Digital Image Processing 2D SYSTEMS & PRELIMINARIES Hamid R. Rabiee Fall 2015
Outline 2 Two Dimensional Fourier & Z-transform Toeplitz & Circulant Matrices Orthogonal & Unitary Matrices Block Matrices Diagonal Forms Kronecker Products Discrete Random Fields Spectral Density Function Rate Distortion Function
Two Dimensional Delta Functions 3 Defined from one-dimensional delta functions: Continuous: Discrete: Which satisfy below equalities: Continuous: Discrete:
Impulse Response 4 Consider a system with output and input sequences y(m,n) & x(m,n). we will define Transformation function of this system as: When x is Kronecker delta at location (m,n ) the output at (m,n) is called Impulse Response of the system:
Linear Systems 5 A system is called linear iff any combination of inputs produce same combination of their respective outputs: For any linear system we have:
Shift Invariant Systems 6 A system is called Spatially Invariant or Shift Invariant if a translation of the input causes a translation of output: Which results in: For any shift invariant system we have:
The Fourier Transform 7 Fourier transform of image x(m,n) is defined as: Which is periodic with period 2π in each argument: Inverse Fourier transform of is defined as: The Fourier transform of shift invariant system is called frequency response.
Properties of Fourier Transform 8
Z-Transform 9 The Z-transform of complex sequence x(m,n) is defined as: It is simple to find Z-transform of output: Inverse of Z-transform is: Contour On ROC What is the relation of Fourier transform & Z-transform?
Properties of Z-Transform 10
Causality & Stability 11 A linear shift Invariant system is stable if: Which implies that ROC of must include unit circles: It is Causal if its impulse response is nonzero only when both arguments are positive.
Optical & Modulation Transfer 12 Functions For a shift invariant imaging system its optical transfer function is: And its modulation transfer function is:
Toeplitz Matrices 13 A matrix is called Toeplitz if or each element is only depend on difference between its indices: For an N N Toeplitz matrix how many elements is needed to completely define it? Toeplitz matrices describe: input-output transformations of one-dimensional linear shift invariant systems correlation matrices of stationary sequences.
Circulant Matrices 14 A matrix C is called circulant if each of its rows (or columns) is a circular shift of the previous row (or column), i.e. c(m, n) = c ((m - n) modulo N). For an N N Circulant matrix how many elements is needed to completely define it? Circulant Matrices describe: input-output behavior of one-dimensional linear periodic systems. correlation matrices of periodic sequences.
Orthogonal and Unitary Matrices 15 Matrix A is orthogonal if or It is unitary if : or Which is unitary? Which is orthogonal?
Diagonal Forms 16 For any Hermitian matrix R there exist a unitary matrix such that: A diagonal matrix with eigenvalues of R Eigenmatrix of R
Kronecker Products 17 Kronecker product of two matrices A and B is defined as: Are and Equal?
Properties of Kronecker Products 18
Block Matrices 19 Block Toeplitz Block Circulant Toeplitz Block Toeplitz Circulant If Both of above Doubly Block Toeplitz If Both of above Doubly Block Circulant
Discrete Random Fields 20 When each sample of a two-dimensional sequence is a random variable, we call it a discrete random field. For a complex random field: We call this random field wide-sense stationary if: It also called shift invariant, translational (or spatial) invariant and homogeneous.
Properties of Random fields 21
Spectral Density Function 22 For stationary random field covariance generating function (CGF) is: Spectral density function or (SDF) is:
Properties of SDF 23
Entropy 24
Rate Distortion Function 25 The distortion function of x (which is a Gaussian random variable with variance and y is its reproduction) is defined as: Where D is:
Relation Between Distortion and 26 Rate
End of Lecture 3 Thank You! Tables, Pictures & equations are taken from Jain Book.