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e178-04L3

# e178-04L3 - Digital Image Processing 2nd ed...

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1 Digital Image Processing, 2nd ed. Digital Image Processing, 2nd ed. www. imageprocessingbook .com © 2001 R. C. Gonzalez & R. E. Woods Objective Objective To provide background material in support of topics in Digital Image Processing that are based on linear system theory. Review Linear Systems Review Linear Systems

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2 Digital Image Processing, 2nd ed. Digital Image Processing, 2nd ed. www. imageprocessingbook .com © 2001 R. C. Gonzalez & R. E. Woods Review: Linear Systems Review: Linear Systems Some Definitions 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.
3 Digital Image Processing, 2nd ed. Digital Image Processing, 2nd ed. www. imageprocessingbook .com © 2001 R. C. Gonzalez & R. E. Woods Review: Linear Systems Review: Linear Systems Some Definitions (Con’t) 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 )}.

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4 Digital Image Processing, 2nd ed. Digital Image Processing, 2nd ed. www. imageprocessingbook .com © 2001 R. C. Gonzalez & R. E. Woods Review: Linear Systems Review: Linear Systems Some Definitions (Con’t) 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 )}.
5 Digital Image Processing, 2nd ed. Digital Image Processing, 2nd ed. www. imageprocessingbook .com © 2001 R. C. Gonzalez & R. E. Woods Review: Linear Systems Review: Linear Systems Some Definitions (Con’t) 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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