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Unformatted text preview: Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com 19922008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Matrices and Vectors Objective To provide background material in support of topics in Digital Image Processing that are based on matrices and/or vectors. Review Matrices and Vectors Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com 19922008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Matrices and Vectors An m n (read "m by n") matrix , denoted by A , is a rectangular array of entries or elements (numbers, or symbols representing numbers) enclosed typically by square brackets, where m is the number of rows and n the number of columns. Some Definitions Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com 19922008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Matrices and Vectors Definitions (Cont) A is square if m = n . A is diagonal if all offdiagonal elements are 0, and not all diagonal elements are 0. A is the identity matrix ( I ) if it is diagonal and all diagonal elements are 1. A is the zero or null matrix ( ) if all its elements are 0. The trace of A equals the sum of the elements along its main diagonal. Two matrices A and B are equal iff the have the same number of rows and columns, and a ij = b ij . Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com 19922008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Matrices and Vectors Definitions (Cont) The transpose A T of an m n matrix A is an n m matrix obtained by interchanging the rows and columns of A . A square matrix for which A T = A is said to be symmetric . Any matrix X for which XA = I and AX = I is called the inverse of A . Let c be a real or complex number (called a scalar ). The scalar multiple of c and matrix A , denoted c A , is obtained by multiplying every elements of A by c . If c =  1, the scalar multiple is called the negative of A . Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com 19922008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Matrices and Vectors Definitions (Cont) A column vector is an m 1 matrix: A row vector is a 1 n matrix: A column vector can be expressed as a row vector by using the transpose: Digital Image Processing, 3rd ed. www.ImageProcessingPlace.com 19922008 R. C. Gonzalez & R. E. Woods Gonzalez & Woods Matrices and Vectors Some Basic Matrix Operations The sum of two matrices A and B (of equal dimension), denoted A + B , is the matrix with elements a ij + b ij . The difference of two matrices, A B , has elements a ij  b ij . The product , AB , of m n matrix A and p q matrix B , is an m q matrix C whose ( i , j )th element is formed by multiplying the entries across the i th row of A times the entries down the j th column of B ; that is, Digital Image Processing, 3rd ed....
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This note was uploaded on 01/27/2010 for the course ECE TECHCOMM taught by Professor Prof.soltanianzdeh during the Spring '10 term at Shahid Beheshti University.
 Spring '10
 Prof.Soltanianzdeh
 Image processing

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