Advanced Training Course on FPGA Design and VHDL for Hardware Simulation and Synthesis. 26 October - 20 November, 2009

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1 Advanced Training Course on FPGA Design and VHDL for Hardware Simulation and Synthesis 26 October - 20 November, 2009 Introduction to two-dimensional digital signal processing Fabio Mammano University of Padua Faculty of Medicine and Surgery Department of Physics via Marzolo Padova Italy

2 Introduction to two-dimensional digital signal processing

3 Fourier transform definition The Fourier transform of a complexvalued function g(x,y) of two independent real variables x and y is defined by, (, ) exp 2 ( ) F g x y g x y j f x f y dxdy The transform so defined is itself a complexvalued function of two independent real variables f and f, generally referred to as spatial frequencies.. For the above integral to exist, function g: 1. must be absolutely integrable over the infinite (x,( y) ) plane 2. must have only a finite number of discontinuities and a finite number of maxima and minima in any finite rectangle 3. must have no infinite discontinuities. Similarly, the inverse Fourier transform of a complexvalued function G(f,,f ) of two independent real variables f and f is defined as 1,, exp 2 ( ) F G f f G f f j f x f y df df Fourier integral theorem. At each point of continuity of g, 1 F F g x y g x y (, ) (, ) (at each point of discontinuity of g,, the result is the angular average of the value of g,, in a small neighborhood of that point). Lecture 1 2

4 The Dirac delta function It is common to represent an idealized point source of light as a twodimensional Dirac Delta function. This is a quantity (actually, a distribution) ) that is zero everywhere except at the origin, where it goes to infinity in a manner so as to encompass a unit volume, that is The defining characteristic of the delta function is its so called sifting property: ( xy, ) x y 0 0 otherwise ( x, y) dxdy 1 gx ( ', y') ( xx', y y') dxdy ' ' gxy (, ) at any point (x, y) of continuity of g. The height of the arrow representing the delta function corresponds to the volume under the function. Lecture 1 3

5 Having an infinite discontinuity, the delta function fails to satisfy condition 3 above for the existence of the Fourier transform. However, it is possible to envisage the delta function as the convergence limit of a sequence of Gaussian pulses such as ( xy, ) lim N x y Ne N Each member function of this defining sequence does satisfy the existence requirements, and each has Fourier transform given by Accordingly, the generalized transform of is found to be F N e e 2 2 f ( f ) 2 N ( x y ) N 2 2 ( f f ) 2 N F ( x, y) lime 1 N As the Gaussian pulses become narrower and taller, their transforms rms grow broader until the pulse is infinitesimal in width, and its transform is infinite in extent, i.e. a constant. Lecture 1 4

6 Let s write explicitly the Fourier integral theorem as Lecture 1 5 g x y F F g x y 1 (, ) (, ) g(, x y) df df expj2 fx fy dx' dy' gx (', y')expj2 fx' fy ' But we also have therefore dx' dy' g( x', y') df df exp j2 f x x' f y y' g( x, y) dx' dy' gx ( ', y') ( xx', yy') which is known the integral representation of the delta function. ( x x ', y y ') exp j2 f x x ' f y y ' df df Hence, we can also write 1 ( xy, ) exp j2 ( f x fy) dfdf F 1

7 Similarly, starting from we conclude that and therefore that G( f, f ) FF G( f, f ) 1 ' ' ' ' ' ' ' ' ' ' ( f f, f f ) exp j2 x f f y f f dxdy Hence, we can also write G( f, f ) df df G( f, f ) dxdy exp j2 x f f y f f ( f, f ) exp j2 fx f y dxdy F 1 Finally, we may use the sifting property to evaluate F x a y b j f x f y x a y bdxdy j f a f b (, ) exp 2 ( ) (, ) exp 2 ( ) Therefore, if the delta spike is shifted off the origin, its transform will change phase but not amplitude, which remains equal to one. Lecture 1 6

8 Spatial frequencies Fg( x,0) g( x,0) F (Fourier transform) f 1 f 1 1 x o f 1 x o x For a transparency resembling a sine target with spatial period x o, there would ideally be only three spots (delta functions) on the transform plane, these being the zero-frequency central peak and the first order or fundamental f 1 = 1 on either side of the center. The zero-frequency component with arises because a photographic slide at points of ideal opacity may produce g = 0, but cannot provide negative values. Thus the mean value (DC-term) is positive. Lecture 1 7

9 In general, we may regard the two- dimensional Fourier transform as a decomposition of a function g(x,y) into a linear combination, with coefficients, G(f, f ),, of complex exponential functions of the form For any particular pair of spatial frequencies, the phase of the exponential is zero or an integer multiple of 2 radians provide that f x f y n, n 0, 1, 2, 3, exp j2 ( f x f y) i.e., for any integer n,, the phase is zero or an integer multiple of 2 along the straight lines in the x-y plane given by: y f f x n f Lines of zero phase for the function tan These lines are separated by a distance L (spatial period) given by 1 L 2 2 f f Lecture 1 8 f f j2 ( fx fy e ). and the normals to these lines form an angle with the x axis such that The corresponding (radial) spatial frequency is w L f f

10 Examples The function g(x,y) ) = cos[2 (2x+3 +3y)] on top, presents with f = 2 cycles per unit distance in x and f = 3 cycles per unit distance in y.. By contrast, g(x,y) ) = cos[2 (3x+2 +2y)] (not shown) has 3 cycles per unit distance in x and 2 cycles per unit distance in y. But along their individual directions, atan f f their (radial) spatial frequencies are the same Lecture 1 9

11 The aperture (pupil) function g(x,y) is defined as: y gxy (, ) 1 0 at points in the domain D otherwise D Thus we may write G( f, f ) F g( x, y) x g( x, y) exp j2 ( f x f y) dxdy For a rectangular aperture,, we have exp j2 ( f x f y) df df D y b/2 a/2 G( f, f ) exp j2 f y dy exp j2 f x dx b/2 a/2 b x where a and b are positive constants. a Lecture 1 10

12 The integral in x is where thus a /2 a /2 a 1 expj2 f x dx exp j d 2 f 2 f a 2 f a a a/2 ja ja 1 j 2 a j e e a expj2 f x dx e e a a/2 a sin fa j f f j f a Defining rect( x) 1 1 x 2 0 otherwise sin x sinc( x) x Lecture 1 11

13 we may rewrite the previous result as a /2 a /2 Consequently b /2 b /2 j fx dx x a j fx dxa fa exp 2 rect exp 2 sinc j f y dy y b j f y dy b fb exp 2 rect exp 2 sinc so that, finally rect( )rect( ) sinc sinc F xa y b ab f a f b b y x Fg( x, y) f f a Lecture 1 12

14 To represent the pupil function for a circular aperture we can exploit the radially symmetrical cylinder (or circle) ) function, which is defined as cyl r 1 r 1 0 otherwise 2 2 cyl x y a Hence, for a circular pupil with radius a,, we write gx, y cyl x y 2 2 a G w and its Fourier transform is the radially symmetric function 2 1 a G w where the radial frequency w is given by w f f 2 J ( aw) aw 2 2 and J 1 is the Bessel function of order 1 (of the first kind). Lecture 1 13

15 J ( x) 0 J ( x) 1 J ( x) 2 x Bessel functions Lecture 1 14

16 gxy (, ) 1 if ( xy, ) belongs to 0 otherwise y D Fg( x, y) D x y D x Apertures (left) and corresponding two-dimensional Fourier transforms amplitudes, color coded according to the color scale bars. Lecture 1 15

17 gxy (, ) Fg( x, y) Left: a famous picture. Right: corresponding two-dimensional Fourier transform. The bright narrow central cross arises from the sharp boundary edges of the picture. Lecture 1 16

18 Images (top) and corresponding two-dimensional Fourier transforms (bottom). Lecture 1 17

19 Fourier transform theorems Linearity theorem. The transform of a weighted sum of two (or more) functions g and h is the identically weighted sum of their individual transform, that is F g h F g F h Proof. This theorem follows directly from the linearity of the integrals i that define the Fourier transform. Shift theorem. Translation in the space domain introduces a linear phase shift in the frequency domain, that is Proof: (, ) (, ) exp 2 ( ) F g x a y b F g x y j f a f b (, ) exp 2 ( ) gx ( ay, b) exp j2 ( f x fy) dxdy g( x ', y') exp j2 f ( x' a) f ( y' b) dx' dy ' F g x y j f a f b Lecture 1 18

20 The Parseval theorem. Proof: 2 2 G( f, f ) F g( x, y) g( x, y) dxdy G( f, f ) df df 2 g( x, y) dxdy g( x, y) g *( x, y) dxdy dxdy ddg(, )expj2 ( x y) dxdy ddg*(, )expj2 ( x y) dxdy ddg(, ) dd G*(, ) expj2 x( ) y( ) dxdy ddg(, ) dd G*(, ) (, ) dd G(, ) Lecture

21 The convolution c of two functions, g and h,, is the result of a commutative operation, denoted by the symbol,, such that cxy (, ) gx ( ', y') hx ( x', y y') dxdy ' ' Convolution theorem. The Fourier transform of the convolution is the product of the individual Fourier transforms, that is Proof: F g h F g F h Fg( x', y') h( xx', y y') dx' dy' gx ( ', y') F hx ( x', y y') dxdy ' ' g( x ', y ') exp j2 ( f x ' f y ') dx ' dy ' F h( x, y) F g F h Lecture 1 20

22 Graphical representation of the 2D convolution process gxy (, ) hxy (, ) y y x x y y hxy (, ) h ( x, y) y gxy (, ) h( x, y) x x x Lecture 1 21

23 Table 1 Lecture 1 22

24 Table 2 Lecture 1 23

25 Proof of the Fourier integral theorem Let F g( x, y) G( f, f ) g( x ', y') exp j2 ( f x ' f y') dx ' dy ' and define g ( x, y) G( f, f )exp j2 ( f x f y) df df R A R where A R is a circle of radius R,, centered at the origin of the frequency plane. Then g ( x, y) df df exp j2 ( f x f y) R A R dx ' dy' g( x ', y') exp j2 ( f x ' f y') Exchanging the order of integration, we may write Lecture 1 24 gr( x, y) dx' dy' g( x', y') dfdf exp j2 f xx' f y y' A R

26 Then, defining r j w we, w f f 2 2 j re, r x x ' y y ' 2 2 the integral over the spatial frequencies becomes A R R 2 rw df df exp j2 f x x ' f y y ' d dw w exp j 0 0 R 2 1 dw w dexp j rwcos R 0 0 Assume that (x,y) is a point of continuity of g.. Then J 2 rr r lim R J1 2 rr gr ( x, y) dx' dy' g( x', y') lim R R r Lecture 1 25

27 Note that for sufficiently large R,, the circularly symmetric function hr () J 1 2 rr vanishes practically everywhere, except in the proximity of r = 0. Therefore it is possible to envisage the delta function also as the convergence limit of the sequence hence J1 2 rr xx', y y' lim R R r lim g R ( x, y) dx ' dy' g( x ', y ') xx', y y' g( x, y) R But, from the definition of g R, we also have r 1 g ( x, y) G( f, f )expj2 ( f x fy) df df F G( f, f ) lim R R therefore (, ) (, ) 1 F F g x y g x y Lecture 1 26

28 Assume now a point of discontinuity of g.. This point can be located anywhere, for instance at the origin, so that J1 2 rr gr (0,0) dxdy g( x, y) R r where r x y 2 2 Again, for sufficiently large R,, the quantity in square brackets is significantly different from zero only in a small area around the origin. In this t small region, g depends approximately on the angle alone, thus J 2 1 gr(0,0) godr rdr r rr Where g o represents the dependence of g from near the origin. Since 1 1 J 2rRRdr J d Lecture 1 27

29 we have shown that 2 1 limgr(0,0) god R 2 i.e. that at each point of discontinuity of g, the two successive e transformations F 1 0 F g( x, y) produce, as a results, the angular average of g in a small neighborhood of that point. Lecture 1 28