1-D MATH REVIEW CONTINUOUS 1-D FUNCTIONS. Kronecker delta function and its relatives. x 0 = 0

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1 -D MATH REVIEW CONTINUOUS -D FUNCTIONS Kronecker delta function and its relatives delta function δ ( 0 ) 0 = 0 NOTE: The delta function s amplitude is infinite and its area is. The amplitude is shown as for convenience in plots. 0 Write the equation that defines the area of a delta function as. Review the definition of delta function in terms of the limit of conventional functions, such as the rectangle function ECE/OPTI533 Digital Image Processing class notes 0 Dr. Robert A. Schowengerdt 2003

2 -D MATH REVIEW even delta pair δδ b = b [ δ ( + b ) + δ( 0 0 b )] 0 = b b - b b 0 - b 0 + b odd delta pair δδ b = b [ δ ( + b ) δ( 0 0 b )] 0 = b b b 0 + b - b 0 - b - b - b ECE/OPTI533 Digital Image Processing class notes Dr. Robert A. Schowengerdt 2003

3 -D MATH REVIEW comb (shah) comb b = b δ 0 n = ( nb ) 0 = 0-2b - b b 0 b 2b b 0-2b 0 -b b 0 +2b Even delta pair, odd delta pair and comb have amplitude of b ECE/OPTI533 Digital Image Processing class notes 2 Dr. Robert A. Schowengerdt 2003

4 -D MATH REVIEW Use of the δ function sifting f ( α )δ( α ) d α = f( 0 0 ) = constant NOTE: Sifting is a convolution, evaluated for a particular shift Finds the value of a function at a specific value of the independent variable (similar to a lookup table) ECE/OPTI533 Digital Image Processing class notes 3 Dr. Robert A. Schowengerdt 2003

5 -D MATH REVIEW sampling f( )δ 0 ( ) = f 0 ( )δ( 0 ) f() = f( 0 ) NOTE: Sampling is a mulitplication Output is a delta function, with area determined by the value of the function at the specified value of the independent variable. uniform sampling f b b ( )comb 0 = f nb 0 n = ( + )δ( 0 nb ) b 0 -b b 0 +2b b 0 -b b 0 +2b NOTE: Must divide comb function by b to retain amplitude of f(). NOTE: f() modulates the comb function. ECE/OPTI533 Digital Image Processing class notes 4 Dr. Robert A. Schowengerdt 2003

6 -D MATH REVIEW shifting g ( ) = f( ) δ ( 0 ) = f ( α )δ( 0 α ) d α = f() f 0 = g() ( ) 0 0 replicating f() g ( ) = f( ) comb b b g()... = b 0 -b b 0-2b 0 -b b NOTE: Must divide by b to retain amplitude of f() ECE/OPTI533 Digital Image Processing class notes 5 Dr. Robert A. Schowengerdt 2003

7 ECE/OPTI533 Digital Image Processing class notes 6 Dr. Robert A. Schowengerdt 2003 What is the value of b in the above graph? function 0.2 as the rect tri -- b = 0 b b b < f() 0.4 is twice as wide 0.6 the tri function For a given b, 0.8 triangle rect tri b < 2.2 rect -- b = 2 b = 2 0 b > 2 rectangle (square pulse) -D MATH REVIEW

8 ECE/OPTI533 Digital Image Processing class notes 7 Dr. Robert A. Schowengerdt 2003 What is the value of b in the above graph? f() sinc-squared sinc 2 ( b) 0.8 sinc sinc squared sinc sinc( b ) = sin[ π b] π b.2 -D MATH REVIEW

9 ECE/OPTI533 Digital Image Processing class notes 8 Dr. Robert A. Schowengerdt 2003 What is the value of b in the above graph? f() gaus.2 gaus(sian) gaus b ( ) = e π b ( )2 -D MATH REVIEW

10 ECE/OPTI533 Digital Image Processing class notes 9 Dr. Robert A. Schowengerdt 2003 What is the value of b in the above graph? f() 0 sin( 2π b) = e j2π ( b) e 2π j ( b) j 0.4 sine cos( 2π b) = cos sin e j2π b ( ) e j + 2 2π ( b) cosine -D MATH REVIEW

11 -D MATH REVIEW CONVOLUTION (-D) Why is it important? Describes the effect of a Linear Shift Invariant (LSI) system on input signals input signal f() system (operator L) output signal g() L is the system operator Description of general system g ( ) = L[ f( ) ] For an LSI system, L is a convolution g ( ) = f( ) h ( ) = f ( α )h ( α ) d α ECE/OPTI533 Digital Image Processing class notes 20 Dr. Robert A. Schowengerdt 2003

12 -D MATH REVIEW Eample f() or f(α) h() or h(α) 2 or α or α ECE/OPTI533 Digital Image Processing class notes 2 Dr. Robert A. Schowengerdt 2003

13 -D MATH REVIEW shift f(α)h(0 - α) h(-α) multiply integrate = 0 h( -α) = h(2 -α) α f(α)h( - α) α f(α)h(2 - α) area = g(0) α area = g() α area = g(2) = 2 h(3 -α) α α f(α)h(3 - α) area = g(3) = 3 h(4 -α) α α f(α)h(4 - α) area = g(4) α α = 4 ECE/OPTI533 Digital Image Processing class notes 22 Dr. Robert A. Schowengerdt 2003

14 -D MATH REVIEW Plot g() g() The shifts in this eample are by integer steps, for illustration convenience ECE/OPTI533 Digital Image Processing class notes 23 Dr. Robert A. Schowengerdt 2003

15 -D MATH REVIEW Recipe Convolution. write both as a function of α f(α) and h(α) 2. flip h (or f) about α = 0 h(-α) 3. shift h (or f) by an amount h( - α) 4. multiply the two functions f(α)h( -α) 5. integrate the product function over all α g() 6. repeat steps 3 through 5 until done ECE/OPTI533 Digital Image Processing class notes 24 Dr. Robert A. Schowengerdt 2003

16 -D MATH REVIEW Convolution Properties -D CONVOLUTION PROPERTIES property commutative f ( ) h ( ) = h ( ) f ( ) distributive = f( ) [ h ( ) + h 2 ( )] f ( ) h ( ) + f ( ) h 2 ( ) associative = f ( ) h ( ) h 2 ( ) [ ] [ f( ) h ( )] h 2 ( ) ECE/OPTI533 Digital Image Processing class notes 25 Dr. Robert A. Schowengerdt 2003

17 -D MATH REVIEW Convolution Eamples -D CONVOLUTION EXAMPLES f() h() g() f() δ() f() f(- 0 ) h() g(- 0 ) f() h(- 0 ) g(- 0 ) rect() rect() tri() sinc() sinc() sinc() gaus() gaus() gaus ECE/OPTI533 Digital Image Processing class notes 26 Dr. Robert A. Schowengerdt 2003

18 -D MATH REVIEW FOURIER TRANSFORMS (-D) Why is it important? For an LSI system, the convolution operator becomes a multiplication operator in the Fourier domain Taking the Fourier transfom of the system equation, Gu ( ) = Fu ( )H ( u) where G(u) is the spectrum of the output signal, F(u) is the spectrum of the input signal, and H(u) is the system transfer function ECE/OPTI533 Digital Image Processing class notes 27 Dr. Robert A. Schowengerdt 2003

19 -D MATH REVIEW In many cases, it is easier to analyze an LSI system in the Fourier domain Forward transform d Fu ( ) = f( )e j2πu Inverse transform f( ) = Fu ( )e j2πu d u ECE/OPTI533 Digital Image Processing class notes 28 Dr. Robert A. Schowengerdt 2003

20 -D MATH REVIEW Fourier Transform Properties f() and F(u) are, in general, comple functions f() real F(u) = F*(-u) F is Hermitian: Re[F(u)] even, Im[F(u)] odd f() real and even Im[F(u)] = 0, i.e. F(u) is real Forward transform is the analysis of f() into its spectrum F(u) Inverse transform is the synthesis of f() from F(u) ECE/OPTI533 Digital Image Processing class notes 29 Dr. Robert A. Schowengerdt 2003

21 -D MATH REVIEW Fourier Transform Pairs -D FOURIER TRANSFORM PAIRS f() F(u) δ(u) δ() cos( 2πu 0 ) δδ ---- u 2 u 0 u δδ cos( 2πu 0 ) sin( 2πu 0 ) j δδ ---- u 2 u 0 u 0 rect() sinc(u) sinc() rect(u) comb() comb(u) gaus() gaus(u) tri() sinc 2 (u) ECE/OPTI533 Digital Image Processing class notes 30 Dr. Robert A. Schowengerdt 2003

22 -D MATH REVIEW Fourier Transform Properties -D FOURIER TRANSFORM PROPERTIES name f() F(u) f() F(u) f(±) F(±u) f ( ) f 2 ( ) F u ( )F 2 u ( ) F(±) f( + u) f ( )f 2 ( ) F ( u ) F 2 ( u ) scaling f(/b) b F(bu) f ( ) f 2 ( ) F ( u )F 2 ( u ) shifting f( ± 0 ) ± e j2πu 0 f( ) Fu ( + u 0 ) ± e j2π 0u Fu ( ) f ( )f 2 ( ) F ( u ) F 2 ( u ) derivative f k ( ) ( ) j2πu ( ) k Fu ( ) j2π ) k f( ) F ( k ) ( u ) linearity a f ( ) + a 2 f 2 ( ) a F ( u ) + a 2 F 2 ( u ) ECE/OPTI533 Digital Image Processing class notes 3 Dr. Robert A. Schowengerdt 2003

23 -D MATH REVIEW LINEAR, SHIFT-INVARIANT (LSI) SYSTEMS (-D) Linear: output of a sum of inputs is equal to the sum of the individual outputs g ( ) = af ( ) h( ) g 2 ( ) = bf 2 ( ) h( ) g ( ) = [ af ( ) + bf 2 ( )] h( ) = ( af ( ) h( ) + bf 2 ( ) h( ) ) = g ( ) + g 2 ( ) shift-invariant: system response does not change over space System equation where h() is the system impulse response g ( ) = f( ) h( ) ECE/OPTI533 Digital Image Processing class notes 32 Dr. Robert A. Schowengerdt 2003

24 -D MATH REVIEW Fourier transform of system equation where H(u) is the system transfer function Gu ( ) = Fu ( )H ( u) H(u) is a comple filter that modifies the spectrum F(u) of f() For comple functions G, F and H ampl G u ( ) [ ] = ampl F u [ ( )]ampl [ H ( u )] phase [ G ( u )] = phase F ( u ) [ ] + phase H u [ ( )] ECE/OPTI533 Digital Image Processing class notes 33 Dr. Robert A. Schowengerdt 2003

25 -D MATH REVIEW -D CASCADED SYSTEMS N cascaded LSI systems f() g() h ()... h N () g ( ) = {[ f( ) h ( )] h 2 ( )} h N ( ) Single system equivalent f() h net () g() h net ( ) = h ( ) h 2 ( ) h N ( ) where h net is the net system impulse response ECE/OPTI533 Digital Image Processing class notes 34 Dr. Robert A. Schowengerdt 2003

26 -D MATH REVIEW FOURIER TRANSFORM EXAMPLES E. sinc(/2) sinc(/3) Convolution in this case is very difficult! Take the Fourier transform 2rect ( 2u ) 3rect ( 3u ) = 6rect ( 3u ) = -/4 /4 u -/6 /6 u -/6 /6 u Take the inverse Fourier transform 6 --sinc(/3) = 2sinc(/3) 3 ECE/OPTI533 Digital Image Processing class notes 35 Dr. Robert A. Schowengerdt 2003

27 -D MATH REVIEW E 2. Find spectrum of square wave with a DC bias f() Write square wave as convolution f ( ) = --rect -- comb Take the Fourier transform Fu ( ) = = sinc 2u 2sinc 2u ( )comb 5u ( ) ( ) comb ( 5u ) spectrum is comb function, modulated by sinc function, sampled at frequency interval u = /5, i.e /period. zeros at u = n/2, n = ±, ±2,... ECE/OPTI533 Digital Image Processing class notes 36 Dr. Robert A. Schowengerdt 2003

28 -D MATH REVIEW if square wave period P = 2 pulse width, we have the classic square wave spectrum at u = 0, ±/P, ±3/P, ±5/P,...? Verify the above statement for an arbitrary period P sinc(2u) and comb(5u) /5-8/5-7/5-6/5-5/5-4/5-3/5-2/5 -/5 /5 2/5 3/5 4/5 5/5 6/5 7/5 8/5-3/2 - -/2 0 /2 3/2 u ECE/OPTI533 Digital Image Processing class notes 37 Dr. Robert A. Schowengerdt 2003

29 -D MATH REVIEW sinc(2u) times comb(5u) /5-7/5-6/5-5/5-4/5-3/5-2/5 -/5-3/2 - -/2 2/5 /5 2/5 3/5 4/5 5/5 6/5 7/5 0 /2 3/2 8/5 comb, modulated by sinc function ECE/OPTI533 Digital Image Processing class notes 38 Dr. Robert A. Schowengerdt 2003

30 -D MATH REVIEW Scaling Property Width of F(u) is inversely proportional to width of f() rect() F sinc(u) u -/2 /2 rect(/2) 2sinc(2u) 2 u - rect(/3) 3sinc(3u) 3 u -3/2 3/2 ECE/OPTI533 Digital Image Processing class notes 39 Dr. Robert A. Schowengerdt 2003

31 -D MATH REVIEW δδ(2) -/2 /2 /2 F cos(πu) u δδ() 2cos(2πu) u - δδ(2/3) 3/2 3cos(3πu) u -3/2 3/2 ECE/OPTI533 Digital Image Processing class notes 40 Dr. Robert A. Schowengerdt 2003

32 -D MATH REVIEW δδ(u)/2 cos(2π) F /2 - u cos(4π) δδ(u/2)/4 /2-2 2 u cos(6π) δδ(u/3)/6 /2-3 3 u ECE/OPTI533 Digital Image Processing class notes 4 Dr. Robert A. Schowengerdt 2003

33 -D MATH REVIEW Superposition Property Fourier transform of sum of functions equals sum of their individual Fourier transforms + cos(2π) δ(u)+δδ(u)/2 F /2 - u cos(2π)+cos(4π) δδ(u)/2+δδ(u/2)/4 / u ECE/OPTI533 Digital Image Processing class notes 42 Dr. Robert A. Schowengerdt 2003

34 -D MATH REVIEW SYSTEM ANALYSIS WITH THE FOURIER TRANSFORM Three applications Application Given Find Spatial Domain Fourier Domain system output f(), h() g() g ( ) = f( ) h( ) Gu ( ) = Fu ( )H ( u) g ( ) = F [ Gu ( )] system identification f(), g() H( u) = Gu ( ) Fu ( ) h() NA ill-conditioned h ( ) = F [ H( u) ] inversion h(), g() Fu ( ) = Gu ( ) H( u) f() NA ill-conditioned f( ) = F [ Fu ( )] ECE/OPTI533 Digital Image Processing class notes 43 Dr. Robert A. Schowengerdt 2003