Multi-Dimensional Signals EE23 Digital Signal Processing Lecture Our world is more complex than D Images: f(x,y) Videos: f(x,y,t) Dynamic 3F scenes: f(x,y,z,t) Medical Imaging 3D Video Computer Graphics We will focus on 2D Continuous-Time 2D functions δ(x,y): Impulse at x=0, y=0 δ(x) : Impulse line (vertical or horizontal?) (x,y) : 2D rect function cos(2π(fxx + fyy) - Spatial harmonic Circularly Symmetic: δ(r): Impulse ring (x,y): Pillbox u(r) (r r 0 ) 2 r o Spatial Frequency What is a spatial frequency? Complex Harmonic: e j( xx+ y y) = e j2 (f xx+f y y) Units (for example): x,y - cm fx,fy - /cm Ωx, Ωy - rad/cm
Spatial Frequency Vinyl Record Transforms a temporal signal to a spatial signal Spatial Frequency CD ROM encodes digital temporal signals to spatial signals http://offtosognefjord.tumblr.com What is the frequency? What is the frequency? (-,) sin(2 (f x x + f y y)) (-,) sin(2 (f x x + f y y)) (-,-) a) fx=2, fy=2 b) fx=, fy=0 (,-) c) fx = 4, fy=0 d) none of the above (-,-) (,-) 2 cycles for 2 cm fx= cm -
What is the frequency? What is the frequency? sin(2 (f x x + f y y)) (-,) or cos(2 (f x x + f y y)) (-,) cos(2 (f x x + f y y)) (-,-) (,-) (-,-) (,-) a) sin, fx=0, fy=2 c) cos, fx = 0, fy=2 a) fx=, fy=2 c) fx = 2, fy= b) cos, fx=0, fy=4 d) none of the above b) fx=4, fy=2 d) fx=2, fy=4 What is the frequency? What is the Temporal Frequency? cos(2 (f x x + f y y)) Vinyl rotates at Hz (-,) (-,) (-,-) (,-) (-,-) (,-) a) fx=, fy=2 c) fx = 2, fy= a) cos(2π8t) c) cos(2π4t) b) fx=4, fy=2 d) fx=2, fy=4 b) cos(2π8t 2 ) d) cos(2π4t 2 )
What is the Temporal Frequency? (-,) Vinyl rotates at Hz Challenge: What is the password? Aliasing! 2sec (-,-) a) cos(2π00t) b) cos(2π00t 2 ) (,-) c) cos(2π40t) d) none of the answers 2D Fourier Transform Z Z F (f x,f y )= f(x, y)e j2 (f xx+f y y) dxdy
sinc(ax)sinc(by) 2D DTFT F (! x,! y )= X X f[n x,n y ]e j(! xn x +! y n y ) n x = n y = apple! x,! y apple sinc(ax)sinc(by) F (apple x, apple y )= I prefer 2nd X X n x = n y = f[n x,n y ]e j2 (apple xn x +apple y n y ) 0.5 apple apple x, apple y apple 0.5 Massaging the DTFT leads to separable transforms in each axis
Separability of 2D DTFT white board... 2D - DFT Similarly to D: Forward: F [k x,k y ]= Inverse: NX MX n x =0 n y =0 f[n x,n y ]e j2 (n xk x /N +n y k y /M ) apple x = k x /N, apple y = k y /M f[n x,n y ]= NM NX MX k x =0 k y =0 F [k x,k y ]e +j2 (n xk x /N +n y k y /M ) 2D - DFT F [k x,k y ]= M NX MX n x =0 n y =0 f[n x,n y ]e j2 (n xk x /N +n y k y /M ) f[n x,n y ] F [k x,k y ] (0,) Properties of 2D DFT Circular Convolution f[n x,n y ] h[n x,n y ]=F [k x,k y ]H[k x,k y ] ny ky Circular shift (0,0) N nx (0,0) kx (,0) Need to fftshift in 2D to get it to look like DTFT. f[(n x m x ) N, (n y m y ) M ]=e j2 (k xm x /N +k y m y /M ) F [k x,k y ]
. Projection w Many Projections - Tomography (A fr:x,j) J(f-xco,e-j>l'ne)rfx:dj / \:. J projection x-ray http://www.youtube.com/watch?v=4gklqhm9ay&feature=related
/ \:. J. Radon Transform w Radon Transform: Sinogram (A fr:x,j) J(f-xco,e-j>l'ne)rfx:dj / \:. J Computed Tomography Projection Slice Theorem (Bracewell) Sinogram cross-section,f F0<. (), ey'" fy{) FBP M. lrd ft spatial frequency x-ray source central-section/ projection slice theorem (Bracewell 56)?voo P: o-tu : Ar fcx,j)dj F( /.;_,o) =- JJ G "' o ='l k, or6f 9 /J-fr{ t,:x-.. f; jj d-xjj =-
Projection Slice Theorem (Bracewell) Discrete Reconstruction?voo P : o-tu : A r G "' o ='l k, or6f 9.. fcx,j)dj F( /.;_,o) =- J J /J-fr{ t,:x-.. f; jj d-xjj =- -\A - (.)0..... ' Discretization /)L{c(2EJ7.7:.[.'. - -- -