Fourier transform. Stefano Ferrari. Università degli Studi di Milano Methods for Image Processing. academic year

Transcription

1 Fourier transform Stefano Ferrari Università degli Studi di Milano Methods for Image Processing academic year Function transforms Sometimes, operating on a class of functions is easier if the functions are expressed in another domain. Hence a given function is transformed in another function on a different domain (the codomain of the transform). Time (or space) functions can be transformed in functions defined on another domain. The operations applied on the original function domain correspond to simpler operations when applied on the transformed function. Once processed, the transformed function can be back-transformed to the original domain by way of a suitable inverse transform. Stefano Ferrari Methods for Image processing a.a. 27/8

2 Function transforms (2) processing original function f P f processed function transform T T inverse transform transformed function T (f ) T (f ) P transformed processed function processing in the T codomain Example: Discrete Cosine Transform 3 f Through the Discrete Cosine Transform, the function f can be expressed in terms of sum of weighted (scaled) cosinusoidal functions. Stefano Ferrari Methods for Image processing a.a. 27/8 2

3 Example: Discrete Cosine Transform (2) x Fourier series Every periodic function, f, with period T, can be represented as a linear combination of sines and cosines: f (t) = n= c n e ι 2πn T t where c n = T T 2 f (t)e 2πn ι T t dt T 2 Note: the base is composed of an infinite set of sines and cosines. Stefano Ferrari Methods for Image processing a.a. 27/8 3

4 Impulse The Dirac delta function, δ, or impulse, is defined as: {, t = δ(t) =, t and δ(t) dt = δ(t t ) t t Impulse (2) The impulse has the interesting property, called the sifting property: f (t) δ(t) dt = f () f (t) δ(t t ) dt = f (t ) Hence, it can be used for sampling a function, f, by convolution. Stefano Ferrari Methods for Image processing a.a. 27/8 4

5 Discrete impulse For working with functions defined only in a discrete set of points (e.g., f : Z R), the discrete impulse can be defined: δ(x x ) δ(x) = {, x =, x δ(x) = x= The sifting property is satisfied also in this case: f (x) δ(x) = f () x= f (x) δ(x x ) = f (x ) x= x x Impulse train The impulse train, s, will be important: s (t) = δ(t n ) n= s is a periodic function with period. Stefano Ferrari Methods for Image processing a.a. 27/8 5

6 Continuous Fourier transform Under mild conditions, for every function f (t) the continuous Fourier transform can be computed as: F{f (t)} = f (t) e ι2πνt dt = F (ν) Since t is integrated, the Fourier transform of f (t) is a function of the variable ν. Hence it is usually indicated as F{f (t)} = F (ν). The Fourier transform describes f (t) as a linear (complex) combination of sines and cosines: F (ν) = f (t) [cos(2πνt) ι sin(2πνt)] dt since: e ιθ = cos θ + ι sin θ Continuous Fourier transform (2) A complementary transform, called inverse Fourier transform can be defined: f (t) = F {F (ν)} = F (ν) e ι2πνt dν It allows to obtain f (t) from F (ν). The F{f (t)} and F {F (ν)} transforms are called the Fourier transforms pair. For some operations, the spectrum of the transform carries valuable information: F (ν) the coefficient of the basis function gives the relative importance of the basis function in the representation. Stefano Ferrari Methods for Image processing a.a. 27/8 6

7 Example: FT of the box function { A, W /2 t W /2 f (t) =, otherwise f (t) A W /2 W /2 t F (ν) = = W /2 W /2 f (t) e ι2πνt dt A e ι2πνt dt = AW sin(πνw ) πνw = AW sinc(νw ) F (ν) AW /W 2/W F (ν) AW /W 2/W ν { sinc(t) =, t = sin πt πt, otherwise 2/W /W 2/W /W ν Example: FT of the box function (2) f (t) F (ν) t ν The box function can be exploited for computing the Fourier transform of δ. When considering only boxes function where AW =, the limit for W is δ: A, the integral is, outside [ W /2, W /2] the function is. Since the main lobe of F (ν) extends over [ /W, /W ], for W it tends to occupy the whole real line, (, ). Hence, the limit of the Fourier transform of these sequence is the constant function. Stefano Ferrari Methods for Image processing a.a. 27/8 7

8 Example: FT of the impulse Using the sifting property of δ, its Fourier transform can be computed directly: F(δ(t)) = F (ν) = δ(t) e ι2πνt dt = e ι2πν = It can be also shown that: F{s (t)} = S(ν) = n= δ ( ν Note: S(ν) is a periodic function with period. n ) Convolution and Fourier transform The convolution between two continuous functions, f and h, is defined as: f (t) h(t) = It can be shown that: f (τ) h(t τ)dτ F{f (t) h(t)} = F (ν) H(ν) Also the opposite holds: F{f (t) h(t)} = F (ν) H(ν) This is called the convolution theorem. Stefano Ferrari Methods for Image processing a.a. 27/8 8

9 Sampling Sampling is a processing step often required for operating on a function with a digital computer. This operation can be modeled as the multiplication of the considered function, f, with the impulse train of a suitable period,, called sampling step: f (t) = f (t) s (t) = The value of each sample results: n= f (t) δ(t n ) f k = f (t) δ(t k ) dt = f (k ), k Z Sampling (2) f (t) s (t) t t f (t) s (t) t f k = f (k ) k Stefano Ferrari Methods for Image processing a.a. 27/8 9

10 Fourier transform of a sampled function Using the convolution theorem, the Fourier transform of a sampled function can be computed as: F (ν) = F{ f (t)} = F{f (t) s (t)} = F (ν) S(ν) From the sifting property of δ: F (ν) = n= F ( ν n ) Hence, the Fourier transform of the sampled function, F is the infinite summation of scaled copies (for a factor of ) of the Fourier transform of the original function, F, shifted by (i.e., F is a periodic function). Fourier transform of a sampled function (2) Although f is a sampled function, F is continuous. It is the summation of scaled copies of a continuous function, F. The separation between the copies is ruled by : oversampling critical sampling undersampling Stefano Ferrari Methods for Image processing a.a. 27/8

11 Sampling theorem We are interested in recovering the original function, f, from its samples, f : Is it always possible? What are the necessary conditions? The Fourier transform of a sampled function, F, provides infinite copies of F : if a single copy of F can be isolated, f can be obtained using the inverse Fourier transform. Since F is composed of shifted copies of F : F have to be different from zero only in a finite interval [ ν max, ν max ] f is a band limited function the separation of of copies. must be sufficient to avoid overlapping Sampling theorem (2) The previous conditions can be summarized in the sampling theorem: > 2 ν max The sampling rate must be at least twice the higher frequency component of f. The critical sampling rate is called the Nyquist rate. Stefano Ferrari Methods for Image processing a.a. 27/8

12 Reconstruction of a sampled function Multiplying F by a box function H: H(ν) = {, νmax ν ν max, otherwise a single copy of F can be recovered. Then, the inverse Fourier transform can be applied to obtain f. f (t) = F {F (ν)} = F {H(ν) F (ν)} = h(t) f (t) It can be shown that: f (t) = n= ( ) t n f (n ) sinc f (t) is equal to f k in t = k ; elsewhere it is obtained using a shifted sinc functions basis. Reconstruction of a sampled function (2) f recovered original k f recovered original k Stefano Ferrari Methods for Image processing a.a. 27/8 2

13 Aliasing If the function is sampled under the Nyquist rate ( < 2 ν max), the periods overlap. The components between 2 and ν max are mixed with other components. This effect is called aliasing. Aliasing (2) For practical cases, aliasing is almost unavoidable. Since only a finite interval of the signal can be considered, the processed function can be modelled multiplying the real (band limited) function by a box function. For the convolution theorem, the Fourier transform of the processed function is equal to the Fourier transform of the original function convolved with the sinc function (that has infinite support). No function that has finite duration can be band limited. Every band limited function must have infinite support. Aliasing effects can be reduced by attenuating the high frequency components (i.e., smoothing) before sampling. Or windowing: use of a different function instead of the box, such that its spectrum vanishes more rapidly than sinc. Stefano Ferrari Methods for Image processing a.a. 27/8 3

14 Aliasing: an example The sampled function looks like a sinusoidal function having a frequency much smaller than the original function. Since the only component of the Fourier transform of the original function is higher than the sampling frequency, its copy in the overlapping period is positioned in a lower frequency. Aliasing: an example (2) F (ν) 2 ν max ν max ν 2 F (ν) 2 ν max ν max ν 2 F (ν) H(ν) 2 ν max ν max ν 2 Stefano Ferrari Methods for Image processing a.a. 27/8 4

15 Discrete Fourier Transform (DFT) F can be expressed in terms of f : F (ν) = From which can be shown: F (ν) = n= f (t)e ι2πνt dt f n e ι2πνn Since F is periodic, all the information carried out by F is contained in a single period. Discrete Fourier Transform (DFT) (2) If M samples of F are considered in one period,, the following frequencies are inspected: ν m = and the samples are: m M, m =,..., M F m = M n= f n e ι2πmn/m The M samples {F m } are computed using only M samples of f. This transform is called the Discrete Fourier Transform. Stefano Ferrari Methods for Image processing a.a. 27/8 5

16 Inverse Discrete Fourier Transform (IDFT) The M samples {f n } can be reconstructed from {F m } using the following transformation: f n = M M m= F m e ι2πmn/m, n =,..., M This transform is called the Inverse Discrete Fourier Transform. Discrete Fourier Transform pair The forward and inverse Fourier transform are usually represented as: F (u) = f (x) = M M x= M u= f (x)e ι2πux/m, u =,..., M F (u)e ι2πux/m, x =,..., M No direct reference to time and frequency is present. {fn } and {F m } are just sequences. It can be shown that F (u) and f (x) are periodic: F (u) = F (u + km) and f (x) = f (x + km), k Z Although if (the original) f is not. Stefano Ferrari Methods for Image processing a.a. 27/8 6

17 Circular convolution The convolution of finite sequence of M elements can be defined through: g(x) = f (x) h(x) = M m= f (m) h(x m) The periodicity of g derives from the periodicity of f and h. This operation is called circular convolution. Through this operation, the convolution theorem (for continuous FT) can be extended to the DFT. The circularity causes the wraparound problem, that will be discussed later. Space and frequency resolution f is composed of M samples taken apart. The sequence covers an interval that is T = M long. In the frequency domain, the samples of F are u = M = T apart. Hence, the DFT is defined over a frequency interval Ω = M u = long. The frequency resolution depends on the length of the sampled interval in the space domain, T. The range of the frequencies covered by the DFT depends on the sampling step,. Stefano Ferrari Methods for Image processing a.a. 27/8 7

18 Homeworks and suggested readings DIP, Sections pp Cool Sound and Water Experiment! Shutter aliasing with dripping mercury Amazing Animated Optical Illusions! Sweet stop motion Stefano Ferrari Methods for Image processing a.a. 27/8 8