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

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

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

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

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
Signal Operations
• Time Shifting: Consider a

signal g(t) and the same

signal delayed by T seconds

which we shall denote by ¢(t).

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).

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).

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.

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
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.

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!

Unit Step
• Definition

Integration of the unit impulse yields the unit step function

which is defined as

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

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.

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