3. Lecture. Fourier Transformation Sampling
|
|
- Virgil Jefferson
- 5 years ago
- Views:
Transcription
1 3. Lecture Fourier Transformation Sampling Some slides taken from Digital Image Processing: An Algorithmic Introduction using Java, Wilhelm Burger and Mark James Burge
2 Separability ² The 2D DFT can be separated in two 1D DFT's: F (u; v) = 1 MN = 1 M M 1 X x=0 M 1 X x=0 N 1 X y=0 f(x; y)e j2¼(ux=m+vy=n) 2 4e j2¼ux=m 1 N N 1 X y=0 f(x; y)e j2¼vy=n ² The DFT can be obtained in two successive applications of 1D transforms 3 5 2
3 Separability ² First compute the 1D DFT's for all the rows ² Second compute the 1D DFT's for all the columns ² And each of the 1D DFT's can be carried out as FFT of course 3
4 Separability ² Also the inverse transform can be separated: f(x; y) = M 1 X u=0 = M 1 X u=0 N 1 X F (u; v)e j2¼(ux=m+vy=n) v=0 2 N 1 j2¼ux=m X 4e v=0 F (u; v)e j2¼vy=n 3 5 4
5 Example: Separability ² Matlab example using the 2D FFT function: fft2(magic(3)) ans = i i i i i i ² Matlab example using two 1D FFT function calls: fft(fft(magic(3)).').' ans = i i i i i i 5
6 Symmetry and periodicity ² If f(x; y) is real (e.g. an image), its Fourier transform is conjugate symmetric ² Additionally the DFT is in nitely periodic 6
7 Symmetry and periodicity ² The same property holds for the 2D case 7
8 Translation in the Fourier domain ² A translation (u 0 ; v 0 ) in the Fourier domain result in F (u u 0 ; v v 0 ) () f(x; y)e j2¼(u 0x+v 0 y)=n ² The origin of the Fourier domain is shifted to the point (u 0 ; v 0 ) ² A special case is u 0 = v 0 = N=2, here the exponential term e j¼(x+y) 1 x+y f(x; y)( 1) x+y () F (u N=2; v N=2) ² The origin will be moved from (0; 0) to the centre of the image 8
9 Example: Translation 9
10 Rotation ² A rotation in the spatial domain rotates the Fourier domain by the same angle and vice versa. 10
11 Examples: DFT image scaling. The rectangular pulse in the image function (a c) creates a strongly oscillating power spectrum (d f), as in the onedimensional case. Stretching the image causes the spectrum to contract and vice versa. 11
12 Examples: DFT oriented, repetitive patterns. The image function (a c) contains patterns with three dominant orientations, which appear as pairs of corresponding frequency spots in the spectrum (c f). Enlarging the image causes the spectrum to contract. 12
13 Examples: DFT image rotation. The original image (a) is rotated by 15deg (b) and 30deg (c). The corresponding (squared) spectrum turns in the same direction and by exactly the same amount (d f). 13
14 Examples: DFT superposition of image patterns. Strong, oriented subpatterns (a c) are easy to identify in the corresponding spectrum (d f). Notice the broadband effects caused by straight structures, such as the dark beam on the wall in (b, e). 14
15 Examples: DFT natural image patterns. Examples of repetitive structures in natural images (a c) that are also visible in the corresponding spectrum (d f). 15
16 Examples: DFT natural image patterns with no dominant orientation. The repetitive patterns contained in these images (a c) have no common orientation or sufficiently regular spacing to stand out locally in the orresponding Fourier spectra (d f). 16
17 Examples: DFT of a print pattern. The regular diagonally oriented raster pattern (a, b) is clearly visible in the corresponding power spectrum (c). It is possible (at least in principle) to remove such patterns by erasing these peaks in the Fourier spectrum and reconstructing the smoothed image from the modified spectrum using the inverse DFT. 17
18 Correcting the geometry Correcting the geometry of the 2D spectrum. Original image (a) with dominant oriented patterns that show up as clear peaks in the corresponding spectrum (b). Because the image and the spectrum are not square (M = N), orientations in the image are not the same as in the actual spectrum (b). After the spectrum is scaled to square size (c), we can clearly observe that the cylinders of this (Harley-Davidson V-Rod) engine are really spaced at a 60 angle. 18
19 Windowing Effects of periodicity in the 2D spectrum. The discrete Fourier transform is computed under the implicit assumption that the image signal is periodic along both dimensions (top). Large differences in intensity at opposite image borders here most notably in the vertical direction lead to broad-band signal components that in this case appear as a bright line along the spectrum s vertical axis (bottom). 19
20 Windowing 20
21 Fourier basis functions ² Fourier transform represents signal in terms of sine and cosine (basis functions) ² Magnitude gives frequency, direction gives orientation ² Fourier basis element e j2¼(ux+vy) = cos(2¼(ux + vy)) j sin(2¼(ux + vy)) 21
22 Fourier basis functions ² Here u and v are larger than in the previous slide 22
23 Fourier basis functions ² And larger still 23
24 Fourier basis functions ² Fourier basis elements e j2¼(ux+vy) = cos(2¼(ux + vy)) j sin(2¼(ux + vy)) 24
25 Convolution and Correlation ² Convolution: Operator on two sequences that represents a ltering operation ² Correlation: Correlation is a measure of similarity between two signals 25
26 ² Discrete Convolution f(x) g(x) = 1 M M 1 X m=0 f(m)g(x m) ² M A + B 1 where A is the length of f and B is the length of g ² f(x; y) g(x; y) = 1 MN M 1 X m=0 N 1 X n=0 f(m; n)g(x m; y n) ² A B and C D are the arrays for f(x; y) and g(x; y) M A + C 1 N B + D 1 26
27 Example: Convolution 27
28 Example: 2D convolution 28
29 Discrete Correlation ² The correlation of two functions f(x) and g(x) is: 1D : f(x) ± g(x) = 1 M M 1 X m=0 2D : f(x; y)±g(x; y) = 1 MN ² is the complex conjugate f (m)g(x + m) M 1 X m=0 N 1 X n=0 f (m; n)g(x+m; y+n) 29
30 Correlation theorem A correlation in the spatial domain is a multiplication with the complex conjugate in the Fourier domain and vice versa. f(x) ± g(x), F (u)g(u) and f(x; y) ± g(x; y), F (u; v)g(u; v) f (x)g(x), F (u) ± G(u) f (x; y)g(x; y), F (u; v) ± G(u; v) 30
31 Example: Correlation 31
32 Example: 2D correlation 32
33 Sampling ² Sampling can be described as the multiplication of a signal with a sequence of impulse functions 33
34 Sampling with the impulse function 34
35 The comb function 35
36 The comb function 36
37 Sampling in Fourier domain ² Multiplication of signal with comb function in time domain, is convolution of them in Fourier domain g(x)iii(x), G(u) III(u) 37
38 Sampling: Fourier domain f(x) A 0 X x 38
39 Reconstruction filters 50 Square pixels 100 Gaussian reconstruction filter spatial Fourier spatial Fourier Bilinear interpolation Perfect reconstruction filter spatial Fourier spatial Fourier
40 Image reconstruction: pixelization square pixels Harmon & Julesz 1973 Gaussian reconstruction 40
41 Sampling: Aliasing effect 41
42 Nyquist theorem ² Nyquist theorem: The sampling frequency must be at least twice the highest frequency! s 2! ² If this is not the case the signal needs to be bandlimited before sampling, e.g. with a lowpass lter 42
43 Aliasing effect ² Sampling without smoothing: Top row shows the images, sampled at every second pixel to get the next; bottom row shows the magnitude spectrum of these images. ² Nyquist criterium not ful lled (aliasing artifacts) 43
44 Aliasing effect ² Sampling with smoothing: We get the next image by smoothing the image with a Gaussian with sigma 1 pixel, then sampling at every second pixel. 44
G52IVG, School of Computer Science, University of Nottingham
Image Transforms Fourier Transform Basic idea 1 Image Transforms Fourier transform theory Let f(x) be a continuous function of a real variable x. The Fourier transform of f(x) is F ( u) f ( x)exp[ j2πux]
More informationSEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis
SEISMIC WAVE PROPAGATION Lecture 2: Fourier Analysis Fourier Series & Fourier Transforms Fourier Series Review of trigonometric identities Analysing the square wave Fourier Transform Transforms of some
More informationECG782: Multidimensional Digital Signal Processing
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu ECG782: Multidimensional Digital Signal Processing Filtering in the Frequency Domain http://www.ee.unlv.edu/~b1morris/ecg782/ 2 Outline Background
More informationFILTERING IN THE FREQUENCY DOMAIN
1 FILTERING IN THE FREQUENCY DOMAIN Lecture 4 Spatial Vs Frequency domain 2 Spatial Domain (I) Normal image space Changes in pixel positions correspond to changes in the scene Distances in I correspond
More informationECG782: Multidimensional Digital Signal Processing
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu ECG782: Multidimensional Digital Signal Processing Spring 2014 TTh 14:30-15:45 CBC C313 Lecture 05 Image Processing Basics 13/02/04 http://www.ee.unlv.edu/~b1morris/ecg782/
More information2. Image Transforms. f (x)exp[ 2 jπ ux]dx (1) F(u)exp[2 jπ ux]du (2)
2. Image Transforms Transform theory plays a key role in image processing and will be applied during image enhancement, restoration etc. as described later in the course. Many image processing algorithms
More informationLecture 13: Implementation and Applications of 2D Transforms
Lecture 13: Implementation and Applications of 2D Transforms Harvey Rhody Chester F. Carlson Center for Imaging Science Rochester Institute of Technology rhody@cis.rit.edu October 25, 2005 Abstract The
More informationFourier Transform 2D
Image Processing - Lesson 8 Fourier Transform 2D Discrete Fourier Transform - 2D Continues Fourier Transform - 2D Fourier Properties Convolution Theorem Eamples = + + + The 2D Discrete Fourier Transform
More informationGBS765 Electron microscopy
GBS765 Electron microscopy Lecture 1 Waves and Fourier transforms 10/14/14 9:05 AM Some fundamental concepts: Periodicity! If there is some a, for a function f(x), such that f(x) = f(x + na) then function
More informationLecture 4 Filtering in the Frequency Domain. Lin ZHANG, PhD School of Software Engineering Tongji University Spring 2016
Lecture 4 Filtering in the Frequency Domain Lin ZHANG, PhD School of Software Engineering Tongji University Spring 2016 Outline Background From Fourier series to Fourier transform Properties of the Fourier
More informationFourier Transforms For additional information, see the classic book The Fourier Transform and its Applications by Ronald N. Bracewell (which is on the shelves of most radio astronomers) and the Wikipedia
More informationDigital Image Processing. Image Enhancement: Filtering in the Frequency Domain
Digital Image Processing Image Enhancement: Filtering in the Frequency Domain 2 Contents In this lecture we will look at image enhancement in the frequency domain Jean Baptiste Joseph Fourier The Fourier
More informationDigital Image Processing
Digital Image Processing Image Transforms Unitary Transforms and the 2D Discrete Fourier Transform DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON What is this
More informationIntroduction to Computer Vision. 2D Linear Systems
Introduction to Computer Vision D Linear Systems Review: Linear Systems We define a system as a unit that converts an input function into an output function Independent variable System operator or Transfer
More informationElec4621 Advanced Digital Signal Processing Chapter 11: Time-Frequency Analysis
Elec461 Advanced Digital Signal Processing Chapter 11: Time-Frequency Analysis Dr. D. S. Taubman May 3, 011 In this last chapter of your notes, we are interested in the problem of nding the instantaneous
More informationToday s lecture. The Fourier transform. Sampling, aliasing, interpolation The Fast Fourier Transform (FFT) algorithm
Today s lecture The Fourier transform What is it? What is it useful for? What are its properties? Sampling, aliasing, interpolation The Fast Fourier Transform (FFT) algorithm Jean Baptiste Joseph Fourier
More informationEmpirical Mean and Variance!
Global Image Properties! Global image properties refer to an image as a whole rather than components. Computation of global image properties is often required for image enhancement, preceding image analysis.!
More informationConvolution Spatial Aliasing Frequency domain filtering fundamentals Applications Image smoothing Image sharpening
Frequency Domain Filtering Correspondence between Spatial and Frequency Filtering Fourier Transform Brief Introduction Sampling Theory 2 D Discrete Fourier Transform Convolution Spatial Aliasing Frequency
More informationContents. Signals as functions (1D, 2D)
Fourier Transform The idea A signal can be interpreted as en electromagnetic wave. This consists of lights of different color, or frequency, that can be split apart usign an optic prism. Each component
More informationContents. Signals as functions (1D, 2D)
Fourier Transform The idea A signal can be interpreted as en electromagnetic wave. This consists of lights of different color, or frequency, that can be split apart usign an optic prism. Each component
More informationContents. Signals as functions (1D, 2D)
Fourier Transform The idea A signal can be interpreted as en electromagnetic wave. This consists of lights of different color, or frequency, that can be split apart usign an optic prism. Each component
More informationIMAGE ENHANCEMENT: FILTERING IN THE FREQUENCY DOMAIN. Francesca Pizzorni Ferrarese
IMAGE ENHANCEMENT: FILTERING IN THE FREQUENCY DOMAIN Francesca Pizzorni Ferrarese Contents In this lecture we will look at image enhancement in the frequency domain Jean Baptiste Joseph Fourier The Fourier
More informationDISCRETE FOURIER TRANSFORM
DD2423 Image Processing and Computer Vision DISCRETE FOURIER TRANSFORM Mårten Björkman Computer Vision and Active Perception School of Computer Science and Communication November 1, 2012 1 Terminology:
More informationKey Intuition: invertibility
Introduction to Fourier Analysis CS 510 Lecture #6 January 30, 2017 In the extreme, a square wave Graphic from http://www.mechatronics.colostate.edu/figures/4-4.jpg 2 Fourier Transform Formally, the Fourier
More informationCITS 4402 Computer Vision
CITS 4402 Computer Vision Prof Ajmal Mian Adj/A/Prof Mehdi Ravanbakhsh, CEO at Mapizy (www.mapizy.com) and InFarm (www.infarm.io) Lecture 04 Greyscale Image Analysis Lecture 03 Summary Images as 2-D signals
More informationDigital Image Processing. Filtering in the Frequency Domain
2D Linear Systems 2D Fourier Transform and its Properties The Basics of Filtering in Frequency Domain Image Smoothing Image Sharpening Selective Filtering Implementation Tips 1 General Definition: System
More informationImage Enhancement in the frequency domain. GZ Chapter 4
Image Enhancement in the frequency domain GZ Chapter 4 Contents In this lecture we will look at image enhancement in the frequency domain The Fourier series & the Fourier transform Image Processing in
More informationImages have structure at various scales
Images have structure at various scales Frequency Frequency of a signal is how fast it changes Reflects scale of structure A combination of frequencies 0.1 X + 0.3 X + 0.5 X = Fourier transform Can we
More informationComputer Vision. Filtering in the Frequency Domain
Computer Vision Filtering in the Frequency Domain Filippo Bergamasco (filippo.bergamasco@unive.it) http://www.dais.unive.it/~bergamasco DAIS, Ca Foscari University of Venice Academic year 2016/2017 Introduction
More informationLecture # 06. Image Processing in Frequency Domain
Digital Image Processing CP-7008 Lecture # 06 Image Processing in Frequency Domain Fall 2011 Outline Fourier Transform Relationship with Image Processing CP-7008: Digital Image Processing Lecture # 6 2
More informationFourier Transform. sin(n# x)), where! = 2" / L and
Fourier Transform Henning Stahlberg Introduction The tools provided by the Fourier transform are helpful for the analysis of 1D signals (time and frequency (or Fourier) domains), as well as 2D/3D signals
More informationIntroduction to the Fourier transform. Computer Vision & Digital Image Processing. The Fourier transform (continued) The Fourier transform (continued)
Introduction to the Fourier transform Computer Vision & Digital Image Processing Fourier Transform Let f(x) be a continuous function of a real variable x The Fourier transform of f(x), denoted by I {f(x)}
More informationDiscrete Fourier Transform
Discrete Fourier Transform DD2423 Image Analysis and Computer Vision Mårten Björkman Computational Vision and Active Perception School of Computer Science and Communication November 13, 2013 Mårten Björkman
More informationReview: Continuous Fourier Transform
Review: Continuous Fourier Transform Review: convolution x t h t = x τ h(t τ)dτ Convolution in time domain Derivation Convolution Property Interchange the order of integrals Let Convolution Property By
More informationLecture 10, Multirate Signal Processing Transforms as Filter Banks. Equivalent Analysis Filters of a DFT
Lecture 10, Multirate Signal Processing Transforms as Filter Banks Equivalent Analysis Filters of a DFT From the definitions in lecture 2 we know that a DFT of a block of signal x is defined as X (k)=
More informationReference Text: The evolution of Applied harmonics analysis by Elena Prestini
Notes for July 14. Filtering in Frequency domain. Reference Text: The evolution of Applied harmonics analysis by Elena Prestini It all started with: Jean Baptist Joseph Fourier (1768-1830) Mathematician,
More informationReview of Linear System Theory
Review of Linear System Theory The following is a (very) brief review of linear system theory and Fourier analysis. I work primarily with discrete signals. I assume the reader is familiar with linear algebra
More informationSyllabus for IMGS-616 Fourier Methods in Imaging (RIT #11857) Week 1: 8/26, 8/28 Week 2: 9/2, 9/4
IMGS 616-20141 p.1 Syllabus for IMGS-616 Fourier Methods in Imaging (RIT #11857) 3 July 2014 (TENTATIVE and subject to change) Note that I expect to be in Europe twice during the term: in Paris the week
More informationSIMG-782 Digital Image Processing Homework 6
SIMG-782 Digital Image Processing Homework 6 Ex. 1 (Circular Convolution) Let f [1, 3, 1, 2, 0, 3] and h [ 1, 3, 2]. (a) Calculate the convolution f h assuming that both f and h are zero-padded to a length
More informationSignal Processing COS 323
Signal Processing COS 323 Digital Signals D: functions of space or time e.g., sound 2D: often functions of 2 spatial dimensions e.g. images 3D: functions of 3 spatial dimensions CAT, MRI scans or 2 space,
More informationImage Processing /6.865 Frédo Durand A bunch of slides by Bill Freeman (MIT) & Alyosha Efros (CMU)
Image Processing 6.815/6.865 Frédo Durand A bunch of slides by Bill Freeman (MIT) & Alyosha Efros (CMU) define cumulative histogram work on hist eq proof rearrange Fourier order discuss complex exponentials
More informationMultiscale Image Transforms
Multiscale Image Transforms Goal: Develop filter-based representations to decompose images into component parts, to extract features/structures of interest, and to attenuate noise. Motivation: extract
More informationLecture 3: Linear Filters
Lecture 3: Linear Filters Professor Fei Fei Li Stanford Vision Lab 1 What we will learn today? Images as functions Linear systems (filters) Convolution and correlation Discrete Fourier Transform (DFT)
More informationAdditional Pointers. Introduction to Computer Vision. Convolution. Area operations: Linear filtering
Additional Pointers Introduction to Computer Vision CS / ECE 181B andout #4 : Available this afternoon Midterm: May 6, 2004 W #2 due tomorrow Ack: Prof. Matthew Turk for the lecture slides. See my ECE
More informationIB Paper 6: Signal and Data Analysis
IB Paper 6: Signal and Data Analysis Handout 5: Sampling Theory S Godsill Signal Processing and Communications Group, Engineering Department, Cambridge, UK Lent 2015 1 / 85 Sampling and Aliasing All of
More informationFiltering in Frequency Domain
Dr. Praveen Sankaran Department of ECE NIT Calicut February 4, 2013 Outline 1 2D DFT - Review 2 2D Sampling 2D DFT - Review 2D Impulse Train s [t, z] = m= n= δ [t m T, z n Z] (1) f (t, z) s [t, z] sampled
More informationLinear Operators and Fourier Transform
Linear Operators and Fourier Transform DD2423 Image Analysis and Computer Vision Mårten Björkman Computational Vision and Active Perception School of Computer Science and Communication November 13, 2013
More informationFrequency Filtering CSC 767
Frequency Filtering CSC 767 Outline Fourier transform and frequency domain Frequency view of filtering Hybrid images Sampling Slide: Hoiem Why does the Gaussian give a nice smooth image, but the square
More informationFourier Sampling. Fourier Sampling. Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2005 Linear Systems Lecture 3.
Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2005 Linear Sstems Lecture 3 Fourier Sampling F Instead of sampling the signal, we sample its Fourier Transform Sample??? F -1 Fourier
More informationDigital Image Processing
Digital Image Processing, 2nd ed. Digital Image Processing Chapter 7 Wavelets and Multiresolution Processing Dr. Kai Shuang Department of Electronic Engineering China University of Petroleum shuangkai@cup.edu.cn
More informationLecture 3: Linear Filters
Lecture 3: Linear Filters Professor Fei Fei Li Stanford Vision Lab 1 What we will learn today? Images as functions Linear systems (filters) Convolution and correlation Discrete Fourier Transform (DFT)
More informationEA2.3 - Electronics 2 1
In the previous lecture, I talked about the idea of complex frequency s, where s = σ + jω. Using such concept of complex frequency allows us to analyse signals and systems with better generality. In this
More informationFourier Series Example
Fourier Series Example Let us compute the Fourier series for the function on the interval [ π,π]. f(x) = x f is an odd function, so the a n are zero, and thus the Fourier series will be of the form f(x)
More informationInformation and Communications Security: Encryption and Information Hiding
Short Course on Information and Communications Security: Encryption and Information Hiding Tuesday, 10 March Friday, 13 March, 2015 Lecture 5: Signal Analysis Contents The complex exponential The complex
More informationImage Processing 1 (IP1) Bildverarbeitung 1
MIN-Fakultät Fachbereich Informatik Arbeitsbereich SAV/BV (KOGS) Image Processing 1 (IP1) Bildverarbeitung 1 Lecture 7 Spectral Image Processing and Convolution Winter Semester 2014/15 Slides: Prof. Bernd
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 12 Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial,
More informationAnnouncements. Filtering. Image Filtering. Linear Filters. Example: Smoothing by Averaging. Homework 2 is due Apr 26, 11:59 PM Reading:
Announcements Filtering Homework 2 is due Apr 26, :59 PM eading: Chapter 4: Linear Filters CSE 52 Lecture 6 mage Filtering nput Output Filter (From Bill Freeman) Example: Smoothing by Averaging Linear
More informationImage Acquisition and Sampling Theory
Image Acquisition and Sampling Theory Electromagnetic Spectrum The wavelength required to see an object must be the same size of smaller than the object 2 Image Sensors 3 Sensor Strips 4 Digital Image
More informationBME 50500: Image and Signal Processing in Biomedicine. Lecture 2: Discrete Fourier Transform CCNY
1 Lucas Parra, CCNY BME 50500: Image and Signal Processing in Biomedicine Lecture 2: Discrete Fourier Transform Lucas C. Parra Biomedical Engineering Department CCNY http://bme.ccny.cuny.edu/faculty/parra/teaching/signal-and-image/
More informationIntroduction to Fourier Analysis Part 2. CS 510 Lecture #7 January 31, 2018
Introduction to Fourier Analysis Part 2 CS 510 Lecture #7 January 31, 2018 OpenCV on CS Dept. Machines 2/4/18 CSU CS 510, Ross Beveridge & Bruce Draper 2 In the extreme, a square wave Graphic from http://www.mechatronics.colostate.edu/figures/4-4.jpg
More informationBiomedical Engineering Image Formation II
Biomedical Engineering Image Formation II PD Dr. Frank G. Zöllner Computer Assisted Clinical Medicine Medical Faculty Mannheim Fourier Series - A Fourier series decomposes periodic functions or periodic
More informationFiltering in the Frequency Domain
Filtering in the Frequency Domain Outline Fourier Transform Filtering in Fourier Transform Domain 2/20/2014 2 Fourier Series and Fourier Transform: History Jean Baptiste Joseph Fourier, French mathematician
More informationDHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A
DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A Classification of systems : Continuous and Discrete
More informationE2.5 Signals & Linear Systems. Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & 2)
E.5 Signals & Linear Systems Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & ) 1. Sketch each of the following continuous-time signals, specify if the signal is periodic/non-periodic,
More informationThe Discrete Fourier Transform
In [ ]: cd matlab pwd The Discrete Fourier Transform Scope and Background Reading This session introduces the z-transform which is used in the analysis of discrete time systems. As for the Fourier and
More informationComputer Vision & Digital Image Processing
Computer Vision & Digital Image Processing Image Restoration and Reconstruction I Dr. D. J. Jackson Lecture 11-1 Image restoration Restoration is an objective process that attempts to recover an image
More informationComputer Vision & Digital Image Processing. Periodicity of the Fourier transform
Computer Vision & Digital Image Processing Fourier Transform Properties, the Laplacian, Convolution and Correlation Dr. D. J. Jackson Lecture 9- Periodicity of the Fourier transform The discrete Fourier
More informationIntroduction to Digital Signal Processing
Introduction to Digital Signal Processing 1.1 What is DSP? DSP is a technique of performing the mathematical operations on the signals in digital domain. As real time signals are analog in nature we need
More informationVisual features: From Fourier to Gabor
Visual features: From Fourier to Gabor Deep Learning Summer School 2015, Montreal Hubel and Wiesel, 1959 from: Natural Image Statistics (Hyvarinen, Hurri, Hoyer; 2009) Alexnet ICA from: Natural Image Statistics
More informationExperimental Fourier Transforms
Chapter 5 Experimental Fourier Transforms 5.1 Sampling and Aliasing Given x(t), we observe only sampled data x s (t) = x(t)s(t; T s ) (Fig. 5.1), where s is called sampling or comb function and can be
More informationEENG580 Game 2: Verbal Representation of Course Content
Intro to DSP Game 2 Verbal Representation of Course Content EENG580 Game 2: Verbal Representation of Course Content Activity summary Overview: Word game similar to Taboo, Catch Phrase, Unspeakable, Battle
More informationMultidimensional digital signal processing
PSfrag replacements Two-dimensional discrete signals N 1 A 2-D discrete signal (also N called a sequence or array) is a function 2 defined over thex(n set 1 of, n 2 ordered ) pairs of integers: y(nx 1,
More informationMath 56 Homework 5 Michael Downs
1. (a) Since f(x) = cos(6x) = ei6x 2 + e i6x 2, due to the orthogonality of each e inx, n Z, the only nonzero (complex) fourier coefficients are ˆf 6 and ˆf 6 and they re both 1 2 (which is also seen from
More informationQuality Improves with More Rays
Recap Quality Improves with More Rays Area Area 1 shadow ray 16 shadow rays CS348b Lecture 8 Pat Hanrahan / Matt Pharr, Spring 2018 pixelsamples = 1 jaggies pixelsamples = 16 anti-aliased Sampling and
More information6.003: Signals and Systems. Sampling and Quantization
6.003: Signals and Systems Sampling and Quantization December 1, 2009 Last Time: Sampling and Reconstruction Uniform sampling (sampling interval T ): x[n] = x(nt ) t n Impulse reconstruction: x p (t) =
More informationI Chen Lin, Assistant Professor Dept. of CS, National Chiao Tung University. Computer Vision: 4. Filtering
I Chen Lin, Assistant Professor Dept. of CS, National Chiao Tung University Computer Vision: 4. Filtering Outline Impulse response and convolution. Linear filter and image pyramid. Textbook: David A. Forsyth
More informationECE Digital Image Processing and Introduction to Computer Vision
ECE592-064 Digital Image Processing and Introduction to Computer Vision Depart. of ECE, NC State University Instructor: Tianfu (Matt) Wu Spring 2017 Outline Recap, image degradation / restoration Template
More informationEE16B - Spring 17 - Lecture 11B Notes 1
EE6B - Spring 7 - Lecture B Notes Murat Arcak 6 April 207 Licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Interpolation with Basis Functions Recall that
More informationEE482: Digital Signal Processing Applications
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE482: Digital Signal Processing Applications Spring 2014 TTh 14:305:45 CBC C222 Lecture 8 Frequency Analysis 14/02/18 http://www.ee.unlv.edu/~b1morris/ee482/
More informationReview of Linear Systems Theory
Review of Linear Systems Theory The following is a (very) brief review of linear systems theory, convolution, and Fourier analysis. I work primarily with discrete signals, but each result developed in
More informationNotes on FFT-based differentiation
Notes on FFT-based differentiation Steven G Johnson, MIT Applied Mathematics Created April, 2011, updated May 4, 2011 Abstract A common numerical technique is to differentiate some sampled function y(x)
More informationFourier Transform and Frequency Domain
Fourier Transform and Frequency Domain http://www.cs.cmu.edu/~16385/ 16-385 Computer Vision Spring 2018, Lecture 3 (part 2) Overview of today s lecture Some history. Fourier series. Frequency domain. Fourier
More informationThe Fractional Fourier Transform with Applications in Optics and Signal Processing
* The Fractional Fourier Transform with Applications in Optics and Signal Processing Haldun M. Ozaktas Bilkent University, Ankara, Turkey Zeev Zalevsky Tel Aviv University, Tel Aviv, Israel M. Alper Kutay
More informationEXAMINATION QUESTION PAPER
Faculty of Science and Technology EXAMINATION QUESTION PAPER Exam in: FYS-2010 Digital Image Processing Date: Monday 26 September 2016 Time: 09.00 13.00 Place: Approved aids: Administrasjonsbygget, Aud.Max.
More informationIndex. p, lip, 78 8 function, 107 v, 7-8 w, 7-8 i,7-8 sine, 43 Bo,94-96
p, lip, 78 8 function, 107 v, 7-8 w, 7-8 i,7-8 sine, 43 Bo,94-96 B 1,94-96 M,94-96 B oro!' 94-96 BIro!' 94-96 I/r, 79 2D linear system, 56 2D FFT, 119 2D Fourier transform, 1, 12, 18,91 2D sinc, 107, 112
More informationChapter 4 Image Enhancement in the Frequency Domain
Chapter 4 Image Enhancement in the Frequency Domain Yinghua He School of Computer Science and Technology Tianjin University Background Introduction to the Fourier Transform and the Frequency Domain Smoothing
More informationNovel Waveform Design and Scheduling For High-Resolution Radar and Interleaving
Novel Waveform Design and Scheduling For High-Resolution Radar and Interleaving Phase Phase Basic Signal Processing for Radar 101 x = α 1 s[n D 1 ] + α 2 s[n D 2 ] +... s signal h = filter if h = s * "matched
More informationLecture 04 Image Filtering
Institute of Informatics Institute of Neuroinformatics Lecture 04 Image Filtering Davide Scaramuzza 1 Lab Exercise 2 - Today afternoon Room ETH HG E 1.1 from 13:15 to 15:00 Work description: your first
More informationTutorial Sheet #2 discrete vs. continuous functions, periodicity, sampling
2.39 utorial Sheet #2 discrete vs. continuous functions, periodicity, sampling We will encounter two classes of signals in this class, continuous-signals and discrete-signals. he distinct mathematical
More informationESS Finite Impulse Response Filters and the Z-transform
9. Finite Impulse Response Filters and the Z-transform We are going to have two lectures on filters you can find much more material in Bob Crosson s notes. In the first lecture we will focus on some of
More informationANALOG AND DIGITAL SIGNAL PROCESSING ADSP - Chapter 8
ANALOG AND DIGITAL SIGNAL PROCESSING ADSP - Chapter 8 Fm n N fnt ( ) e j2mn N X() X() 2 X() X() 3 W Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2 W W 3 W W x () x () x () 2 x ()
More informationWhy does a lower resolution image still make sense to us? What do we lose? Image:
2D FREQUENCY DOMAIN The slides are from several sources through James Hays (Brown); Srinivasa Narasimhan (CMU); Silvio Savarese (U. of Michigan); Bill Freeman and Antonio Torralba (MIT), including their
More information! Introduction. ! Discrete Time Signals & Systems. ! Z-Transform. ! Inverse Z-Transform. ! Sampling of Continuous Time Signals
ESE 531: Digital Signal Processing Lec 25: April 24, 2018 Review Course Content! Introduction! Discrete Time Signals & Systems! Discrete Time Fourier Transform! Z-Transform! Inverse Z-Transform! Sampling
More informationFourier transform. Stefano Ferrari. Università degli Studi di Milano Methods for Image Processing. academic year
Fourier transform Stefano Ferrari Università degli Studi di Milano stefano.ferrari@unimi.it Methods for Image Processing academic year 27 28 Function transforms Sometimes, operating on a class of functions
More informationWavelets and Multiresolution Processing
Wavelets and Multiresolution Processing Wavelets Fourier transform has it basis functions in sinusoids Wavelets based on small waves of varying frequency and limited duration In addition to frequency,
More informationA. Relationship of DSP to other Fields.
1 I. Introduction 8/27/2015 A. Relationship of DSP to other Fields. Common topics to all these fields: transfer function and impulse response, Fourierrelated transforms, convolution theorem. 2 y(t) = h(
More informationSampling. Alejandro Ribeiro. February 8, 2018
Sampling Alejandro Ribeiro February 8, 2018 Signals exist in continuous time but it is not unusual for us to process them in discrete time. When we work in discrete time we say that we are doing discrete
More informationLecture 28 Continuous-Time Fourier Transform 2
Lecture 28 Continuous-Time Fourier Transform 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/6/14 1 Limit of the Fourier Series Rewrite (11.9) and (11.10) as As, the fundamental
More informationSound & Vibration Magazine March, Fundamentals of the Discrete Fourier Transform
Fundamentals of the Discrete Fourier Transform Mark H. Richardson Hewlett Packard Corporation Santa Clara, California The Fourier transform is a mathematical procedure that was discovered by a French mathematician
More informationLecture 14: Convolution and Frequency Domain Filtering
Lecture 4: Convolution and Frequency Domain Filtering Harvey Rhody Chester F. Carlson Center for Imaging Science Rochester Institute of Technology rhody@cis.rit.edu October 7, 005 Abstract The impulse
More information