Fourier Transforms 1D
|
|
- Brooke Cannon
- 5 years ago
- Views:
Transcription
1 Fourier Transforms 1D 3D Image Processing Torsten Möller
2 Overview Recap Function representations shift-invariant spaces linear, time-invariant (LTI) systems complex numbers Fourier Transforms Transform pairs properties Convolution Theorem Understanding sampling in the Fourier space 2
3 What is a (digital) image? An image is made of pixels (=picture elements) the coordinate values are discretized Laurent Condat / Torsten Möller 3
4 Quantization The pixel values are quantized: they belong to a discrete set of values, generally represented by integers between 0 and N-1 coded on 8 bits 255 possible val. coded on 4 bits Laurent Condat / Torsten Möller coded on 2 bits 4 possible val. 4
5 2D lattices An image is defined on a lattice. The most common is the Cartesian (a.k.a square) lattice. But other lattices exist and have interesting properties. Laurent Condat / Torsten Möller 5
6 Image quality Common defaults in images: blur (motion blur, out of focus blur...) ringing aliasing (staircasing, Moiré patterns) Laurent Condat / Torsten Möller 6
7 Image quality Low-frequency Moiré artifacts appear when high-frequency content is sampled in an incorrect way. Laurent Condat / Torsten Möller 7
8 Image quality We need to be able to measure the difference between two images, for instance an original image and a degraded one. Classical difference measures between two images I 1 and I 2 : mean absolute error: MAE = 1 P W P H WH k x =1 k y =1 I 1[k x,k y ] I 2 [k x,k y ] mean square error: MSE = 1 P W P H WH k x =1 k y =1 I 1[k x,k y ] I 2 [k x,k y ] peak signal to noise ratio (db): PSNR = 10 log 10 MSE There exist much more sophisticated quality measures and difference measures for images (SSIM...) Laurent Condat / Torsten Möller 8
9 Image enhancement Intensity Transformations image negatives log transforms gamma (power-law) transforms contrast stretching intensity-level slicing bit-plane slicing 9
10 Histogram equalization Idea -- stretch histogram non-uniformally such that final histogram is a uniform distribution p s (s) =p r (r) dr ds s = T (r) = Z r 0 p r (w)dw 10
11 Histogram matching original matched equalization 11
12 Local histogram equalization 12
13 Mechanics of filtering Correlation: w(x, y)? f(x, y) = ax bx w(s, t) f(x s, y t) Convolution: w(x, y) f(x, y) = s= a ax t= b bx w(s, t) f(x s, y t) s= a t= b 13
14 Median filtering (denoising) 14
15 Sharpening enhance / highlight transition in intensity how to find transition? unsharp masking / highboost filtering first / second order derivatives in 1D multi-d: gradient magnitude Laplacian 15
16 Overview Recap Function representations shift-invariant spaces linear, time-invariant (LTI) systems complex numbers Fourier Transforms Transform pairs properties Convolution Theorem Understanding sampling in the Fourier space 16
17 How to represent a function? on a computer, can only store a bunch of numbers, not a continuous function! brute-force idea: sampling 17
18 What is sampling? (mathematically speaking) modeled through an impulse not really a function, but a distribution: Z 1 1 (t) = 1 if t =0 (t)dt =1 0 if t 6= 0 18
19 The sifting property picking a value off from f: more general: Z 1 1 Z 1 1 f(t) (t)dt = f(0) f(t) (t t 0 )dt = f(t 0 ) 19
20 The impulse train pick up multiple values of f at once: s T (t) = 1X 1 (t n T ) 20
21 What is sampling? f(t) s T (t) = 1X f(n T ) (t n T ) 1 1X 1 f[n] (t n T ) 21
22 Sampling: step by step 1. break function into pieces & measure 22
23 Sampling: step by step 1. break function into pieces & measure 2. reconstruct function from measurements = 23
24 Sampling: step by step 1. break function into pieces & measure 2. reconstruct function from measurements = 24
25 Sampling: step by step f(t) = 25
26 Sampling: step by step Z 1 1 f(t) (t t 0 )dt (t t 0 ) f(t) = 26
27 Sampling: step by step Z 1 1 f(t) (t t 0 )dt (t t 0 ) f(t) Z 1 1 f(t) (t (t 0 T ))dt (t (t 0 T )) = 27
28 Sampling: step by step Z 1 1 f(t) (t t 0 )dt (t t 0 ) f(t) Z 1 1 = Z 1 1 f(t) (t (t 0 T ))dt f(t) (t (t 0 2 T ))dt (t (t 0 T )) (t (t 0 2 T )) 28
29 Sampling: step by step Z 1 1 f(t) (t t 0 )dt (t t 0 ) f(t) Z 1 1 = Z 1 1 f(t) (t (t 0 T ))dt f(t) (t (t 0 2 T ))dt (t (t 0 T )) (t (t 0 2 T )) Z 1 1 f(t) (t (t 0 3 T ))dt (t (t 0 3 T )) 29
30 Sampling: step by step f(t) Z 1 1 Z 1 1 = Z 1 1 f(t) (t t 0 )dt (t t 0 ) f(t) (t (t 0 T ))dt f(t) (t (t 0 2 T ))dt (t (t 0 T )) (t (t 0 2 T )) f[0] (t t 0 ) Z 1 1 f(t) (t (t 0 3 T ))dt (t (t 0 3 T )) 30
31 Sampling: step by step f(t) Z 1 1 Z 1 1 = Z 1 1 f(t) (t t 0 )dt (t t 0 ) f(t) (t (t 0 T ))dt f(t) (t (t 0 2 T ))dt (t (t 0 T )) (t (t 0 2 T )) f[0] (t t 0 ) f[1] (t (t 0 T )) Z 1 1 f(t) (t (t 0 3 T ))dt (t (t 0 3 T )) 31
32 Sampling: step by step f(t) Z 1 1 Z 1 1 = Z 1 1 Z 1 1 f(t) (t t 0 )dt (t t 0 ) f(t) (t (t 0 T ))dt f(t) (t (t 0 2 T ))dt f(t) (t (t 0 3 T ))dt (t (t 0 T )) (t (t 0 2 T )) (t (t 0 3 T )) f[0] (t t 0 ) f[1] (t (t 0 T )) f[2] (t (t 0 2 T )) 32
33 Sampling: step by step f(t) Z 1 1 Z Z 1 1 f(t) (t t 0 )dt (t t 0 ) f(t) (t (t 0 T ))dt f(t) (t (t 0 2 T ))dt f(t) (t (t 0 3 T ))dt (t (t 0 T )) = Z 1 (t (t 0 2 T )) (t (t 0 3 T )) f[0] (t t 0 ) f[1] (t (t 0 T )) f[2] (t (t 0 2 T )) f[3] (t (t 0 3 T )) 33
34 A more general view on sampling in summary: f(t) = f(t) = more generally: f(t) = f(t) = 1X n= 1 1X n= 1 1X n= 1 1X n= 1 Z 1 1 f(t) (t n T )dt f[n] (t n T ) Z 1 1 f(t) (t n T )dt c[n] (t n T ) (t n T ) (t n T ) 34
35 A more general view on sampling in summary: f(t) = f(t) = 1X n= 1 Z 1 1 even more general: f(t) = f(t) = 1X n= 1 1X n= 1 1X n= 1 f(t) (t n T )dt f[n] (t n T ) Z 1 1 f(t) (t n T )dt c[n] (t n T ) (t n T ) (t n T ) 35
36 Sampling: generalization I = 36
37 Sampling: generalization I (t) = 37
38 Sampling: generalization II (t) (t) = 38
39 A more general view on sampling in summary: f(t) f(t) 1X n= 1 1X n= 1 Z 1 1 f(t) (t n T )dt c[n] (t n T ) (t n T ) ѱ also known as a point-spread function ; common for image acquisition Φ also known as the reconstruction function 39
40 Shift-invariant spaces All functions f that can be represented as f(t) = considered being part of a shift-invariant space V(Φ) Function f(φ) is characterized by coefficients c[n] positions nδt we can store this in a computer! Remaining questions: 1X n= 1 What is a good Φ and ѱ? c[n] (t n T ) 40
41 Overview Recap Function representations shift-invariant spaces linear, time-invariant (LTI) systems complex numbers Fourier Transforms Transform pairs properties Convolution Theorem Understanding sampling in the Fourier space 41
42 What are systems? Engineers like to figure out stuff in boxes (ever heard of reverse engineering? Example: Radio incoming: radio waves outcoming: sound /02/general-information-of-radiowave.html
43 What are systems? Engineers like to figure out stuff in boxes (ever heard of reverse engineering? Example: Radio incoming: radio waves outcoming: sound r(t) s(t) 43
44 Time-invariant system Example: Radio if the same radio waves come in tomorrow, it ll produce the same sound! r(t) s(t) 44
45 Time-invariant system Example: Radio if the same radio waves come in tomorrow, it ll produce the same sound! r(t t 0 ) s(t t 0 ) 45
46 Linear system Example: Radio When two stations are sending at the same frequency, then you hear both at once! r 1 (t) s 1 (t) r 2 (t) s 2 (t) r 1 (t)r 2 (t) s 1 (t)s 2 (t) 46
47 LTI systems How to characterize such systems? All we need to know is how the system response to an impulse! Black Box System (x x 0 ) (x x 0 ) 47
48 LTI systems Sidenote If response to the impulse is finite (bounded) = FIR infinite (not bounded) = IIR Black Box System (x x 0 ) (x x 0 ) 48
49 LTI systems (t) (t) Black Box System 49
50 LTI systems (t) 1X f s (t) = n= 1 f[n] (t n T ) Black Box System 50
51 LTI systems f s (t) = 1X n= 1 f[n] (t n T ) f r (t) = Black Box System Also known as a 1X convolution! n= 1 f[n] (t n T ) 51
52 LTI s for sinecosine Let s plug in sine and cosine waves and see what happens to them in an LTI system! Why sine cosine? carry amplitude and frequency hence, they are closer to our radio waves! however, they are just special! First -- need to learn about complex numbers! 52
53 Overview Recap Function representations shift-invariant spaces linear, time-invariant (LTI) systems complex numbers Fourier Transforms Transform pairs properties Convolution Theorem Understanding sampling in the Fourier space 53
54 Complex numbers most general: C = xjy components of C: Re(C) = x Im(C) = y Im C w y Re x Also expressed in terms of frequency (angle) and amplitude: C = C (cos! j sin!)= C e j! C = p x 2 y 2 54
55 Conjugate complex numbers complex: C = xjy Im conjugate: C* = x-jy C w x y -y Re 55
56 LTI systems f s (t) = 1X n= 1 f[n] (t n T ) Black Box System f r (t) = 1X n= 1 f[n] (t n T ) 56
57 LTI systems f s (t) = 1X n= 1 f[n] (t n T ) Let s try f r (t) = Black Box System f[n] =e j!n 1X f[n] (t n T ) n= 1 57
58 Steps of the algebra I f r (t) = f r (k T )= f r [k] = f r [k] = 1X n= 1 1X n= 1 1X m= 1 1X m= 1 f[n] (t n T ) f[n] ((k n) T ) f[k m] (m T ) e (k m)j! T (m T ) 58
59 Steps of the algebra II 1X f r [k] = e (k m)j! T (m T ) m= 1 1 f r [k] =e kj! T X e mj! T (m T ) f r [k] =f s [k] (!) m= 1 59
60 LTI systems sinecosine are eigenfunctions of an LTI system (a convolution) The eigenvalue popping out is the Fourier Transform of the LTI Black Box System e j!n T e j!n T (!) (!) = 1X e mj! T (m T ) m= 1 60
61 Overview Recap Function representations shift-invariant spaces linear, time-invariant (LTI) systems complex numbers Fourier Transforms Transform pairs properties Convolution Theorem Understanding sampling in the Fourier space 61
62 What is a Fourier Transform? an eigenvalue of an LTI? Huh? Let s go back to representation of functions: f(t) f(t) 1X n= 1 1X n= 1 Z 1 1 What is a good Φ/ѱ? f(t) (t n T )dt c[n] (t n T ) (t n T ) Current choices were focused on a spatial location! (also known as spatial domain) 62
63 Sampling: Spatial Domain Rep (t) (t) = 63
64 What is a Fourier Transform? w2 w1 f 1 f 2 w4 w3 f 3 f
65 What is a Fourier Transform? Let s go back to (spatial) representation of functions: f(t) f(t) c n = 1 T f(t) Z T/2 1X 1X n= 1 1X n= 1 Fourier series into Frequency Domain: n= 1 T/2 Z 1 1 f(t)e j 2 n T c n e j 2 n T t f(t) (t n T )dt c[n] (t n T ) t dt (t n T ) 65
66 Leap of faith? we can represent ANY function through a sum of sine cosine! Fourier 1807/1822 not very intuitive, even for the mathematical elite of the time 66
67 Some intuition 67
68 There are 4 Fourier Transforms! Recall Fourier series: f(t) is periodic with period T! General Fourier Transform requires no periodicity: F (!) = f(t) = Z 1 1 Z 1 1 c n = 1 T f(t) Z T/2 1X n= 1 T/2 f(t)e j 2 n T c n e j 2 n T f(t)e j2!t dt F (!)e j2!t dt t t 68
69 DFT the most important one Discrete Fourier Transform (DFT) requires periodicity in both transform pairs F m = M 1 X n=0 f n = 1 M f n e j2 mn/m M 1 X m=0 F m e j2 mn/m 69
70 All Fourier Transforms Spatial Domain Frequency Domain FT FS Fourier Series DFT Discrete FT DTFT Discrete Time FT f(t) = f(t) = Z 1 1 F (!)e j2!t dt continuous f n = 1 M 1X n= 1 MX 1 m=0 c n e j 2 n T t continuous periodic f n = 1 2 Z F m e j2 mn/m discrete periodic discrete F (!) = c n = 1 T F m = F (!)e j!n d! F (!) = Z 1 1 Z T/2 MX 1 n=0 T/2 f(t)e j2!t dt continuous discrete f n e j2 mn/m 1X n= 1 f(t)e j 2 n T t dt discrete periodic f n e j!n d! continuous periodic 70
71 A little more intuition 71
72 Visualization of the spectrum Generally, we look at the amplitude of an image transform; hence we take a logarithmic scale to represent the values as gray-values. high frequencies original image Laurent Condat / Torsten Möller its amplitude spectrum low frequencies 72
73 Interpretation The lines correspond to discontinuities, with perpendicular orientation. The horizontal and vertical lines come from the implicit periodic boundary conditions. For natural images, most of the information is concentrated in the low-frequency region. Low frequencies correspond to the slowly varying components, whereas high frequencies correspond to fast gray level changes (edges...) The diagonal line results from the discontinuity induced by the hat. Laurent Condat / Torsten Möller 73
74 Interpretation The lines correspond to discontinuities, with perpendicular orientation. The horizontal and vertical lines come from the implicit periodic boundary conditions. For natural images, most of the information is concentrated in the low-frequency region. Low frequencies correspond to the slowly varying components, whereas high frequencies correspond to fast gray level changes (edges...) Laurent Condat / Torsten Möller 74
75 Interpretation The small lines correspond to the windows and flags. Laurent Condat / Torsten Möller 75
76 Some Examples Since the image content is periodic, the spectrum is discrete. If the period of the signal increases, the distance between the frequencies decreases, and vice versa. Laurent Condat / Torsten Möller 76
77 Some Examples The 2D periodicity of the image induces the 2D periodicity of the FT Since the image content is periodic, the spectrum is discrete. Laurent Condat / Torsten Möller 77
78 Texture = periodic pattern The Fourier spectrum is well suited for describing the directionality of textures Laurent Condat / Torsten Möller 78
79 A little more intuition 79
80 A little more intuition 80
81 A little more intuition 81
82 Not very intuitive 82
83 Overview Recap Function representations shift-invariant spaces linear, time-invariant (LTI) systems complex numbers Fourier Transforms Transform pairs properties Convolution Theorem Understanding sampling in the Fourier space 83
84 What happens to an impulse? it is basically a constant! c n = 1 T Z T/2 T/2 (t)e j 2 n T t c n = 1 T e0 c n = 1 T 84
85 What about a shifted impulse? the shifts remain as frequencies c n = 1 T Z T/2 T/2 (t t 0 )e j 2 n T t c n = 1 T e j 2 n T t 0 85
86 What happens to an impulse train? Impulse train is periodic apply Fourier series, will not do the math here, see book: s T (t) = S T (!) = 1 T 1X (t n T ) 1 1X n (! T ) n= 1 distance between impulses grows inversely 86
87 What happens to a box? it is the well-known sinc function sinc(t) = sin t t 87
88 Overview Recap Function representations shift-invariant spaces linear, time-invariant (LTI) systems complex numbers Fourier Transforms Transform pairs properties Convolution Theorem Understanding sampling in the Fourier space 88
89 LTI systems f s (t) = 1X n= 1 f[n] (t n T ) Black Box System f r (t) = 1X n= 1 f[n] (t n T ) 89
90 LTI systems sinecosine are eigenfunctions of an LTI system (a convolution) The eigenvalue popping out is the Fourier Transform of the LTI Black Box System e j!n T e j!n T (!) (!) = 1X e mj! T (m T ) m= 1 90
91 What is the Fourier Transform of a convolution? doing this in the continuous domain: f r (t) =f (t) = Z 1 1 Z applez F r (!) = f( )h(t )d e j2!t dt Z applez = f( ) h(t )e j2!t dt d Z = f( ) H(!)e j2! d Z = H(!) f( )e j2! d f( )h(t )d = H(!)F (!) 91
92 What is the Fourier Transform of a convolution? convolution == multiplication: f (t) () F (!)H(!) multiplication == convolution: f(t) (t) () F (!) H(!) 92
93 Overview Recap Function representations shift-invariant spaces linear, time-invariant (LTI) systems complex numbers Fourier Transforms Transform pairs properties Convolution Theorem Understanding sampling in the Fourier space 93
94 What was a convolution again? in summary: f(t) f(t) 1X n= 1 1X n= 1 Z 1 1 f(t) (t n T )dt c[n] (t n T ) (t n T ) ѱ also known as a point-spread function ; common for image acquisition Φ also known as the reconstruction function 94
95 What was a convolution again? (t) (t) = = 95
96 Convolution the movie from wikipedia: 96
97 What was sampling again? f(t) s T (t) = 1X f(n T ) (t n T ) 1 1X 1 f[n] (t n T ) 97
98 Sampling in the Fourier Domain f(t)s T (t) () F (!) S T (!) F (!) S T (!) =F (!) 1 T = 1 T 1X n= 1 1X n= 1 F (! (! n T ) n T ) 98
99 Sampling in the Fourier Domain 1 T > 2! max 99
100 How do you properly recover? H(!) () Sinc(t) f(t) =f s (t) Sinc(t) 1X t n T = f[n]sinc n T n= 1 100
101 Aliasing 101
102 Aliasing 102
103 Aliasing 103
104 Next time repeating all this for 2D (designing) filters in the Fourier Domain 104
חלק מהשקפים מעובדים משקפים של פרדו דוראנד, טומס פנקהאוסר ודניאל כהן-אור קורס גרפיקה ממוחשבת 2009/2010 סמסטר א' Image Processing
חלק מהשקפים מעובדים משקפים של פרדו דוראנד, טומס פנקהאוסר ודניאל כהן-אור קורס גרפיקה ממוחשבת 2009/2010 סמסטר א' Image Processing 1 What is an image? An image is a discrete array of samples representing
More informationSpectrum Analyser Basics
Hands-On Learning Spectrum Analyser Basics Peter D. Hiscocks Syscomp Electronic Design Limited Email: phiscock@ee.ryerson.ca June 28, 2014 Introduction Figure 1: GUI Startup Screen In a previous exercise,
More informationSampling. Sampling. CS 450: Introduction to Digital Signal and Image Processing. Bryan Morse BYU Computer Science
Sampling CS 450: Introduction to Digital Signal and Image Processing Bryan Morse BYU Computer Science Introduction Sampling f(t) Continuous t f(t) Discrete t Introduction Sampling Sampling a continuous
More informationDigital Signal. Continuous. Continuous. amplitude. amplitude. Discrete-time Signal. Analog Signal. Discrete. Continuous. time. time.
Discrete amplitude Continuous amplitude Continuous amplitude Digital Signal Analog Signal Discrete-time Signal Continuous time Discrete time Digital Signal Discrete time 1 Digital Signal contd. Analog
More informationFourier Integral Representations Basic Formulas and facts
Engineering Mathematics II MAP 436-4768 Spring 22 Fourier Integral Representations Basic Formulas and facts 1. If f(t) is a function without too many horrible discontinuities; technically if f(t) is decent
More informationProblem Set #1 Problem Set Due: Friday, April 12
1 EE102B Pring 2018-19 Signal Processing and Linear Systems II Pauly Problem Set #1 Problem Set Due: Friday, April 12 In the following problems, assume that δ T (t) = δ(t nt ) n = is an infinite array
More informationECE438 - Laboratory 4: Sampling and Reconstruction of Continuous-Time Signals
Purdue University: ECE438 - Digital Signal Processing with Applications 1 ECE438 - Laboratory 4: Sampling and Reconstruction of Continuous-Time Signals October 6, 2010 1 Introduction It is often desired
More informationFundamentals of DSP Chap. 1: Introduction
Fundamentals of DSP Chap. 1: Introduction Chia-Wen Lin Dept. CSIE, National Chung Cheng Univ. Chiayi, Taiwan Office: 511 Phone: #33120 Digital Signal Processing Signal Processing is to study how to represent,
More informationCourse Web site:
The University of Texas at Austin Spring 2018 EE 445S Real- Time Digital Signal Processing Laboratory Prof. Evans Solutions for Homework #1 on Sinusoids, Transforms and Transfer Functions 1. Transfer Functions.
More informationThe following exercises illustrate the execution of collaborative simulations in J-DSP. The exercises namely a
Exercises: The following exercises illustrate the execution of collaborative simulations in J-DSP. The exercises namely a Pole-zero cancellation simulation and a Peak-picking analysis and synthesis simulation
More informationLinear Time Invariant (LTI) Systems
Linear Time Invariant (LTI) Systems Superposition Sound waves add in the air without interacting. Multiple paths in a room from source sum at your ear, only changing change phase and magnitude of particular
More informationCM3106 Solutions. Do not turn this page over until instructed to do so by the Senior Invigilator.
CARDIFF UNIVERSITY EXAMINATION PAPER Academic Year: 2013/2014 Examination Period: Examination Paper Number: Examination Paper Title: Duration: Autumn CM3106 Solutions Multimedia 2 hours Do not turn this
More informationELEC 310 Digital Signal Processing
ELEC 310 Digital Signal Processing Alexandra Branzan Albu 1 Instructor: Alexandra Branzan Albu email: aalbu@uvic.ca Course information Schedule: Tuesday, Wednesday, Friday 10:30-11:20 ECS 125 Office Hours:
More informationInverse Filtering by Signal Reconstruction from Phase. Megan M. Fuller
Inverse Filtering by Signal Reconstruction from Phase by Megan M. Fuller B.S. Electrical Engineering Brigham Young University, 2012 Submitted to the Department of Electrical Engineering and Computer Science
More informationModule 8 : Numerical Relaying I : Fundamentals
Module 8 : Numerical Relaying I : Fundamentals Lecture 28 : Sampling Theorem Objectives In this lecture, you will review the following concepts from signal processing: Role of DSP in relaying. Sampling
More informationA Novel Approach towards Video Compression for Mobile Internet using Transform Domain Technique
A Novel Approach towards Video Compression for Mobile Internet using Transform Domain Technique Dhaval R. Bhojani Research Scholar, Shri JJT University, Jhunjunu, Rajasthan, India Ved Vyas Dwivedi, PhD.
More informationStudy of White Gaussian Noise with Varying Signal to Noise Ratio in Speech Signal using Wavelet
American International Journal of Research in Science, Technology, Engineering & Mathematics Available online at http://www.iasir.net ISSN (Print): 2328-3491, ISSN (Online): 2328-3580, ISSN (CD-ROM): 2328-3629
More informationMultichannel Satellite Image Resolution Enhancement Using Dual-Tree Complex Wavelet Transform and NLM Filtering
Multichannel Satellite Image Resolution Enhancement Using Dual-Tree Complex Wavelet Transform and NLM Filtering P.K Ragunath 1, A.Balakrishnan 2 M.E, Karpagam University, Coimbatore, India 1 Asst Professor,
More informationwith - < n < +. There are two types of approximations associated with the sampling process. finite precision of the ADC finite sampling frequency.
EE345M/EE380L.6 Lecture 0. Lecture 0 objectives are to: Introduce basic principles involved in digital filtering, Define the and use it to analyze filters, Develop digital filter implementations "hello",
More informationBioengineering 508: Physical Aspects of Medical Imaging Nature of Medical Imaging. Nature of Medical Imaging
Bioengineering 508: Physical Aspects of Medical Imaging http://courses.washington.edu/bioen508/ Bioengineering 508: Physical Aspects of Medical Imaging Organizer: Paul Kinahan, PhD Adam Alessio, PhD Ruth
More informationFFT Laboratory Experiments for the HP Series Oscilloscopes and HP 54657A/54658A Measurement Storage Modules
FFT Laboratory Experiments for the HP 54600 Series Oscilloscopes and HP 54657A/54658A Measurement Storage Modules By: Michael W. Thompson, PhD. EE Dept. of Electrical Engineering Colorado State University
More informationMusical Sound: A Mathematical Approach to Timbre
Sacred Heart University DigitalCommons@SHU Writing Across the Curriculum Writing Across the Curriculum (WAC) Fall 2016 Musical Sound: A Mathematical Approach to Timbre Timothy Weiss (Class of 2016) Sacred
More informationExperiment 2: Sampling and Quantization
ECE431, Experiment 2, 2016 Communications Lab, University of Toronto Experiment 2: Sampling and Quantization Bruno Korst - bkf@comm.utoronto.ca Abstract In this experiment, you will see the effects caused
More informationEMBEDDED ZEROTREE WAVELET CODING WITH JOINT HUFFMAN AND ARITHMETIC CODING
EMBEDDED ZEROTREE WAVELET CODING WITH JOINT HUFFMAN AND ARITHMETIC CODING Harmandeep Singh Nijjar 1, Charanjit Singh 2 1 MTech, Department of ECE, Punjabi University Patiala 2 Assistant Professor, Department
More informationThe Effect of Time-Domain Interpolation on Response Spectral Calculations. David M. Boore
The Effect of Time-Domain Interpolation on Response Spectral Calculations David M. Boore This note confirms Norm Abrahamson s finding that the straight line interpolation between sampled points used in
More informationNanoGiant Oscilloscope/Function-Generator Program. Getting Started
Getting Started Page 1 of 17 NanoGiant Oscilloscope/Function-Generator Program Getting Started This NanoGiant Oscilloscope program gives you a small impression of the capabilities of the NanoGiant multi-purpose
More information2-Dimensional Image Compression using DCT and DWT Techniques
2-Dimensional Image Compression using DCT and DWT Techniques Harmandeep Singh Chandi, V. K. Banga Abstract Image compression has become an active area of research in the field of Image processing particularly
More informationReduced complexity MPEG2 video post-processing for HD display
Downloaded from orbit.dtu.dk on: Dec 17, 2017 Reduced complexity MPEG2 video post-processing for HD display Virk, Kamran; Li, Huiying; Forchhammer, Søren Published in: IEEE International Conference on
More information10:15-11 am Digital signal processing
1 10:15-11 am Digital signal processing Data Conversion & Sampling Sampled Data Systems Data Converters Analog to Digital converters (A/D ) Digital to Analog converters (D/A) with Zero Order Hold Signal
More informationLab 5 Linear Predictive Coding
Lab 5 Linear Predictive Coding 1 of 1 Idea When plain speech audio is recorded and needs to be transmitted over a channel with limited bandwidth it is often necessary to either compress or encode the audio
More informationAppendix D. UW DigiScope User s Manual. Willis J. Tompkins and Annie Foong
Appendix D UW DigiScope User s Manual Willis J. Tompkins and Annie Foong UW DigiScope is a program that gives the user a range of basic functions typical of a digital oscilloscope. Included are such features
More informationZONE PLATE SIGNALS 525 Lines Standard M/NTSC
Application Note ZONE PLATE SIGNALS 525 Lines Standard M/NTSC Products: CCVS+COMPONENT GENERATOR CCVS GENERATOR SAF SFF 7BM23_0E ZONE PLATE SIGNALS 525 lines M/NTSC Back in the early days of television
More informationDELTA MODULATION AND DPCM CODING OF COLOR SIGNALS
DELTA MODULATION AND DPCM CODING OF COLOR SIGNALS Item Type text; Proceedings Authors Habibi, A. Publisher International Foundation for Telemetering Journal International Telemetering Conference Proceedings
More informationUpgrading E-learning of basic measurement algorithms based on DSP and MATLAB Web Server. Milos Sedlacek 1, Ondrej Tomiska 2
Upgrading E-learning of basic measurement algorithms based on DSP and MATLAB Web Server Milos Sedlacek 1, Ondrej Tomiska 2 1 Czech Technical University in Prague, Faculty of Electrical Engineeiring, Technicka
More informationProfessor Laurence S. Dooley. School of Computing and Communications Milton Keynes, UK
Professor Laurence S. Dooley School of Computing and Communications Milton Keynes, UK The Song of the Talking Wire 1904 Henry Farny painting Communications It s an analogue world Our world is continuous
More informationA SVD BASED SCHEME FOR POST PROCESSING OF DCT CODED IMAGES
Electronic Letters on Computer Vision and Image Analysis 8(3): 1-14, 2009 A SVD BASED SCHEME FOR POST PROCESSING OF DCT CODED IMAGES Vinay Kumar Srivastava Assistant Professor, Department of Electronics
More informationAutomatic music transcription
Music transcription 1 Music transcription 2 Automatic music transcription Sources: * Klapuri, Introduction to music transcription, 2006. www.cs.tut.fi/sgn/arg/klap/amt-intro.pdf * Klapuri, Eronen, Astola:
More informationResearch and Development Report
BBC RD 1995/12 Research and Development Report ARCHIVAL RETRIEVAL: Techniques for image enhancement J.C.W. Newell, B.A., D.Phil. Research and Development Department Technical Resources THE BRITISH BROADCASTING
More informationMurdoch redux. Colorimetry as Linear Algebra. Math of additive mixing. Approaching color mathematically. RGB colors add as vectors
Murdoch redux Colorimetry as Linear Algebra CS 465 Lecture 23 RGB colors add as vectors so do primary spectra in additive display (CRT, LCD, etc.) Chromaticity: color ratios (r = R/(R+G+B), etc.) color
More informationLecture 18: Exam Review
Lecture 18: Exam Review The Digital World of Multimedia Prof. Mari Ostendorf Announcements HW5 due today, Lab5 due next week Lab4: Printer should be working soon. Exam: Friday, Feb 22 Review in class today
More informationHugo Technology. An introduction into Rob Watts' technology
Hugo Technology An introduction into Rob Watts' technology Copyright Rob Watts 2014 About Rob Watts Audio chip designer both analogue and digital Consultant to silicon chip manufacturers Designer of Chord
More informationResearch Article. ISSN (Print) *Corresponding author Shireen Fathima
Scholars Journal of Engineering and Technology (SJET) Sch. J. Eng. Tech., 2014; 2(4C):613-620 Scholars Academic and Scientific Publisher (An International Publisher for Academic and Scientific Resources)
More informationComputer Vision for HCI. Image Pyramids. Image Pyramids. Multi-resolution image representations Useful for image coding/compression
Computer Vision for HCI Image Pyramids Image Pyramids Multi-resolution image representations Useful for image coding/compression 2 1 Image Pyramids Operations: General Theory Two fundamental operations
More informationMixer Conversion Loss
3/7/005 Mixer Conversion Loss.doc 1/6 Mixer Conversion Loss Let s examine the typical application of a mixer. v ( t ) v ( t ) IF v ( t ) Generally, the signal delivered to the Local Oscillator port is
More informationAudio Processing Exercise
Name: Date : Audio Processing Exercise In this exercise you will learn to load, playback, modify, and plot audio files. Commands for loading and characterizing an audio file To load an audio file (.wav)
More informationSwept-tuned spectrum analyzer. Gianfranco Miele, Ph.D
Swept-tuned spectrum analyzer Gianfranco Miele, Ph.D www.eng.docente.unicas.it/gianfranco_miele g.miele@unicas.it Video section Up until the mid-1970s, spectrum analyzers were purely analog. The displayed
More informationDr. David A. Clifton Group Leader Computational Health Informatics (CHI) Lab Lecturer in Engineering Science, Balliol College
Dr. David A. Clifton Group Leader Computational Health Informatics (CHI) Lab Lecturer in Engineering Science, Balliol College 1. Introduction to Fourier analysis, the Fourier series 2. Sampling and Aliasing
More informationELEC 691X/498X Broadcast Signal Transmission Fall 2015
ELEC 691X/498X Broadcast Signal Transmission Fall 2015 Instructor: Dr. Reza Soleymani, Office: EV 5.125, Telephone: 848 2424 ext.: 4103. Office Hours: Wednesday, Thursday, 14:00 15:00 Time: Tuesday, 2:45
More informationScope of the course. Video processing. G. de Haan. Schedule lectures 5P530. This is our field. Week 1 Week 2 Week 3 Week 4.
1 2 Video processing G. de Haan Scope of the course 3 24 Hz 25 Hz 30 Hz Scope of the course 50 Hz 2:1 60 Hz 2:1 CIF QCIF 1-25Hz WEB 72 Hz 85 Hz 95 Hz This is our field Theory (most repetition) pplications
More informationContents. EEM401 Digital Signal Processing. Textbook. Examples of Typical Signals - ECG. Examples of Typical Signals - Speech
Contents EEM401 Digital Signal Processing Contents http://www.ee.hacettepe.edu.tr/ usezen/eem401/ Dr. Umut Sezen Department of Electrical and Electronic Engineering, Hacettepe University Discrete-Time
More informationECE 45 Homework 2. t x(τ)dτ. Problem 2.2 Find the Bode plot (magnitude and phase) and label all critical points of the transfer function
UC San Diego Spring 2018 ECE 45 Homework 2 Problem 2.1 Are the following systems linear? Are they time invariant? (a) x(t) [ System (a)] 2x(t 3) (b) x(t) [ System (b)] x(t)+t (c) x(t) [ System (c)] (x(t)+1)
More information2. AN INTROSPECTION OF THE MORPHING PROCESS
1. INTRODUCTION Voice morphing means the transition of one speech signal into another. Like image morphing, speech morphing aims to preserve the shared characteristics of the starting and final signals,
More informationLecture 2 Video Formation and Representation
2013 Spring Term 1 Lecture 2 Video Formation and Representation Wen-Hsiao Peng ( 彭文孝 ) Multimedia Architecture and Processing Lab (MAPL) Department of Computer Science National Chiao Tung University 1
More informationFilm Sequence Detection and Removal in DTV Format and Standards Conversion
TeraNex Technical Presentation Film Sequence Detection and Removal in DTV Format and Standards Conversion 142nd SMPTE Technical Conference & Exhibition October 20, 2000 Scott Ackerman DTV Product Manager
More informationMusic Source Separation
Music Source Separation Hao-Wei Tseng Electrical and Engineering System University of Michigan Ann Arbor, Michigan Email: blakesen@umich.edu Abstract In popular music, a cover version or cover song, or
More informationSteganographic Technique for Hiding Secret Audio in an Image
Steganographic Technique for Hiding Secret Audio in an Image 1 Aiswarya T, 2 Mansi Shah, 3 Aishwarya Talekar, 4 Pallavi Raut 1,2,3 UG Student, 4 Assistant Professor, 1,2,3,4 St John of Engineering & Management,
More informationImage Resolution and Contrast Enhancement of Satellite Geographical Images with Removal of Noise using Wavelet Transforms
Image Resolution and Contrast Enhancement of Satellite Geographical Images with Removal of Noise using Wavelet Transforms Prajakta P. Khairnar* 1, Prof. C. A. Manjare* 2 1 M.E. (Electronics (Digital Systems)
More informationDCI Requirements Image - Dynamics
DCI Requirements Image - Dynamics Matt Cowan Entertainment Technology Consultants www.etconsult.com Gamma 2.6 12 bit Luminance Coding Black level coding Post Production Implications Measurement Processes
More informationEmbedded System Hardware
Embedded System Hardware Peter Marwedel Informatik 12 Germany 2009/11/10 12 Structure of this course Application Knowledge 2: Specification Design repository 3: ES-hardware 6: Application mapping 4: system
More informationThe Cocktail Party Effect. Binaural Masking. The Precedence Effect. Music 175: Time and Space
The Cocktail Party Effect Music 175: Time and Space Tamara Smyth, trsmyth@ucsd.edu Department of Music, University of California, San Diego (UCSD) April 20, 2017 Cocktail Party Effect: ability to follow
More informationColour Reproduction Performance of JPEG and JPEG2000 Codecs
Colour Reproduction Performance of JPEG and JPEG000 Codecs A. Punchihewa, D. G. Bailey, and R. M. Hodgson Institute of Information Sciences & Technology, Massey University, Palmerston North, New Zealand
More informationDigitizing and Sampling
F Digitizing and Sampling Introduction................................................................. 152 Preface to the Series.......................................................... 153 Under-Sampling.............................................................
More informationRecap: Representation. Subtle Skeletal Differences. How do skeletons differ? Target Poses. Reference Poses
Animation by Example Lecture 2: Motion Signal Processing Michael Gleicher University of Wisconsin- Madison www.cs.wisc.edu/~gleicher www.cs.wisc.edu/graphics Recap: Representation Represent human as hierarchical
More informationPlease feel free to download the Demo application software from analogarts.com to help you follow this seminar.
Hello, welcome to Analog Arts spectrum analyzer tutorial. Please feel free to download the Demo application software from analogarts.com to help you follow this seminar. For this presentation, we use a
More informationAudio-Based Video Editing with Two-Channel Microphone
Audio-Based Video Editing with Two-Channel Microphone Tetsuya Takiguchi Organization of Advanced Science and Technology Kobe University, Japan takigu@kobe-u.ac.jp Yasuo Ariki Organization of Advanced Science
More informationDVG-5000 Motion Pattern Option
AccuPel DVG-5000 Documentation Motion Pattern Option Manual DVG-5000 Motion Pattern Option Motion Pattern Option for the AccuPel DVG-5000 Digital Video Calibration Generator USER MANUAL Version 1.00 2
More informationni.com Digital Signal Processing for Every Application
Digital Signal Processing for Every Application Digital Signal Processing is Everywhere High-Volume Image Processing Production Test Structural Sound Health and Vibration Monitoring RF WiMAX, and Microwave
More informationDigital Audio: Some Myths and Realities
1 Digital Audio: Some Myths and Realities By Robert Orban Chief Engineer Orban Inc. November 9, 1999, rev 1 11/30/99 I am going to talk today about some myths and realities regarding digital audio. I have
More informationProblem Weight Total 100
EE 350 Problem Set 4 Cover Sheet Fall 2016 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: Submission deadlines: Turn in the written solutions by 4:00 pm on Tuesday October 4
More informationModule 1: Digital Video Signal Processing Lecture 5: Color coordinates and chromonance subsampling. The Lecture Contains:
The Lecture Contains: ITU-R BT.601 Digital Video Standard Chrominance (Chroma) Subsampling Video Quality Measures file:///d /...rse%20(ganesh%20rana)/my%20course_ganesh%20rana/prof.%20sumana%20gupta/final%20dvsp/lecture5/5_1.htm[12/30/2015
More informationNON-UNIFORM KERNEL SAMPLING IN AUDIO SIGNAL RESAMPLER
NON-UNIFORM KERNEL SAMPLING IN AUDIO SIGNAL RESAMPLER Grzegorz Kraszewski Białystok Technical University, Electrical Engineering Faculty, ul. Wiejska 45D, 15-351 Białystok, Poland, e-mail: krashan@teleinfo.pb.bialystok.pl
More informationContents. xv xxi xxiii xxiv. 1 Introduction 1 References 4
Contents List of figures List of tables Preface Acknowledgements xv xxi xxiii xxiv 1 Introduction 1 References 4 2 Digital video 5 2.1 Introduction 5 2.2 Analogue television 5 2.3 Interlace 7 2.4 Picture
More informationComparative Analysis of Wavelet Transform and Wavelet Packet Transform for Image Compression at Decomposition Level 2
2011 International Conference on Information and Network Technology IPCSIT vol.4 (2011) (2011) IACSIT Press, Singapore Comparative Analysis of Wavelet Transform and Wavelet Packet Transform for Image Compression
More informationAn Introduction to the Sampling Theorem
An Introduction to the Sampling Theorem An Introduction to the Sampling Theorem With rapid advancement in data acquistion technology (i.e. analog-to-digital and digital-to-analog converters) and the explosive
More informationCh. 1: Audio/Image/Video Fundamentals Multimedia Systems. School of Electrical Engineering and Computer Science Oregon State University
Ch. 1: Audio/Image/Video Fundamentals Multimedia Systems Prof. Ben Lee School of Electrical Engineering and Computer Science Oregon State University Outline Computer Representation of Audio Quantization
More information4.4 The FFT and MATLAB
4.4. THE FFT AND MATLAB 69 4.4 The FFT and MATLAB 4.4.1 The FFT and MATLAB MATLAB implements the Fourier transform with the following functions: fft, ifft, fftshift, ifftshift, fft2, ifft2. We describe
More informationVoice Controlled Car System
Voice Controlled Car System 6.111 Project Proposal Ekin Karasan & Driss Hafdi November 3, 2016 1. Overview Voice controlled car systems have been very important in providing the ability to drivers to adjust
More information5.7 Gabor transforms and spectrograms
156 5. Frequency analysis and dp P(1/2) = 0, (1/2) = 0. (5.70) dθ The equations in (5.69) correspond to Equations (3.33a) through (3.33c), while the equations in (5.70) correspond to Equations (3.32a)
More informationHello, welcome to the course on Digital Image Processing.
Digital Image Processing Prof. P. K. Biswas Department of Electronics and Electrical Communications Engineering Indian Institute of Technology, Kharagpur Module 01 Lecture Number 05 Signal Reconstruction
More informationLaboratory Assignment 3. Digital Music Synthesis: Beethoven s Fifth Symphony Using MATLAB
Laboratory Assignment 3 Digital Music Synthesis: Beethoven s Fifth Symphony Using MATLAB PURPOSE In this laboratory assignment, you will use MATLAB to synthesize the audio tones that make up a well-known
More informationProcessing. Electrical Engineering, Department. IIT Kanpur. NPTEL Online - IIT Kanpur
NPTEL Online - IIT Kanpur Course Name Department Instructor : Digital Video Signal Processing Electrical Engineering, : IIT Kanpur : Prof. Sumana Gupta file:///d /...e%20(ganesh%20rana)/my%20course_ganesh%20rana/prof.%20sumana%20gupta/final%20dvsp/lecture1/main.htm[12/31/2015
More informationInvestigation of Digital Signal Processing of High-speed DACs Signals for Settling Time Testing
Universal Journal of Electrical and Electronic Engineering 4(2): 67-72, 2016 DOI: 10.13189/ujeee.2016.040204 http://www.hrpub.org Investigation of Digital Signal Processing of High-speed DACs Signals for
More informationDATA COMPRESSION USING THE FFT
EEE 407/591 PROJECT DUE: NOVEMBER 21, 2001 DATA COMPRESSION USING THE FFT INSTRUCTOR: DR. ANDREAS SPANIAS TEAM MEMBERS: IMTIAZ NIZAMI - 993 21 6600 HASSAN MANSOOR - 993 69 3137 Contents TECHNICAL BACKGROUND...
More informationDigital Image and Fourier Transform
Lab 5 Numerical Methods TNCG17 Digital Image and Fourier Transform Sasan Gooran (Autumn 2009) Before starting this lab you are supposed to do the preparation assignments of this lab. All functions and
More informationDesign of Speech Signal Analysis and Processing System. Based on Matlab Gateway
1 Design of Speech Signal Analysis and Processing System Based on Matlab Gateway Weidong Li,Zhongwei Qin,Tongyu Xiao Electronic Information Institute, University of Science and Technology, Shaanxi, China
More informationFigure 1: Feature Vector Sequence Generator block diagram.
1 Introduction Figure 1: Feature Vector Sequence Generator block diagram. We propose designing a simple isolated word speech recognition system in Verilog. Our design is naturally divided into two modules.
More informationSupplemental Material for Gamma-band Synchronization in the Macaque Hippocampus and Memory Formation
Supplemental Material for Gamma-band Synchronization in the Macaque Hippocampus and Memory Formation Michael J. Jutras, Pascal Fries, Elizabeth A. Buffalo * *To whom correspondence should be addressed.
More informationRegion Adaptive Unsharp Masking based DCT Interpolation for Efficient Video Intra Frame Up-sampling
International Conference on Electronic Design and Signal Processing (ICEDSP) 0 Region Adaptive Unsharp Masking based DCT Interpolation for Efficient Video Intra Frame Up-sampling Aditya Acharya Dept. of
More informationTransform Coding of Still Images
Transform Coding of Still Images February 2012 1 Introduction 1.1 Overview A transform coder consists of three distinct parts: The transform, the quantizer and the source coder. In this laboration you
More informationCSC475 Music Information Retrieval
CSC475 Music Information Retrieval Monophonic pitch extraction George Tzanetakis University of Victoria 2014 G. Tzanetakis 1 / 32 Table of Contents I 1 Motivation and Terminology 2 Psychacoustics 3 F0
More informationPS User Guide Series Seismic-Data Display
PS User Guide Series 2015 Seismic-Data Display Prepared By Choon B. Park, Ph.D. January 2015 Table of Contents Page 1. File 2 2. Data 2 2.1 Resample 3 3. Edit 4 3.1 Export Data 4 3.2 Cut/Append Records
More informationPERCEPTUAL QUALITY ASSESSMENT FOR VIDEO WATERMARKING. Stefan Winkler, Elisa Drelie Gelasca, Touradj Ebrahimi
PERCEPTUAL QUALITY ASSESSMENT FOR VIDEO WATERMARKING Stefan Winkler, Elisa Drelie Gelasca, Touradj Ebrahimi Genista Corporation EPFL PSE Genimedia 15 Lausanne, Switzerland http://www.genista.com/ swinkler@genimedia.com
More informationError Resilience for Compressed Sensing with Multiple-Channel Transmission
Journal of Information Hiding and Multimedia Signal Processing c 2015 ISSN 2073-4212 Ubiquitous International Volume 6, Number 5, September 2015 Error Resilience for Compressed Sensing with Multiple-Channel
More informationThe Engineer s Guide to
HANDBOOK SERIES The Engineer s Guide to By John Watkinson The Engineer s Guide to Compression John Watkinson Snell & Wilcox Ltd. 1996 All rights reserved Text and diagrams from this publication may be
More informationResearch Article Design and Analysis of a High Secure Video Encryption Algorithm with Integrated Compression and Denoising Block
Research Journal of Applied Sciences, Engineering and Technology 11(6): 603-609, 2015 DOI: 10.19026/rjaset.11.2019 ISSN: 2040-7459; e-issn: 2040-7467 2015 Maxwell Scientific Publication Corp. Submitted:
More informationChapter 2 Signals. 2.1 Signals in the Wild One-Dimensional Continuous Time Signals
Chapter 2 Signals Lasciate ogni speranza, voi ch entrate. Dante Alighieri, The Divine Comedy We all send and receive signals. A letter or a phone call, a raised hand, a hunger cry signals are our information
More informationSignal Processing with Wavelets.
Signal Processing with Wavelets. Newer mathematical tool since 199. Limitation of classical methods of Descretetime Fourier Analysis when dealing with nonstationary signals. A mathematical treatment of
More informationAPPLICATIONS OF DIGITAL IMAGE ENHANCEMENT TECHNIQUES FOR IMPROVED
APPLICATIONS OF DIGITAL IMAGE ENHANCEMENT TECHNIQUES FOR IMPROVED ULTRASONIC IMAGING OF DEFECTS IN COMPOSITE MATERIALS Brian G. Frock and Richard W. Martin University of Dayton Research Institute Dayton,
More informationErrata to the 2nd, 3rd, and 4th printings, A Technical Introduction to Digital Video
Charles Poynton tel +1 416 486 3271 fax +1 416 486 3657 poynton @ poynton.com www.inforamp.net/ ~ poynton Errata to the 2nd, 3rd, and 4th printings, A Technical Introduction to Digital Video This note
More informationDesign Approach of Colour Image Denoising Using Adaptive Wavelet
International Journal of Engineering Research and Development ISSN: 78-067X, Volume 1, Issue 7 (June 01), PP.01-05 www.ijerd.com Design Approach of Colour Image Denoising Using Adaptive Wavelet Pankaj
More information