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Welcome back.

[UNKNOWN] systems or systems theory, we're typically interested

in the number of properties that systems might have.

For example, we are interested if the systems are stable.

an important properties, since you might say that

in general unstable systems are of not much use.

Whether they are causal, [UNKNOWN] and so on.

All of these are independant of each other.

One property does not require another property,

or does not lead to another property.

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Out of all possible 2D systems, we're primarily

interested in systems that have two important properties.

They're both linear and spatially invariant.

That's forming LSI systems.

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Linearity means that the, a sum of signals is that the input of a linear system, the

system can process each signal separately, and add up the processed signals.

Spatially variance, means that, It is irrelevant where

the origin of the coordinate system is located.

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So far we've talked about signals, two dimensional,

multi-dimensional signals in general, we have examples of

some important signals such as the delta, and

the cosine that will be using throughout this class.

We are interested in processing or manipulating such signals through systems.

So x in 1 and 2, is the input image to start the system.

Y is the output, which is equal to a transformed version of the input.

A manipulated version of the input.

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is equal to 255 minus x and 1 and 2.

Assuming these images are eightbits per pixel.

Therefore the range from 0 to 55.

What such a system does is changing the polarity

of the input is turning black values into white, and white values into black.

So a negative into a positive or a positive into a negative.

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And what the system does is stretching the intensity values of the input image.

We'll see actually quite a few systems of

this nature when I talk about image enhancement.

Similarly, we have a system, that provides an output which is equal to the average

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And yet as a final example, I can have a system that gives me the output,

the medium of the input values again in the neighborhood.

When we talk about systems we're interested in a number

of their properties, some of which are mentioned here.

Whether, if the system is stable, whether it is has memory, or is memory less,

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All these five here listed properties are independent of each other.

A system can have all of them, none of them, just one or two, and so on.

And out of all these five properties we're going to

focus next on systems that have the last two properties.

And [UNKNOWN] systems, we'll be referring to them as linear

and spatially invariant systems, are quite useful, are used very widely.

And it's relatively straightforward to describe such systems, both in

the spatial domain as well as in the frequency domain.

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A two-imensional system is linear if

it satisfies the homogeneity property shown here.

In other words, if at the input of the system might put the

weighted sum of two signals, the weight is alpha one and alpha two.

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And if at the output, I find the weighted sum of the individual outputs.

So alpha 1, the response of the system to x1 plus alpha

2, the response of the system to x2, then the system is linear.

This is clearly a very useful properties, because in many applications,

I have to process the sum, weighted sum of individual images.

And it might be easier to process each image

individually and add up the responses, or looking at it

the reverse way, in some sense, I can take

any signal and decompose it into simpler symbols, simpler images.

Which then I process individually, and again I add up the individual responses.

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From this equation it's clear that if alpha 2 is equal to 0

then the response of the system to alpha 1 x 1

n 1 and 2 is simply alpha 1 The response of the system, x1.

So if an image is multiplied by scalar, I don't need to be

concerned, I can process the image and then multiply the output by the scalar.

Similarly, If alpha1 equals minus alpha2, and x1

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alpha 1 the response to x 1.

Minus alpha 1 the response to x 1.

Which is equal to zero.

So, in other words, we see that the system is linear when I

put the zero in the input, I find a zero in the output.

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Now, this is a property that can be shared also by non-linear systems, therefore,

therefore it's a property that I cannot use to prove that the system is linear.

But I can use to prove that the system is non-linear.

And this is a simple example, we can proceed there the system, that

we looked at the previous slide, so using the notation here YN1

is the response of the system to X X as an input, right?

And we define the system as 255 minus X, N1, N2, right?

So, it's like the system takes an eight bit

image and inverts it, finds the negative of it, right?

So clearly if I put a zero as the input of the system, the output

equals 255, hich is different than zero, and therefore this system that

finds the negative of an image is non-linear.

Generally speaking is rather forward to utilize

this homogenize property in this equation that

you see here on top, and prove or disprove that the system is linear.

And this property and everything that we

covered here, this light applies to twodimensional Systems

and signals as well as one dimensional, three

dimensional, five dimensional signals and systems in general.

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Let us consider again the two dimensional system T, x n one

and two they input y, n one and two to the output.

[UNKNOWN] system if when I sift the input

By k1, k2, I find that the output is shifted

by the same amounts, k1, k2, then, the system is spatially invariant.

Another way to express this, is that to say that the system does not care

about the location of the axis, does not care where the 0,0 point is located.

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Now this property is independent of linearities, so if we consider

the system we looked at earlier,

which for an input x n1, n2

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Therefore, the system is spatially invariant, SI.

So, the system takes the negative of an

image is no linear, but is spatially Spatially invariant.

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As another example, if I look at this, this system that multiplies

the input by a time varying gain, so this.

C is a gain, but changes according to the location of the pixel n1 and n2.

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It is rather straightforward for you to verify that.

So the system is linear, but is

not spatially

invariant or it is spacially varying.

That's another way to express it.

Again, this particular property faults through

everything we talked about here for,

calls to for one dimensional, three

dimensional, multi-dimensional in general systems and signals.

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Now, such systems can be completely described by a signal that

we'll, we'll be referring to as the impulse response of the system.

As the name implies, if I put a delta at the input of sight

system and measure the output which I'll denote by h, and oneand two.

This is again the response of the system to an impulse, and we will refer

to it as the impulse

response of the LSI system.

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This is not just a mathematical construct, but in

many cases, I can utilize the system, such as a

camera and point the camera to a printout that is,

a black background and a white spot in the middle.

Or I can point the telescope that is orbiting

outside the atmosphere, such as the Hubble Space Telescope, to

a distant star in the dark sky measure that

is force and that's the impulse response of the system.

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K1 minus infinity, infinity, k2 minus infinity, infinity, x k1,

k2, h, n1 minus k1, n2 minus k2.

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I'll show some examples of performant convolution in the following slides.

It's easy to verify by substituting variables

that the convolution he's the communitive property so in other words

convolution of x with a equals convolution of a with x.