Review_of_Linear_Systems

Review_of_Linear_Systems - Digital Image Processing 3rd ed...

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Unformatted text preview: Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com © 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Review of Linear Systems Objective To provide background material in support of topics in Digital Image Processing that are based on linear system theory. Review Linear Systems Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com © 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Review of Linear Systems 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. Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com © 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Review of Linear Systems Some Definitions (Con’t) 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 where H is the system operator , defined as a mapping or assignment of a member of the set of possible outputs { g ( x )} to each member of the set of possible inputs { f ( x )}. In other words, the system operator completely characterizes the system response for a given set of inputs { f ( x )}. Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com © 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Review of Linear Systems Some Definitions (Con’t) An operator H is called a linear operator for a class of inputs { f ( x )} if for all f i ( x ) and f j ( x ) belonging to { f ( x )}, where the a 's are arbitrary constants and is the output for an arbitrary input f i ( x ) ∈ { f ( x )}. Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com © 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Review of Linear Systems Some Definitions (Con’t) The system described by a linear operator is called a linear system (with respect to the same class of inputs as the operator). The property that performing a linear process on the sum of inputs is the same that performing the operations individually and then summing the results is called the property of additivity ....
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Review_of_Linear_Systems - Digital Image Processing 3rd ed...

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