Image Processing Eng. Ahmed H. Abo absa E-mail: a.absa@up.ps

1. Image ProcessingEng. Ahmed H. Aboabsa E-mail: a.absa@up.edu.ps

2. Outline the lecture • Signal Operations • Time Shifting • Time Scaling • Time Inversion. • Important Functions • Mean value, Mean square value, variance, standard deviation. • Signal and Vector • Correlation

3. Signal Operations • Time Shifting: Consider a signal g(t) and the same signal delayed by T seconds which we shall denote by ¢(t).

4. Time Scaling: The compression or expansion of a signal in time Example:in below Figures a and b shows the signals g(t) and z(t), respectively. Sketch: (a) g(3t); (b) z(t /2).

5. Time Inversion: Time inversion may be considered a special case of time scaling with a = -1. • To invert g(t), we rotate this frame 180 deg about the vertical axis. • Example:For the signal g(t) shown in the figure, sketch g(-t).

6. x(t) x T time, t Mean • The mean value,x , is the height of the rectangular area having the same area as that under the function x(t) • Can also be defined as the first moment of the p.d.f.

7. x x(t) x T time, t Mean square value, variance, standard deviation • Mean square value • Variance: (average of the square of the deviation of x(t) from the mean valuex) • Standard deviation, x, is the square root of the variance

8. Unit Impulse • Definition: The unit impulse δ(t) is not a function in the ordinary sense. It is defined by the integral relation and is called a generalized function. The unit impulse is not defined in terms of its values, but is defined by how it acts inside an integral when multiplied by a smooth function f(t). To see that the area of the unit impulse is 1, choose f(t) = 1 in the definition. We represent the unit impulse schematically as shown below; the number next to the impulse is its area.

9. Unit Impulse (cont.) • Unit impulse — narrow pulse approximation To obtain an intuitive feeling for the unit impulse, it is often helpful to imagine a set of rectangular pulses where each puls has width εand height 1/ εso that its area is 1. The unit impulse is the quintessential tall and narrow pulse!

10. Unit Step • Definition Integration of the unit impulse yields the unit step function which is defined as

11. Unit Impulse vs. Unit Step • As an example of the method for dealing with generalized functions consider the generalized function • Since u(t) is discontinuous, its derivative does not exist as an ordinary function, but it does as a generalized function. To see what x(t) means, put it in an integral with a smooth testing function and apply the usual integration-by-parts theorem

12. Unit Impulse vs. Unit Step (cont.) • The result is that which, from the definition of the unit impulse, implies that That is, the unit impulse is the derivative of the unit step in a generalized function sense.

13. Combination of Building Block Signals

14. Plotting the signal • Plot • t<-2  f(t)=0 • -2<t<-1  f(t)=3[t+2] • -1<t<1  f(t)=-3t • 1<t<3  f(t)=-3 • 3<t< f(t)=0

15. Signal and Vector • A vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are defined and satisfy certain natural axioms. • Signal represented by weighted sum of vectors • Concept of orthogonality • Sin, cos, FFT • Exp(-jz), DFT • X, Taylor series • DCT (JPEG, MPEG, MP3) • Subspace • Wavelet (not quite orthogonal)

16. Correlation • indicates the strength and direction of a linear relationship between two random variables

17. Questions?