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. 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. We'll be introducing them in this segment. These two properties define an important subset of all possible 2d systems. 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. As we'll see in the next two segments, we can describe and implement LSI systems efficiently in the spatial domain through convolution. We can also describe them in the frequency domain as we'll see in week three. They're therefore friendly systems. In that they reveal their properties to us in a rather straightforward way, but the same time, extremely useful and widely used in a number of applications. So, let us see how LSI systems can be defined mathematically, and also let us look at some simple examples. 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. Some examples of such signals y n1 n2 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. We going to have another system that performs this transformation to the input values according to this function. 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 of the input values in a neighborhood denoted here by n. So this denotes a neighborhood of input values. 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, whether it's causal, whether it's linear, and whether it's spatial invariant. 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. 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. 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. 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 equals x2. Since this property holds for any x1, x2. And then you wait. Then, I see that the response to alpha one x one, minus alpha one x one, which is equal to zero, right? So the response to the zero signal Is equal to 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. 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. 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. This properties important by itself, but even more important when combined with linearity as we're going to see right away. Now this property is independent of linearities, so if we consider the system we looked at earlier, which for an input x n1, n2 generates as output 255 minus x n1, n2. So the system finds a negative of an eight bit image. We saw that this system is non-linear. Now, if I shift the input to the system by x n1 minus k1. So, I shifted by k1, k2. And I put this as input to the system. The output is equal to 255 minus x n1 minus k1, n2 minus k2, which is clearly Equal to the shifted output. Therefore, the system is spatially invariant, SI. So, the system takes the negative of an image is no linear, but is spatially Spatially invariant. 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. 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. Let us look now at systems that are both linear and spatially invariant LSI systems. Such systems are used widely, and we have developed very useful and convenient tools to describe and analyze them. 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. 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. Knowing the impulse response now again I can completely describe the system which means for any input X n1, n2. I can find the output of the system, and this is equal to the convolution of the input, with the impulse response of the system. So this double star here denotes the two dimensional discrete convolution. So linear systems or filters, as we often refer to, are perform convolution, discrete convolution. The convolution of x with h, as we will show, is equal to this super position sum. K1 minus infinity, infinity, k2 minus infinity, infinity, x k1, k2, h, n1 minus k1, n2 minus k2. 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.
