Properties of the Fourier Transform CS/BIOEN 4640: Image Processing - PowerPoint PPT Presentation

Properties of the Fourier Transform CS/BIOEN 4640: Image Processing Basics March 22, 2012

Review: Euler’s Representation Im ◮ A complex number can be given as an angle φ and a radius r r ◮ Think 2D polar coordinates φ ◮ Exponential form: Re 0 re i φ = r cos ( φ ) + i ( r sin ( φ ))

Review: Fourier Transform Given a complex-valued function g : R → C , Fourier transform produces a function of frequency ω : � ∞ 1 � � √ G ( ω ) = g ( x ) · cos ( ω x ) − i · sin ( ω x ) dx 2 π −∞ � ∞ 1 g ( x ) · e − i ω x dx = √ 2 π −∞

Inverse Fourier Transform The Fourier transform is invertible. That is, given the Fourier transform G ( ω ) we can reconstruct the original function g as � ∞ 1 G ( ω ) · e i ω x d ω g ( x ) = √ 2 π −∞ We use the notation: G = F{ g } Fourier transform: g = F − 1 { G } Inverse Fourier transform:

The Dirac Delta Definition The Dirac delta or impulse is defined as � ∞ δ ( x ) = 0 for x � = 0 , δ ( x ) dx = 1 and −∞ ◮ The Dirac delta is not a function ◮ It is undefined at x = 0 . ◮ Has the property � ∞ f ( x ) δ ( x ) dx = f ( 0 ) for any function f −∞

The Dirac Delta Even though the Dirac delta is not a function, we will plot it like this: 2.0 1.5 δ ( x ) 1.0 0.5 0.0 −1.0 −0.5 0.0 0.5 1.0 x

Shifting and Scaling Deltas 2.0 2.0 1.5 1.5 δ ( x − 0.5 ) ) 2 δ ( x ) 1.0 1.0 0.5 0.5 0.0 0.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 x x A shifted impulse is A scaled impulse is given by δ ( x − a ) given by s · δ ( x ) � ∞ −∞ s · δ ( x ) dx = s

Fourier Transform Pairs: Cosine 1.0 1.5 1.0 0.5 0.5 −5 −3 −1 1 3 5 −5 −3 −1 1 3 5 −0.5 −0.5 −1.0 −1.0 −1.5 G ( ω ) = � π g ( x ) = cos ( ω 0 x ) 2 · ( δ ( ω + ω 0 ) + δ ( ω − ω 0 )) (Here ω 0 = 3 )

Fourier Transform Pairs: Sine 1.0 1.5i 1.0i 0.5 0.5i −5 −3 −1 1 3 5 −5 −3 −1 1 3 5 −0.5i −0.5 −1.0i −1.0 −1.5i G ( ω ) = i � π g ( x ) = sin ( ω 0 x ) 2 · ( δ ( ω + ω 0 ) − δ ( ω − ω 0 )) (Here ω 0 = 5 ) BOOK TYPO

Fourier Transform Pairs: Gaussian 1.0 1.0 0.5 0.5 −9 −7 −5 −3 −1 1 3 5 7 9 −9 −7 −5 −3 −1 1 3 5 7 9 −0.5 −0.5 −1.0 −1.0 � � � � − x 2 − ω 2 g ( x ) = 1 σ exp G ( ω ) = exp 2 σ 2 2 · ( 1 /σ 2 ) (Here σ = 3 )

Fourier Transform Pairs: Box 1.5 1.5 1.0 1.0 0.5 0.5 −9 −7 −5 −3 −1 1 3 5 7 9 −9 −7 −5 −3 −1 1 3 5 7 9 −0.5 −0.5 � if | x | < b 1 G ( ω ) = 2 sin ( b ω ) g ( x ) = √ 2 π ω 0 otherwise (Here b = 2 ) BOOK TYPO

Properties: Linearity Scaling: F{ c · g ( x ) } = c · G ( ω ) Addition: F{ g 1 ( x ) + g 2 ( x ) } = G 1 ( ω ) + G 2 ( ω )

Properties for Real-Valued Functions ◮ If g is a real-valued function, then G ( ω ) = G ∗ ( − ω ) ◮ If g is real-valued and even : g ( x ) = g ( − x ) , then G ( ω ) is real-valued and even ◮ If g is real-valued and odd : g ( x ) = − g ( − x ) , then G ( ω ) is purely imaginary and odd

Properties: Similarity Stretching a signal horizontally leads to a shrinking of the Fourier spectrum: � ω F{ g ( s · x ) } = 1 � | s | · G s And vice versa, shrinking the signal causes a stretching in the Fourier spectrum

Properties: Shift A horizontal shift of the signal results in a phase shift of the Fourier transform: F{ g ( x + d ) } = e − i ω d · G ( ω ) ◮ Notice magnitude | G ( ω ) | stays the same ◮ This is a rotation in the complex plane by − ω d

Properties: Convolution Convolution becomes multiplication in the Fourier domain: √ F{ g ( x ) ∗ h ( x ) } = 2 π G ( ω ) · H ( ω ) And vice versa, multiplication becomes convolution: 1 √ F{ g ( x ) · h ( x ) } = G ( ω ) ∗ H ( ω ) 2 π