20-08-2012, 04:49 PM
Digital Image Processing
Some Definitions
With reference to the following figure, we define a system as a unit that converts an input function f(x) into an output (or response) function g(x), where x is an independent variable, such as time or, as in the case of images, spatial position. We assume for simplicity that x is a continuous variable, but the results that will be derived are equally applicable to discrete variables.
It is required that the system output be determined completely by the input, the system properties, and a set of initial conditions. From the figure in the previous page, we write
System Characterization
is called the impulse response of H. In other words, h(x, ) is the response of the linear system to a unit impulse located at coordinate x (the origin of the impulse is the value of that produces (0); in this case, this happens when = x).
This expression is called the convolution integral. It states that the response of a linear, fixed-parameter system is completely characterized by the convolution of the input with the system impulse response. As will be seen shortly, this is a powerful and most practical result.
These results are the basis for all the filtering work done in Chapter 4, and some of the work in Chapter 5 of Digital Image Processing. Those chapters extend the results to two dimensions, and illustrate their application in considerable detail.
Some Definitions
With reference to the following figure, we define a system as a unit that converts an input function f(x) into an output (or response) function g(x), where x is an independent variable, such as time or, as in the case of images, spatial position. We assume for simplicity that x is a continuous variable, but the results that will be derived are equally applicable to discrete variables.
It is required that the system output be determined completely by the input, the system properties, and a set of initial conditions. From the figure in the previous page, we write
System Characterization
is called the impulse response of H. In other words, h(x, ) is the response of the linear system to a unit impulse located at coordinate x (the origin of the impulse is the value of that produces (0); in this case, this happens when = x).
This expression is called the convolution integral. It states that the response of a linear, fixed-parameter system is completely characterized by the convolution of the input with the system impulse response. As will be seen shortly, this is a powerful and most practical result.
These results are the basis for all the filtering work done in Chapter 4, and some of the work in Chapter 5 of Digital Image Processing. Those chapters extend the results to two dimensions, and illustrate their application in considerable detail.