1Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com© 2001 R. C. Gonzalez & R. E. WoodsObjectiveObjectiveTo provide background material in support of topics in Digital Image Processing that are based on linear system theory.ReviewLinear SystemsReviewLinear Systems2Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com© 2001 R. C. Gonzalez & R. E. WoodsReview: Linear SystemsReview: Linear SystemsSome DefinitionsSome DefinitionsWith 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.3Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com© 2001 R. C. Gonzalez & R. E. WoodsReview: Linear SystemsReview: Linear SystemsSome 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 writewhere 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 responsefor a given set of inputs {f(x)}.4Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com© 2001 R. C. Gonzalez & R. E. WoodsReview: Linear SystemsReview: Linear SystemsSome Definitions (Con’t)Some Definitions (Con’t)An operator H is called a linear operator for a class of inputs {f(x)} iffor all fi(x) and fj(x) belonging to {f(x)}, where the a's are arbitrary constants andis the output for an arbitrary input fi(x) ∈{f(x)}.[ ( ) ( )] [ ( )] [ ( )]() ()ii j j i i j jii j jHfx fx Hfx Hfxgx g xαα α ααα+= +=+5Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com© 2001 R. C. Gonzalez & R. E. WoodsReview: Linear SystemsReview: Linear SystemsSome 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. The property that the response of a linear system to a constant times an input is the same as the response to the original inputmultiplied by a constant is called the property of homogeneity.6Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com© 2001 R. C. Gonzalez & R. E. WoodsReview: Linear SystemsReview: Linear SystemsSome Definitions (Con’t)Some Definitions (Con’t)An operator H is called time invariant (if x represents time), spatially invariant (if x is a spatial variable), or simply fixed parameter, for some class of inputs {f(x)} iffor all fi(x) ∈{f(x)} and for all x0. A system described by a fixed-parameter operator is said to be a fixed-parameter system. Basically all this means is that offsetting the independent variable of the input by x0causes the same offset in the independent variable of the output. Hence, the input-output relationship remains the same.7Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com© 2001 R. C. Gonzalez & R. E. WoodsReview: Linear SystemsReview: Linear SystemsSome Definitions (Con’t)Some Definitions (Con’t)An operator H is said to be causal, and hence the system described by H is a causal system, if there is no output before there is an input. In other words,Finally, a linear system H is said to be stable if its response to any bounded input is bounded. That is, ifwhere K and c are constants.8Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com© 2001 R. C. Gonzalez & R. E. WoodsReview: Linear SystemsReview: Linear SystemsSome Definitions (Con’t)Some Definitions (Con’t)Example: Suppose that operator H is the integral operator between the limits −∞ and x. Then, the output in terms of the input is given bywhere w is a dummy variable of integration. This system is linear because9Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com© 2001 R. C. Gonzalez & R. E. WoodsReview: Linear SystemsReview: Linear SystemsSome Definitions (Con’t)Some Definitions (Con’t)We see also that the system is fixed parameter because where d(w + x0) = dw because x0is a constant. Following similar manipulation it is easy to show that this system also iscausal and stable.0000() ()()[( )]xxxgxxfwdwfsxdsHfx x+−∞−∞+==+=+∫∫10Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com© 2001 R. C. Gonzalez & R. E. WoodsReview: Linear SystemsReview: Linear SystemsSome Definitions (Con’t)Some Definitions (Con’t)Example: Consider now the system operator whose output is the inverse of the input so that In this case,so this system is not linear. The system, however, is fixed parameter and causal.11Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com© 2001 R. C. Gonzalez & R. E. WoodsReview: Linear SystemsReview: Linear SystemsLinear System Characterization-ConvolutionLinear System Characterization-ConvolutionA unit impulse function, denoted δ(x − a), is defined by the expressionFrom the previous sections, the output of a system is given by g(x) = H[f(x)]. But, we can express f(x) in terms of the impulse function just defined, so12Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com© 2001 R. C. Gonzalez & R. E. WoodsReview: Linear SystemsReview: Linear SystemsSystem Characterization (Con’t)System Characterization (Con’t)Extending the property of addivity to integrals (recall that an integral can be approximated by limiting summations) allows us to writeBecause f(α) is independent of x, and using the homogeneity property, it follows that13Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com©