10-信息光学2.pdfVIP

#2 Review of Linear Systems and Fourier Transforms 1 Systems Imaging p1(x1, y1) System S{p1(x1, y1)} = p2 (x2 , y2 ) S{ } A system accepts an input signal and produces an output signal. Mathematically, a system can be described using an operator S{ } that maps a set of input functions onto a set of output functions. For imaging systems, the inputs and outputs are generally two dimensional complex-valued functions. 2 Examples of linear and nonlinear systems Linear System Multiply by 5 S{p(x1, y1) + q(x1, y1)} = 5p(x1, y1) + 5q(x1, y1) Linear since the input signals interact Nonlinear independently System Square 2 2 S{p(x1, y1) + q(x1, y1)} = p (x1, y1) +q (x1, y1) + 2 p(x1, y1)q(x1, y1) Not linear since the input signals interact with one another in this term. 3 Linear systems satisfy superposition and scaling properties Suppose we have a signal that can be composed of a sum of “elementary” functions. Response to an individual elementary function: Response to an input signal composed of these scaled elementary functions (inputted at the same time into the system): S{ap(x1, y1) + bq(x1, y1)} = aS{p(x1, y1)}+ bS{q(x1, y1)} where a, b are constants (can be complex-valued) 4 Properties of Linear Systems The system treats each of the elementary functions p(x1,y1) and q(x1,y1) independently. S{ap(x1, y1) +