Theory of signals and images I. Dr. Victor Castaneda

Transcription

1 Theory of signals and images I Dr. Victor Castaneda

2 Image as a function Think of an image as a function, f, f: R 2 R I=f(x, y) gives the intensity at position (x, y) The image only is defined over a rectangle, with a finite range: f: [a,b]x[c,d] [0,1] A color image is just three dimensions: I =f (x, y, c R,G,B )

3 Image Image Discrete Continuity model 1D function

4 Image as a function

5 Linear Systems Let define a new image g in terms of image f We can transform either the domain or the range of f Range transformation: Preserve the range but change the domain of f : other operations operate on both the domain and the range of f

6 Linear Systems Sum: two function can be added: Scaling: a function can be multiplied by a constant

7 a a Linear Systems Linear Shift Invariant Systems (LSIS) Linearity: f 1 1 g 2 f 2 g f f g g Shift invariance: f x a g x a

8 Example of LSIS Linear Systems Defocused image ( g ) is a processed version of the focused image ( f ) g f Ideal lens is a LSIS x f LSIS g x Linearity: Brightness variation Shift invariance: Scene movement (not valid for lenses with non-linear distortions)

9 Linear Systems Example (Rotation Matrix) Let vector x y and R θ the rotation matrix where rotated vector will be: x R x y = g x, y = R θ R y = R 11 R 12 x R 21 R 22 y Check if this function is linear R θ = cos θ sin θ sin θ cos θ

10 Linear Systems Convolution LSIS is doing convolution; convolution is linear and shift invariant f h f g d x h f x g h h x kernel h

11 Convolution Linear Systems f g f g Eric Weinstein s Math World

12 f g h h f g What h will give us g = f? Dirac Delta Function (Unit Impulse) x dx x f dx x x f f dx x f x f d x f x g x h d x h Shifting property: Linear Systems Convolution

13 Linear Systems Convolution Commutative a b b a Associative a b c a b c Cascade system f h1 2 h g f h h 1 2 g f h 2 h 1 g

14 Convolution Linear Systems A lot of filters are based on the concvolution Matrix convolution is an operation between two matrices. image, I ( kernel, K K I ) [x i, j i ] = (-1 * 222) K I James Matthews, ( 0 * 170) + ( 1 * 149) + (-2 * 173) + ( 0 * 147) + ( 2 * 205) + (-1 * 149) + ( 0 * 198) + ( 1 * 221) = 63

15 Image Filtering Signal to Noise Ratio (SNR) Modeling: Noise is usually assumed to be additive and random The observed intensity is the sum of the true intensity and a spurious and random signal. Signal-to-noise ratio, or SNR Ratio between std of signal and noise

16 Image Filtering Type of noises: Salt and pepper Spurious noise White noise Normally zero mean Gaussian distribution And others

17 Objective improve SNR Challenges Image Filtering blurs object boundaries and smears out important structures

18 Serie de Fourier A finales del siglo XVIII Jan Baptiste Joseph Fourier ( ) descubrió un método que permite aproximar funciones periódicas mediante combinación lineal de funciones trigonométricas sencillas.

19 Serie de Fourier Definición: Se llama serie de Fourier de una función f(x) en el intervalo [ L, L] a: Donde los coeficientes a 0, a n y b n deben ser determinados.

20 Los coeficientes a 0, a n y b n están dados por: Serie de Fourier

21 Serie de Fourier

22 Serie de Fourier Ejemplo: consideremos la función:

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24 f(x) = -1 entre -π y 0 f(x) = 1 entre 0 y π

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32 Amplitud: A k = a k 2 + b k 2 Fase: φ k = tan 1 b k a k Donde:

33 Fourier Transform Frequency domain transformations (Fourier)

34 Fourier Transform Any periodic function can be rewritten as a weighted sum of Sines and Cosines of different frequencies, called Fourier Series.

35 A sum of sinusoids Building box Fourier Transform A sin( x Assumptions Periodic signals More coefficients makes signal closer to the original.

36 Fourier Transform We want to understand the frequency of our signal. So, let s reparametrize the signal by instead of x: Spatial domain f(x) Fourier Transform F() Frequency domain For every from 0 to inf, F() holds the amplitude A and phase of the corresponding sine A sin( x How can F hold both? Complex number trick! A F ( ) R ( ) ii ( ) 2 R ( ) I ( ) 2 tan 1 I ( ) R ( ) F() Inverse Fourier Transform f(x)

37 Fourier Transform Example = + =

38 Fourier Transform Example = + =

39 Fourier Transform Example = + =

40 Fourier Transform Example = + =

41 Fourier Transform Example = + =

42 Fourier Transform Example = A k 1 1 s i n ( 2 k t ) k

43 Example Fourier Transform

44 Fourier Transform Fourier transform and convolution Spatial Domain (x) Frequency Domain (u) g f h G FH g fh G F H So, we can find g(x) by Fourier transform g f h IFT FT FT G F H

45 Fourier Transform Fourier transform is symmetrical x and y direction FFT

46 Fourier Transform F F F F F F = F F F

47 Fourier Transform

48 Wavelets Transform

49 Wavelets Transform Imagine a sinusoidal signal with a small discontinuity: Fourier does not see the discontinuity. Wavelet shows exactly the location of the discontinuity in time. Fourier coeficients Wavelet coeficients

50 Wavelets Transform Matematically, Fourier analysis representsed by the Fourier transform divide the original signal in a sum of sinusoidal signals. Wavelets transform is defined as a sum all time of the signal multiplied by a scale, changing the wavelet function. The wavelet coefficient result are then in terms of scale and position.

51 Scaling of wavelet Wavelets Transform Scale a wavelet means shrink or elongate is denominated scale factor In sinusoidal the scale factor is easy to see.

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53 Questions?