Compressing a 1D Discrete Signal
|
|
- Mavis Perkins
- 5 years ago
- Views:
Transcription
1 Compressing a D Discrete Signal Divide the signal into 8blocks. Subtract the sample mean from each value. Compute the 8 8covariancematrixforthe blocks. Compute the eigenvectors of the covariance matrix. Compute the KL transform of each block. Threshold each transformed block to create long runs of zeros. Compute differences of KL transform coefficients in raster scan order. Run length encode differences. Huffman-code run-lengths. Huffman-code labels.
2 Average Over all Initial Positions Given a D discrete signal f of length, construct an 8 8covariancematrixforthe8consecutive values starting at all initial positions n: C = = [ ] T [ ] fn+0 f n+7 fn+0 f n+7 n=0 f n+0 f n+0 f n+0 f n+ f n+0 f n+7 f n+ f n+0 f n+ f n+ f n+ f n f n+7 f n+0 f n+7 f n+ f n+7 f n+7. Because the covariance matrix is an average over all initial positions, there can be no dependence on n: f n+i f n+ j = f m+i f m+ j. It follows that C is circulant and the eigenvectors of C are sampled harmonic signals!
3 Compressing a D Discrete Signal (revised) Divide the signal into 8blocks. Subtract the sample mean from each value. Compute DFT of each block. Threshold each transformed block to create long runs of zeros. Compute differences of thresholded DFT coefficients in raster scan order. Run length encode differences. Huffman-code run-lengths. Huffman-code labels.
4 Symmetric Extension Given a vector f of length construct a vector g of length such that { f (n) if 0 n g(n)= f ( n ) if n. The DFT of g shifted by one-half is: ( G(m) = jπ(n + g(n) exp ) n=0 for 0 m. Recall that ( jπ(n + exp ) ( π(n + = cos ) ( π(n + + j sin ). ow, because g is symmetric: ( π(n + g(n) j sin ) = 0. n=0 Consequently G(m) = n=0 ( π(n + g(n) cos ).
5 Symmetric Extension (contd.) Dividing the sum into two parts and substituting f (n) for g(n) in the first part and f ( n ) for g(n) in the second part yields G(m)= n=0 n= We now observe that ( π(n + f (n)cos ) + ( π(n + f ( n )cos ). ( π(n + f ( n )cos ) ( π(n + = f ( )cos ) when n. It follows that [ ( G(m) = π(n + f (n) cos ) ( π(n + + cos )] n=0 ( = π(n + f (n)cos ). n=0
6 Discrete Cosine Transform The Discrete Cosine Transform is defined as follows: ( F(m) = π(n + f (n)cos ). n=0 Because the DCT is a real valued transform there is no need for complex arithmetic. Unlike f which had a discontinuity at f (0), g is continuous at both g(0) and g()! For this reason, the energy in the DCT representation of a function is localized in fewer non-zero values.
7 Periodicity of DFT and DCT Figure : The DCT is the unique half of the real part of the DFT of a length symmetric extension of a length signal shifted by one-half.
8 Periodicity of D DFT and DCT Figure : Periodicity of the D Discrete Fourier Transform (DFT). Figure 3: Periodicity of D Discrete Cosine Transform (DCT).
9 JPEG Divide image into 8 8blocks. Center gray values (i.e., subtract 8). Compute the DCT of each block. Quantize DCT coefficients using quantization table. Compute differences of quantized DCT coefficients along zigzag path. Run-length encode differences. Huffman-code run-lengths. Huffman-code labels. Joint Photographic Experts Group
10 Example Block Figure 4: JPEG compression (top) and decompression (bottom) algorithms.
11 Subtracting 8 f = Figure 5: Subtracting 8 from blocks.
12 DCT of 8 8Block 4 7 x=0 7 y=0 F(u,v)= ( ) ( ) π(x + )u π(y + )v f (x,y)cos cos 6 6 F = Figure 6: Discrete cosine transform of blocks
13 JPEG Basis Functions Figure 7: JPEG basis functions.
14 Block Quantization Q = G(u,v)= F(u,v) Q(u,v) G = Figure 8: Quantize blocks
15 Difference Encode along Zigzag Path Figure 9: Zigzag scan order. Figure 0: Difference encode blocks. Table : AC Coefficient Encoding Scheme. 4bits 4bits 8bits zero run length category amplitude
16 Rescaling Block F(u,v)=G(u,v) Q(u,v) Figure : Rescale blocks. F =
17 Inverse DCT 4 7 u=0 7 v=0 f (x,y)= [ ] π(x + )u F(u,v)cos cos 6 [ ] π(y + )v 6 f = Figure : Inverse discrete cosine transform of blocks
18 Reconstruction Errors f f =
19 Adding Figure 3: Add 8 to blocks.
20 Conclusion The DCT is really the KL transform of the block in disguise! The quantization table tells us which eigenvectors are most important perceptually. Compression ratios of 8 to without perceptible differences are typical.
Compressing a 1D Discrete Signal
Compressing a D Discrete Signal Divide the signal into 8blocks. Subtract the sample mean from each value. Compute the 8 8covariancematrixforthe blocks. Compute the eigenvectors of the covariance matrix.
More informationIMAGE COMPRESSION-II. Week IX. 03/6/2003 Image Compression-II 1
IMAGE COMPRESSION-II Week IX 3/6/23 Image Compression-II 1 IMAGE COMPRESSION Data redundancy Self-information and Entropy Error-free and lossy compression Huffman coding Predictive coding Transform coding
More informationIMAGE COMPRESSION IMAGE COMPRESSION-II. Coding Redundancy (contd.) Data Redundancy. Predictive coding. General Model
IMAGE COMRESSIO IMAGE COMRESSIO-II Data redundancy Self-information and Entropy Error-free and lossy compression Huffman coding redictive coding Transform coding Week IX 3/6/23 Image Compression-II 3/6/23
More informationImage Compression. Fundamentals: Coding redundancy. The gray level histogram of an image can reveal a great deal of information about the image
Fundamentals: Coding redundancy The gray level histogram of an image can reveal a great deal of information about the image That probability (frequency) of occurrence of gray level r k is p(r k ), p n
More informationOverview. Analog capturing device (camera, microphone) PCM encoded or raw signal ( wav, bmp, ) A/D CONVERTER. Compressed bit stream (mp3, jpg, )
Overview Analog capturing device (camera, microphone) Sampling Fine Quantization A/D CONVERTER PCM encoded or raw signal ( wav, bmp, ) Transform Quantizer VLC encoding Compressed bit stream (mp3, jpg,
More informationMultimedia Networking ECE 599
Multimedia Networking ECE 599 Prof. Thinh Nguyen School of Electrical Engineering and Computer Science Based on lectures from B. Lee, B. Girod, and A. Mukherjee 1 Outline Digital Signal Representation
More informationencoding without prediction) (Server) Quantization: Initial Data 0, 1, 2, Quantized Data 0, 1, 2, 3, 4, 8, 16, 32, 64, 128, 256
General Models for Compression / Decompression -they apply to symbols data, text, and to image but not video 1. Simplest model (Lossless ( encoding without prediction) (server) Signal Encode Transmit (client)
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 informationRLE = [ ; ], with compression ratio (CR) = 4/8. RLE actually increases the size of the compressed image.
MP/BME 574 Application Solutions. (2 pts) a) From first principles in class, we expect the entropy of the checkerboard image to be since this is the bit depth of the image and the frequency of each value
More informationImage Data Compression
Image Data Compression Image data compression is important for - image archiving e.g. satellite data - image transmission e.g. web data - multimedia applications e.g. desk-top editing Image data compression
More informationL. Yaroslavsky. Fundamentals of Digital Image Processing. Course
L. Yaroslavsky. Fundamentals of Digital Image Processing. Course 0555.330 Lec. 6. Principles of image coding The term image coding or image compression refers to processing image digital data aimed at
More informationImage Compression - JPEG
Overview of JPEG CpSc 86: Multimedia Systems and Applications Image Compression - JPEG What is JPEG? "Joint Photographic Expert Group". Voted as international standard in 99. Works with colour and greyscale
More informationWaveform-Based Coding: Outline
Waveform-Based Coding: Transform and Predictive Coding Yao Wang Polytechnic University, Brooklyn, NY11201 http://eeweb.poly.edu/~yao Based on: Y. Wang, J. Ostermann, and Y.-Q. Zhang, Video Processing and
More informationTransform coding - topics. Principle of block-wise transform coding
Transform coding - topics Principle of block-wise transform coding Properties of orthonormal transforms Discrete cosine transform (DCT) Bit allocation for transform Threshold coding Typical coding artifacts
More informationObjective: Reduction of data redundancy. Coding redundancy Interpixel redundancy Psychovisual redundancy Fall LIST 2
Image Compression Objective: Reduction of data redundancy Coding redundancy Interpixel redundancy Psychovisual redundancy 20-Fall LIST 2 Method: Coding Redundancy Variable-Length Coding Interpixel Redundancy
More informationTransform Coding. Transform Coding Principle
Transform Coding Principle of block-wise transform coding Properties of orthonormal transforms Discrete cosine transform (DCT) Bit allocation for transform coefficients Entropy coding of transform coefficients
More informationDepartment of Electrical Engineering, Polytechnic University, Brooklyn Fall 05 EL DIGITAL IMAGE PROCESSING (I) Final Exam 1/5/06, 1PM-4PM
Department of Electrical Engineering, Polytechnic University, Brooklyn Fall 05 EL512 --- DIGITAL IMAGE PROCESSING (I) Y. Wang Final Exam 1/5/06, 1PM-4PM Your Name: ID Number: Closed book. One sheet of
More informationJustify all your answers and write down all important steps. Unsupported answers will be disregarded.
Numerical Analysis FMN011 10058 The exam lasts 4 hours and has 13 questions. A minimum of 35 points out of the total 70 are required to get a passing grade. These points will be added to those you obtained
More informationPrincipal Components Analysis
Principal Components Analysis Santiago Paternain, Aryan Mokhtari and Alejandro Ribeiro March 29, 2018 At this point we have already seen how the Discrete Fourier Transform and the Discrete Cosine Transform
More informationBasic Principles of Video Coding
Basic Principles of Video Coding Introduction Categories of Video Coding Schemes Information Theory Overview of Video Coding Techniques Predictive coding Transform coding Quantization Entropy coding Motion
More informationat Some sort of quantization is necessary to represent continuous signals in digital form
Quantization at Some sort of quantization is necessary to represent continuous signals in digital form x(n 1,n ) x(t 1,tt ) D Sampler Quantizer x q (n 1,nn ) Digitizer (A/D) Quantization is also used for
More informationBASICS OF COMPRESSION THEORY
BASICS OF COMPRESSION THEORY Why Compression? Task: storage and transport of multimedia information. E.g.: non-interlaced HDTV: 0x0x0x = Mb/s!! Solutions: Develop technologies for higher bandwidth Find
More informationLec 04 Variable Length Coding (VLC) in JPEG
ECE 5578 Multimedia Communication Lec 04 Variable Length Coding (VLC) in JPEG Zhu Li Dept of CSEE, UMKC Z. Li Multimedia Communciation, 2018 p.1 Outline Lecture 03 ReCap VLC JPEG Image Coding Framework
More informationSYDE 575: Introduction to Image Processing. Image Compression Part 2: Variable-rate compression
SYDE 575: Introduction to Image Processing Image Compression Part 2: Variable-rate compression Variable-rate Compression: Transform-based compression As mentioned earlier, we wish to transform image data
More informationMultimedia & Computer Visualization. Exercise #5. JPEG compression
dr inż. Jacek Jarnicki, dr inż. Marek Woda Institute of Computer Engineering, Control and Robotics Wroclaw University of Technology {jacek.jarnicki, marek.woda}@pwr.wroc.pl Exercise #5 JPEG compression
More informationDigital Speech Processing Lecture 10. Short-Time Fourier Analysis Methods - Filter Bank Design
Digital Speech Processing Lecture Short-Time Fourier Analysis Methods - Filter Bank Design Review of STFT j j ˆ m ˆ. X e x[ mw ] [ nˆ m] e nˆ function of nˆ looks like a time sequence function of ˆ looks
More informationEE16B - Spring 17 - Lecture 12A Notes 1
EE6B - Spring 7 - Lecture 2A Notes Murat Arcak April 27 Licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4. International License. Sampling and Discrete Time Signals Discrete-Time
More informationLec 05 Arithmetic Coding
ECE 5578 Multimedia Communication Lec 05 Arithmetic Coding Zhu Li Dept of CSEE, UMKC web: http://l.web.umkc.edu/lizhu phone: x2346 Z. Li, Multimedia Communciation, 208 p. Outline Lecture 04 ReCap Arithmetic
More informationCompression and Coding
Compression and Coding Theory and Applications Part 1: Fundamentals Gloria Menegaz 1 Transmitter (Encoder) What is the problem? Receiver (Decoder) Transformation information unit Channel Ordering (significance)
More informationIntroduction to Video Compression H.261
Introduction to Video Compression H.6 Dirk Farin, Contact address: Dirk Farin University of Mannheim Dept. Computer Science IV L 5,6, 683 Mannheim, Germany farin@uni-mannheim.de D.F. YUV-Colorspace Computer
More information2. the basis functions have different symmetries. 1 k = 0. x( t) 1 t 0 x(t) 0 t 1
In the next few lectures, we will look at a few examples of orthobasis expansions that are used in modern signal processing. Cosine transforms The cosine-i transform is an alternative to Fourier series;
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 informationCSE 408 Multimedia Information System Yezhou Yang
Image and Video Compression CSE 408 Multimedia Information System Yezhou Yang Lots of slides from Hassan Mansour Class plan Today: Project 2 roundup Today: Image and Video compression Nov 10: final project
More informationImage and Multidimensional Signal Processing
Image and Multidimensional Signal Processing Professor William Hoff Dept of Electrical Engineering &Computer Science http://inside.mines.edu/~whoff/ Image Compression 2 Image Compression Goal: Reduce amount
More informationLecture 4 - Spectral Estimation
Lecture 4 - Spectral Estimation The Discrete Fourier Transform The Discrete Fourier Transform (DFT) is the equivalent of the continuous Fourier Transform for signals known only at N instants separated
More informationOn Compression Encrypted Data part 2. Prof. Ja-Ling Wu The Graduate Institute of Networking and Multimedia National Taiwan University
On Compression Encrypted Data part 2 Prof. Ja-Ling Wu The Graduate Institute of Networking and Multimedia National Taiwan University 1 Brief Summary of Information-theoretic Prescription At a functional
More informationJPEG and JPEG2000 Image Coding Standards
JPEG and JPEG2000 Image Coding Standards Yu Hen Hu Outline Transform-based Image and Video Coding Linear Transformation DCT Quantization Scalar Quantization Vector Quantization Entropy Coding Discrete
More informationCompression. Encryption. Decryption. Decompression. Presentation of Information to client site
DOCUMENT Anup Basu Audio Image Video Data Graphics Objectives Compression Encryption Network Communications Decryption Decompression Client site Presentation of Information to client site Multimedia -
More informationCompression. Reality Check 11 on page 527 explores implementation of the MDCT into a simple, working algorithm to compress audio.
C H A P T E R 11 Compression The increasingly rapid movement of information around the world relies on ingenious methods of data representation, which are in turn made possible by orthogonal transformations.the
More informationNumerical Analysis. Carmen Arévalo Lund University Arévalo FMN011
Numerical Analysis Carmen Arévalo Lund University carmen@maths.lth.se Discrete cosine transform C = 2 n 1 2 1 2 1 2 cos π 2n cos 3π 2n cos (2n 1)π 2n cos 6π 2n cos 2(2n 1)π 2n cos 2π 2n... cos (n 1)π 2n
More informationBasics of DCT, Quantization and Entropy Coding
Basics of DCT, Quantization and Entropy Coding Nimrod Peleg Update: April. 7 Discrete Cosine Transform (DCT) First used in 97 (Ahmed, Natarajan and Rao). Very close to the Karunen-Loeve * (KLT) transform
More informationCS 4495 Computer Vision. Frequency and Fourier Transforms. Aaron Bobick School of Interactive Computing. Frequency and Fourier Transform
CS 4495 Computer Vision Frequency and Fourier Transforms Aaron Bobick School of Interactive Computing Administrivia Project 1 is (still) on line get started now! Readings for this week: FP Chapter 4 (which
More informationBasics of DCT, Quantization and Entropy Coding. Nimrod Peleg Update: Dec. 2005
Basics of DCT, Quantization and Entropy Coding Nimrod Peleg Update: Dec. 2005 Discrete Cosine Transform (DCT) First used in 974 (Ahmed, Natarajan and Rao). Very close to the Karunen-Loeve * (KLT) transform
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 informationHARMONIC VECTOR QUANTIZATION
HARMONIC VECTOR QUANTIZATION Volodya Grancharov, Sigurdur Sverrisson, Erik Norvell, Tomas Toftgård, Jonas Svedberg, and Harald Pobloth SMN, Ericsson Research, Ericsson AB 64 8, Stockholm, Sweden ABSTRACT
More informationExamples of the Fourier Theorem (Sect. 10.3). The Fourier Theorem: Continuous case.
s of the Fourier Theorem (Sect. 1.3. The Fourier Theorem: Continuous case. : Using the Fourier Theorem. The Fourier Theorem: Piecewise continuous case. : Using the Fourier Theorem. The Fourier Theorem:
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 5 Other Coding Techniques Instructional Objectives At the end of this lesson, the students should be able to:. Convert a gray-scale image into bit-plane
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 informationCompression. What. Why. Reduce the amount of information (bits) needed to represent image Video: 720 x 480 res, 30 fps, color
Compression What Reduce the amount of information (bits) needed to represent image Video: 720 x 480 res, 30 fps, color Why 720x480x20x3 = 31,104,000 bytes/sec 30x60x120 = 216 Gigabytes for a 2 hour movie
More informationon a per-coecient basis in large images is computationally expensive. Further, the algorithm in [CR95] needs to be rerun, every time a new rate of com
Extending RD-OPT with Global Thresholding for JPEG Optimization Viresh Ratnakar University of Wisconsin-Madison Computer Sciences Department Madison, WI 53706 Phone: (608) 262-6627 Email: ratnakar@cs.wisc.edu
More informationThe Fourier Transform (and more )
The Fourier Transform (and more ) imrod Peleg ov. 5 Outline Introduce Fourier series and transforms Introduce Discrete Time Fourier Transforms, (DTFT) Introduce Discrete Fourier Transforms (DFT) Consider
More informationOverview. Signals as functions (1D, 2D) 1D Fourier Transform. 2D Fourier Transforms. Discrete Fourier Transform (DFT) Discrete Cosine Transform (DCT)
Fourier Transform Overview Signals as functions (1D, 2D) Tools 1D Fourier Transform Summary of definition and properties in the different cases CTFT, CTFS, DTFS, DTFT DFT 2D Fourier Transforms Generalities
More informationWavelets & Mul,resolu,on Analysis
Wavelets & Mul,resolu,on Analysis Square Wave by Steve Hanov More comics at http://gandolf.homelinux.org/~smhanov/comics/ Problem set #4 will be posted tonight 11/21/08 Comp 665 Wavelets & Mul8resolu8on
More informationCMPT 365 Multimedia Systems. Final Review - 1
CMPT 365 Multimedia Systems Final Review - 1 Spring 2017 CMPT365 Multimedia Systems 1 Outline Entropy Lossless Compression Shannon-Fano Coding Huffman Coding LZW Coding Arithmetic Coding Lossy Compression
More information4. Quantization and Data Compression. ECE 302 Spring 2012 Purdue University, School of ECE Prof. Ilya Pollak
4. Quantization and Data Compression ECE 32 Spring 22 Purdue University, School of ECE Prof. What is data compression? Reducing the file size without compromising the quality of the data stored in the
More informationJPEG Standard Uniform Quantization Error Modeling with Applications to Sequential and Progressive Operation Modes
JPEG Standard Uniform Quantization Error Modeling with Applications to Sequential and Progressive Operation Modes Julià Minguillón Jaume Pujol Combinatorics and Digital Communications Group Computer Science
More information8/19/16. Fourier Analysis. Fourier analysis: the dial tone phone. Fourier analysis: the dial tone phone
Patrice Koehl Department of Biological Sciences National University of Singapore http://www.cs.ucdavis.edu/~koehl/teaching/bl5229 koehl@cs.ucdavis.edu Fourier analysis: the dial tone phone We use Fourier
More informationFault Tolerance Technique in Huffman Coding applies to Baseline JPEG
Fault Tolerance Technique in Huffman Coding applies to Baseline JPEG Cung Nguyen and Robert G. Redinbo Department of Electrical and Computer Engineering University of California, Davis, CA email: cunguyen,
More informationCOSC460 Honours Report. A Fast Discrete Tchebichef Transform Algorithm for Image Compression
COSC460 Honours Report A Fast Discrete Tchebichef Transform Algorithm for Image Compression November 2006 Kiyoyuki Nakagaki kna23@student.canterbury.ac.nz Supervisor : Dr. Ramakrishnan Mukundan mukundan@canterbury.ac.nz
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 informationFourier analysis of discrete-time signals. (Lathi Chapt. 10 and these slides)
Fourier analysis of discrete-time signals (Lathi Chapt. 10 and these slides) Towards the discrete-time Fourier transform How we will get there? Periodic discrete-time signal representation by Discrete-time
More informationEE67I Multimedia Communication Systems
EE67I Multimedia Communication Systems Lecture 5: LOSSY COMPRESSION In these schemes, we tradeoff error for bitrate leading to distortion. Lossy compression represents a close approximation of an original
More informationWavelet Scalable Video Codec Part 1: image compression by JPEG2000
1 Wavelet Scalable Video Codec Part 1: image compression by JPEG2000 Aline Roumy aline.roumy@inria.fr May 2011 2 Motivation for Video Compression Digital video studio standard ITU-R Rec. 601 Y luminance
More informationEE5356 Digital Image Processing
EE5356 Digital Image Processing INSTRUCTOR: Dr KR Rao Spring 007, Final Thursday, 10 April 007 11:00 AM 1:00 PM ( hours) (Room 111 NH) INSTRUCTIONS: 1 Closed books and closed notes All problems carry weights
More informationPolynomial Methods for Component Matching and Verification
Polynomial Methods for Component Matching and Verification James Smith Stanford University Computer Systems Laboratory Stanford, CA 94305 1. Abstract Component reuse requires designers to determine whether
More informationIntelligent Visual Prosthesis
Orientation sensor (IMU) Intelligent Visual Prosthesis Depth image-based obstacle detection Depth camera Wideangle RGB camera Simultaneous object recognition, localization, and hand tracking New projects:
More informationEntropy Encoding Using Karhunen-Loève Transform
Entropy Encoding Using Karhunen-Loève Transform Myung-Sin Song Southern Illinois University Edwardsville Sept 17, 2007 Joint work with Palle Jorgensen. Introduction In most images their neighboring pixels
More informationA study of image compression techniques, with specific focus on weighted finite automata
A study of image compression techniques, with specific focus on weighted finite automata Rikus Muller Thesis presented in partial fulfilment of the requirements for the Degree of Master of Science at the
More informationSource Coding for Compression
Source Coding for Compression Types of data compression: 1. Lossless -. Lossy removes redundancies (reversible) removes less important information (irreversible) Lec 16b.6-1 M1 Lossless Entropy Coding,
More informationImage Filtering, Edges and Image Representation
Image Filtering, Edges and Image Representation Capturing what s important Req reading: Chapter 7, 9 F&P Adelson, Simoncelli and Freeman (handout online) Opt reading: Horn 7 & 8 FP 8 February 19, 8 A nice
More informationEE5356 Digital Image Processing. Final Exam. 5/11/06 Thursday 1 1 :00 AM-1 :00 PM
EE5356 Digital Image Processing Final Exam 5/11/06 Thursday 1 1 :00 AM-1 :00 PM I), Closed books and closed notes. 2), Problems carry weights as indicated. 3), Please print your name and last four digits
More informationModule 4 MULTI- RESOLUTION ANALYSIS. Version 2 ECE IIT, Kharagpur
Module 4 MULTI- RESOLUTION ANALYSIS Lesson Theory of Wavelets Instructional Objectives At the end of this lesson, the students should be able to:. Explain the space-frequency localization problem in sinusoidal
More informationQuantization. Introduction. Roadmap. Optimal Quantizer Uniform Quantizer Non Uniform Quantizer Rate Distorsion Theory. Source coding.
Roadmap Quantization Optimal Quantizer Uniform Quantizer Non Uniform Quantizer Rate Distorsion Theory Source coding 2 Introduction 4 1 Lossy coding Original source is discrete Lossless coding: bit rate
More informationImage Transforms. Digital Image Processing Fundamentals of Digital Image Processing, A. K. Jain. Digital Image Processing.
Digital Image Processing Fundamentals of Digital Image Processing, A. K. Jain 2D Orthogonal and Unitary Transform: Orthogonal Series Expansion: {a k,l (m,n)}: a set of complete orthonormal basis: N N *
More informationMultidimensional Signal Processing
Multidimensional Signal Processing Mark Eisen, Alec Koppel, and Alejandro Ribeiro Dept. of Electrical and Systems Engineering University of Pennsylvania aribeiro@seas.upenn.edu http://www.seas.upenn.edu/users/~aribeiro/
More informationImage Compression. Qiaoyong Zhong. November 19, CAS-MPG Partner Institute for Computational Biology (PICB)
Image Compression Qiaoyong Zhong CAS-MPG Partner Institute for Computational Biology (PICB) November 19, 2012 1 / 53 Image Compression The art and science of reducing the amount of data required to represent
More informationFast Fourier Transform
Fast Fourier Transform December 8, 2016 FFT JPEG RGB Y C B C R (luma (brightness), chroma 2 (color)) chroma resolution is reduced image is split in blocks 8 8 pixels JPEG RGB Y C B C R (luma (brightness),
More informationNoise Reduction of JPEG Images by Sampled-Data H Optimal ε Filters
SICE Annual Conference 25 in Okayama, August 8-1, 25 Okayama University, Japan Noise Reduction of JPEG Images by Sampled-Data H Optimal ε Filters H. Kakemizu,1, M. Nagahara,2, A. Kobayashi,3, Y. Yamamoto,4
More informationCh. 15 Wavelet-Based Compression
Ch. 15 Wavelet-Based Compression 1 Origins and Applications The Wavelet Transform (WT) is a signal processing tool that is replacing the Fourier Transform (FT) in many (but not all!) applications. WT theory
More informationWhere and how can linear algebra be useful in practice.
Where and how can linear algebra be useful in practice. Jiří Fiala All examples are simplified to demonstrate the use of tools of linear algebra. Real implementations are much more complex. Systems of
More informationRate Bounds on SSIM Index of Quantized Image DCT Coefficients
Rate Bounds on SSIM Index of Quantized Image DCT Coefficients Sumohana S. Channappayya, Alan C. Bovik, Robert W. Heath Jr. and Constantine Caramanis Dept. of Elec. & Comp. Engg.,The University of Texas
More informationFrequency2: Sampling and Aliasing
CS 4495 Computer Vision Frequency2: Sampling and Aliasing Aaron Bobick School of Interactive Computing Administrivia Project 1 is due tonight. Submit what you have at the deadline. Next problem set stereo
More informationCompression and Coding. Theory and Applications Part 1: Fundamentals
Compression and Coding Theory and Applications Part 1: Fundamentals 1 What is the problem? Transmitter (Encoder) Receiver (Decoder) Transformation information unit Channel Ordering (significance) 2 Why
More informationMore on Fourier Series
More on Fourier Series R. C. Trinity University Partial Differential Equations Lecture 6.1 New Fourier series from old Recall: Given a function f (x, we can dilate/translate its graph via multiplication/addition,
More informationSurvey of JPEG compression history analysis
Survey of JPEG compression history analysis Andrew B. Lewis Computer Laboratory Topics in security forensic signal analysis References Neelamani et al.: JPEG compression history estimation for color images,
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 informationSeismic compression. François G. Meyer Department of Electrical Engineering University of Colorado at Boulder
Seismic compression François G. Meyer Department of Electrical Engineering University of Colorado at Boulder francois.meyer@colorado.edu http://ece-www.colorado.edu/ fmeyer IPAM, MGA 2004 Seismic Compression
More informationA Complete Video Coding Chain Based on Multi-Dimensional Discrete Cosine Transform
RADIOENGINEERING, VOL. 19, NO. 3, SEPTEMBER 2010 421 A Complete Video Coding Chain Based on Multi-Dimensional Discrete Cosine Transform Tomas FRYZA Department of Radio Electronics, Brno University of Technology,
More informationCompression and Coding. Theory and Applications Part 1: Fundamentals
Compression and Coding Theory and Applications Part 1: Fundamentals 1 Transmitter (Encoder) What is the problem? Receiver (Decoder) Transformation information unit Channel Ordering (significance) 2 Why
More information18.085: Summer 2016 Due: 3 August 2016 (in class) Problem Set 8
Problem Set 8 Unless otherwise specified, you may use MATLAB to assist with computations. provide a print-out of the code used and its output with your assignment. Please 1. More on relation between Fourier
More informationThe Frequency Domain, without tears. Many slides borrowed from Steve Seitz
The Frequency Domain, without tears Many slides borrowed from Steve Seitz Somewhere in Cinque Terre, May 2005 CS194: Image Manipulation & Computational Photography Alexei Efros, UC Berkeley, Fall 2016
More informationContents. Acknowledgments
Table of Preface Acknowledgments Notation page xii xx xxi 1 Signals and systems 1 1.1 Continuous and discrete signals 1 1.2 Unit step and nascent delta functions 4 1.3 Relationship between complex exponentials
More informationChapter 4 Discrete Fourier Transform (DFT) And Signal Spectrum
Chapter 4 Discrete Fourier Transform (DFT) And Signal Spectrum CEN352, DR. Nassim Ammour, King Saud University 1 Fourier Transform History Born 21 March 1768 ( Auxerre ). Died 16 May 1830 ( Paris ) French
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 informationTTIC 31230, Fundamentals of Deep Learning David McAllester, April Information Theory and Distribution Modeling
TTIC 31230, Fundamentals of Deep Learning David McAllester, April 2017 Information Theory and Distribution Modeling Why do we model distributions and conditional distributions using the following objective
More informationObjectives of Image Coding
Objectives of Image Coding Representation of an image with acceptable quality, using as small a number of bits as possible Applications: Reduction of channel bandwidth for image transmission Reduction
More informationMidterm Summary Fall 08. Yao Wang Polytechnic University, Brooklyn, NY 11201
Midterm Summary Fall 8 Yao Wang Polytechnic University, Brooklyn, NY 2 Components in Digital Image Processing Output are images Input Image Color Color image image processing Image Image restoration Image
More informationCEG4311 Digital Image Processing Dec. 21, Professor: Eric Dubois
This exam has 23 pages 1 CEG4311 Digital Image Processing Dec. 21, 2004 Final exam Duration: 3 hours Professor: Eric Dubois Closed-book exam: you may not use any books, notes or calculator. Answer all
More informationProyecto final de carrera
UPC-ETSETB Proyecto final de carrera A comparison of scalar and vector quantization of wavelet decomposed images Author : Albane Delos Adviser: Luis Torres 2 P a g e Table of contents Table of figures...
More informationMysteries Around Graph Laplacian Eigenvalue 4
Mysteries Around Graph Laplacian Eigenvalue 4 Yuji Nakatsukasa, Naoki Saito, Ernest Woei Department of Mathematics University of California, Davis ICIAM 2011, Vancouver, BC, Canada July 22, 2011 saito@math.ucdavis.edu
More information