Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 LECTURE 1: M-D SIGNALS AND TRANSFORMS • M-D Signals – Finite-Extent Signals and Periodic Signals – Symmetric Signals – Special signals • M-D Transforms – – – – – MD-FT for Continuous Signals MD-FT for Discrete Signals MD-DFT MD-DCT MD-Wavelet Chapter 1 Multi-dimensional Signals and Systems 1 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 Finite-Extent Signals Finite-extent signals are defined over a finite support. • Quarter-plane (QP) support • Half-plane (HP) support • Non-symmetric half-plane (NSHP) support • Wedge support n2 n1 Chapter 1 Multi-dimensional Signals and Systems 2 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 Periodic Signals ̃ • Definition: ̃ | • Arbitrary periodicity • Rectangular-periodicity: Chapter 1 Multi-dimensional Signals and Systems and diagonal 3 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 Periodic Extension of Finite-Extent Signals • Finite-extent signals and periodic signals are isomorphic to each other. – Given a periodic signal, the main period is finite-extent – Given a finite extent signal, we can always define its periodic extension ̃ , Chapter 1 Multi-dimensional Signals and Systems , 4 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 Symmetric Signals • Two-fold (NSHP) symmetry , • Four-fold (QP) symmetry , , , , • Circular symmetry – A signal , is circularly symmetric if it is only a from the origin. Circular function of distance symmetry implies four-fold symmetry. Chapter 1 Multi-dimensional Signals and Systems 5 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 Separable Signals • An MD signal (function) is separable if , ,⋯, ⋯ , can be represented by • A finite support 2D signal a matrix . If the signal is separable, then the matrix can be written as the outer product , where the vectors and denote samples of 1D signals and , respectively. • While a general matrix has degrees of freedom, the outer product has degrees of freedom. Chapter 1 Multi-dimensional Signals and Systems 6 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 Special Signals: 2D Kronecker Delta • MD Kronecker-delta signal , 1 0 ,⋯, ⋯ 0 otherwise • 2D discrete time signals can be expressed as shifted and weighted sum of Kronecker delta signals. s(1,1) s(0,1) s(3,1) s(0,0) n2 s(4,2) s(1,0) n1 Chapter 1 Multi-dimensional Signals and Systems 7 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 Special Signals: 2D Spatial Frequency Patterns • Horizontal pattern , cos , cos • Vertical pattern • 45-degree orientation , • Complex exponential cos – A 2D discrete complex exponential signal , is rectangularly periodic in where and , , , and with period , if and are unitless integers, and the units of are radians. Chapter 1 Multi-dimensional Signals and Systems 8 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 2D Fourier Transform of Continuous Signals • Forward transform , , , • Inverse transform 1 , 2 , , • MD Fourier transform is complex , , , • Frequency variables 2 , cycles/mm, Chapter 1 Multi-dimensional Signals and Systems , , radians/mm 9 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 Convergence of the Fourier Transform The 2D signal , may have an infinite extent. • Uniform convergence: The double integral converges uniformly and is a continuous function of and if ∬ • • , ∞; i.e., , is absolutely integrable. , exists but has discontinuities, then a Mean-square convergence: If weaker form of convergence applies. For example, , is not absolutely integrable, but its Fourier transform sin ⁄ sin ⁄ converges in the mean square sense. We observe the Gibbs effect around points of discontinuity. Generalized convergence: In some cases, neither uniform nor mean-square convergence applies, but , may still be defined using the Dirac delta function , . For example, , 1 for all , , is not absolutely integrable, but its Fourier transform is defined in the generalized , , . sense as Chapter 1 Multi-dimensional Signals and Systems 10 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 2D Fourier Transform: Coordinate Transforms • Let , , denote an affine transformation of coordinates. If , is the 2D Fourier transform of , , then the 2D Fourier transform , is given by of 1 , , 1, , • Translation: 0, • Rotation: 0. , sin,; where cos, 1. , cos Chapter 1 Multi-dimensional Signals and Systems sin , sin cos 11 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 Projection Slice Theorem • Let , have Fourier transform denote the Radon transform of , cos which projects Then, , sin , , , and defined by sin cos to a line through the origin with angle . cos , sin denotes the 1D Fourier transform of for where each angle . • The projection-slice theorem is fundamental to how several medical imaging modalities, e.g., computer tomography, works. Chapter 1 Multi-dimensional Signals and Systems 12 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 Projection Slice Thm. (cont’d) • Proof for the case Projection of , 0 on the horizontal axis is defined by , Taking the 1D Fourier transform of the projection yields , , Chapter 1 Multi-dimensional Signals and Systems ,0 13 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 2D Fourier Transform of Discrete Signals • Forward transform is periodic with period 2 , , • Inverse transform 1 , 2 • Properties: – , real implies – , , 2 , , has conjugate symmetry. is two‐fold symmetric implies ∗ , → , Chapter 1 Multi-dimensional Signals and Systems , ∗ is real , 14 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 2D Discrete Fourier Transform • 2D DFT can be obtained by sampling , of a finite-extent signal , or by computing the Fourier series coefficients of the periodic extension ̃ , , , • Normalized frequency variables 2 and ∆ ∆ 2 Chapter 1 Multi-dimensional Signals and Systems and 2 ∆ ∆ 2 15 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 Computation of 2D Discrete Fourier Transform • Since 2D complex exponentials are separable, 2D DFT can be computed as a cascade of two 1D DFTs, first on the rows of , , then on the columns of , as , , where , is the 1D DFT over the row Chapter 1 Multi-dimensional Signals and Systems , of the image. 16 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 Computation of 2D IDFT • Inverse 2D DFT can be computed using the forward FFT algorithm by first conjugating , , then computing 2D forward DFT, and again taking the complex conjugate of the result, since we have 1 , 1 Chapter 1 Multi-dimensional Signals and Systems , ∗ ∗ , 17 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 Properties of 2D Discrete Fourier Transform • 2D DFT is rectangularly periodic , , , • , , real implies for all is Hermitian symmetry | 0,0 | | 1,0 | | 2,0 | | 3,0 | 2,0 1,0 | 0,1 | | 1,1 | | 2,1 | | 3,1 | | 4,1 | | 5,1 | | 0,2 | | 1,2 | | 2,2 | | 3,2 | | 4,2 | | 5,2 | | 0,3 | | 1,3 | | 2,3 | | 3,3 | 2,3 1,3 0,2 5,2 4,2 3,2 2,2 0,1 5,1 4,1 3,1 2,1 0 1,0 2,0 0 0,1 1,1 2,1 3,1 4,1 5,1 0,2 1,2 2,2 3,2 4,2 5,2 0 1,3 2,3 0 2,3 1,3 2,0 1,2 1,1 1,0 0,2 5,2 4,2 3,2 2,2 1,2 0,1 5,1 4,1 3,1 2,1 1,1 Chapter 1 Multi-dimensional Signals and Systems , 18 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 Properties of 2D DFT (cont’d) • Circular Shift , ↔ , • Circular Convolution , ⊛⊛ , ↔ , , • Parseval’s Theorem: , Chapter 1 Multi-dimensional Signals and Systems 1 , 19 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 2D Discrete Cosine Transform • Symmetric extension for type II DCT is given by 2 1 , 0 • The algorithm to compute -point DCT using 2 -point FFT is as follows: of 2 -point signal 1) Form 2 -point symmetrically extended signal . 2) Compute , 0, … , 2 1, the 2 -point DFT of 3) ⁄ , 0, … , 1 . 1, where • The high frequency coefficients of the DCT contains less energy compared with those of the DFT due to symmetric extension. Chapter 1 Multi-dimensional Signals and Systems 20 Prof. A. Murat Tekalp Digital Video Processing, 2E, Prentice Hall, 2015 2D Discrete Wavelet Transform (DWT) • 2D DWT is a multi-scale image representation. • Discussion of 2D DWT is delayed until we study FIR filtering and multi-scale image representations in Chapter 3. Chapter 1 Multi-dimensional Signals and Systems 21 @Copyright 2015 Prof. A. Murat Tekalp Digital Video Processing Fall 2015 Display of 2D Signals and Transforms • Two-dimensional functions can be displayed as gray-scale plots, isometric (surface) plots, or contour plots, which are supported by MATLAB. • Appropriate scaling is important for gray-scale plots, e.g., to display 2-D Fourier transform of images, or to compare multiple images. Common approaches for scaling are linear min/max scaling and nonlinear scaling. For example, log scaling for Fourier magnitude is given by • • , log 1 , where D() denotes the display image and F() denotes the actual Fourier transform. Surface plots give the appearance of 3D drawing. A wire mesh or a shaded solid can represent the surface. Some portions of the surface may be occluded. This is very useful to display point spread functions and frequency response of systems. In contour plots, all points that have a specific value are connected to form a continuous line. This type of plot is useful locating minima and maxima of twodimensional functions. Chapter 1 Multi-dimensional Signals and Systems 22
