - odd numbe, 2 - een numbes, 2 - odd numbes 2 H /2; H2 2 V C, 2 /2 /2; 2 /2 V C2 -odd, 2 - een 2 - een, 2 - odd numbe /2; 2, 2 /2 Types of the DFT spect symmety fo one nd two-dimensionl el-lued signls

3 Centeing DFT spectum 2 H /2; H2 V C, 2 /2 /2; 2 /2 V /2; 2 /2, 2 /2 C C2 V /2; H2 2 H C V C2

4 2 + s i n ( 2 * p i * / ( 2 8 / 6 ) ) D F T ( + s i n ( 2 * p i * / ( 2 8 / 6 ) ) ) C e n t e e d D F T s i n ( 2 * p i * * 5.5 / 2 8 ) D F T ( + s i n ( 2 * p i * * 5.5 /( 2 8 ) ) ) C e n te e d D F T DFT spect of sinusoidl signls with intege nd nonintege numbe of peiods (x) u x x ( f) Shifted DFT: smpling digm of continuous signl nd its Fouie spectum f f

5 Chest X-Ry MRI imge Angiogm DFT spectum.^.2 DFT spectum.^.2 DFT spectum.^.2 DCT spectum.^.2 DCT spectum.^.2 DCT spectum.^.2 HAAR spectum.^.2 HAAR spectum.^.2 HAAR spectum.^.2 Imges nd thei DFT, DCT nd HAAR spect

6 Displcsp.m:Input imge Selected fgment DFT spectum of the fgment DCT spectum of the fgment Wlsh spectum of the fgment H spectum of the fgment Cumsum(Spectum),DFT:(.9);DCT-c(.46);Wlsh-g(.8);H-b(.2) DFT-ed DCT-cin Wlshgeen H-blue 2 3 Index of odeed spectl coefficients Compison of enegy compction popeties of DFT, DCT, Wlsh nd H Tnsfoms

7 T e s t s i g n l: s i n ( 2 * p i *. 2 / ( 5 2 * 2 ) ) L o c l D F T s p e c t u m ; W i n d o w L o c l D C T s p e c t u m ; W i n d o w S i g n l s m p le s L c s p te s t:t e s t s ig n l L o c l D F T s p e c tu m ; W in d o w Locl D C T spectum ; W indow Time-fequency DFT nd DCT epesenttion of chip signl (uppe) nd electocdiogm (lowe)