INTRODUCTION AND BACKGROUND

1. INTRODUCTION

2. NEED/MOTIVATION FOR COMPRESSION

3. BASIC DEFINITIONS

4. STRATEGIES FOR COMPRESSION

5. INFORMATION THEORY PRELIMINARIES

1. INTRODUCTION

• What is Compression?
  • It is a process of deriving more compact (i.e., smaller) representations of data
  
  
• Goal of Compression
  • Significant reduction in the data size to reduce the storage/bandwidth requirements
  
  
• Constraints on Compression
  • Perfect or near-perfect reconstruction (lossless/lossy)
  
  
• Strategies for Compression
  • Reducing redundancies
  • Exploiting the characteristics of human vision

• Massive Amounts of Data Involved in Storage/Transmission of Text, Sound, Images, and Videos in Many Applications
  
• Applications
  • Medical imaging
  • Teleradiology
  • Space/Satellite imaging
  • Multimedia
  • Digital Video: entertainment, home use
  • Digital photography
  
  
• Concrete Figures
  • A typical hospital generates close to 1 terabits per year
  • NASA's EOS will generate 1 terabytes per day
    
  • One 2-hour video = 1.3 terabits
  • Video transmission speed = 180Mb/sec
  • With MPEG1 (1.5Mb/s), need compression ratio of 120
  • With MPEG2 (4-10Mb/s), need comp. ratio of 18-45

• Coding: Compression

• Codeword: A binary string representing either the whole coded data or one coded data symbol

• Coded Bitstream: the binary string representing the whole coded data.

• Lossless Compression: 100% accurate reconstruction of the original data

• Lossy Compression: The reconstruction involves errors which may or may not be tolerable

• Bit Rate: Average number of bits per original data element after compression
  

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• Signal-to-Noise Ratio (SNR) in the case of lossy compression.
  Let I be an original signal (e.g., an image), and R be its lossily reconstructed counterpart. SNR is defined to be:
  SNR=10 log₁₀ (||I||²/||I-R||²) = 20 log₁₀ (||I||/||I-R||) where for any vector/matrix/set of number E={x₁, x₂, ... , x_N}, ||E||²=x₁² + x₂² + ... + x_N².
  • The unit of SNR is "decibel" (or dB for short).
  • So, if SNR = 23, we say the SNR is 23 dB.

• Mean-Square Error (MSE): ||I-R||²/N

• Relative Mean-Square Error (RMSE): ||I-R||²/||I||²

• Therefore, SNR = -10 log₁₀ RMSE.
  • So the smaller the error, the higher the SNR. In particular, the higher the SNR, the better the quality of the reconstructed data.
  • Exercise: Prove that if RMSE is decreased by a factor of 10, then SNR increases by 10 decibels.
  • It is this nice fact that justifies the multiplicative factor in the definition of the SNR.

• An image is a matrix of numbers, each number called a pixel (short for picture element)

• a binaty image (or black-and-white B/W image) is an image where every pixel can have one of two values only (typically 0 and 1).

• a grayscale image is a non-color image but there various shades of gray. That is what we typically refer to informally when we talk aboiut the (old) back-and-wite TVs/cameras/photos.

• A color image is an image where every pixel can have a color.
  Theorem (Newton): Every color is a combination of three colors (such as Red, Green and Blue), called the basic colors.
  • Therefore, in color images, every pixel is represented with three (numerical) components, such as (R,G,B), where R represents how much red there is in that pixel, G how much green, and B how much blue.

• The spatial resolution of an image is:
  • the total number of pixels in the image (like you hear a megapixel image, or a 6-megapixel camera, ...); or
  • the number of pixels per row and per column, like when one says this image is 512 x 1000 image, which means it has 512 rows and 1000 columns, which also means that every row has 1000 pixels and every column has 512 pixels; or
  • number of pixels per inch; for binary images (like in fax machines, basic scanners, and old dot-matrix printers), it is called "dot per inch (dpi)".

  • Note: For a fixed physical size image, the higher the spatial resolution, the more (and smaller) the pixels are, and thus the better the quality (and detail) of the image.

• The density resolution (or bit depth) is the number of bits per pixel. The higher the density resolution, the more colors (or shades of gray) can be represented, and thus the crisper or more detailed the image is.

• A video is a sequence of images. Every image in the sequence is called a frame.

• A video is captured/displayed at a certain rate, called the frame rate.
  • A typical rate that does not show jerkiness is 30 frames per second (fps). For higher definition (of motion), the rate can be higher. Lowere rates, likes 20 fps and even 15 fps, were used before when communications and computers were slower.
  • Lower than 15 fps rates would be quite jerky and unacceptable.
• A sound/audio (digital) signal is a sequence of values, called samples, where every sample is the intensity of the recorded sound at the corresponding moment in time.

• The sampling rate of a sound is the number of samples per second.
  • The CD quality sampling rate is 44.1K samples per second (or 44.1 KHz), usually at 16 bits per sample, though 24 bits per sample is now common.
  • In digital sound used for miniDV, digital TV, and DVD, the sampling rate is 48 KHz.
  • In DVD-Audio and in Blu-ray audio tracks, the sampling rate is 96 KHz or 192 KHz.

• When the sampling rate is infinity, the signal becomes what is called "analog signal".
  • When signal is captured as an analog signal, it can be converted to a digital signal by an analog-to-digital converter (also called A/D converter or digitizer)
  • If a digital signal is fed into a digital-to-analog converter (also called D/A converter), the output is obviously an analog signal.
  • Modulators are D/A converters, and demodulators are A/D converters. So, a modem is both an A/D and D/A converter.

• Symbol-Level Representation Redundancy
  • Different symbols occur with different frequencies
  • Variable-length codes vs. fixed-length codes
  • Frequent symbols are better coded with short codes
  • Infrequent symbols are coded with long codes
  • Example Techniques: Huffman Coding
  
  
• Block-Level Representation Redundancy
  • Different blocks of data occur with varying frequencies
  • Better then to code blocks than individual symbols
  • The block size can be fixed or variable
  • The block-code size can be fixed or variable
  • Frequent blocks are better coded with short codes
  • Example techniques: Block-oriented Huffman, Run-Length Encoding (RLE), Arithmetic Coding, Lempil-Ziv (LZ)
  
  
• Inter-Pixel Spatial Redundancy
  • Neighboring pixels tend to have similar values
  • Neighboring pixels tend to exhibit high correlations
  • Techniques: Decorrelation and/or processing in the frequency domain
  • Spatial decorrelation converts correlations into symbol- or block-redundancy
  • Frequency domain processing addresses visual redundancy (see the next slide)
  
  
• Inter-Pixel Temporal Redundancy (in Video)
  • Often, the majority of corresponding pixels in successive video-frames are identical over long spans of frames
  • Due to motion, blocks of pixels change in position but not in values between successive frames
  • Thus, block-oriented motion-compensated redundancy reduction techniques are used for video compression
  
  
• Visual Redundancy
  • The human visual system (HVS) has certain limitations that make many image contents invisible. Those contents, termed visually redundant, are the target of removal in lossy compression.
    
  • In fact, the HVS can see within a small range of spatial frequencies: 1-60 cycles/arc-degree

    (Plot by hand the contrast sensitivity function) 
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    	

  • Approach for reducing visual redundancy in lossy compression
    1. Transform: Convert the data to the frequency domain
    2. Quantize: Under-represent the high frequencies
    3. Losslessly compress the quantized data

• Discrete Memoryless Source S: A data generator where the alphabet is finite and the symbols generated are independent of one another. Assume the alphabet is {a₁,a₂,...,a_n}
• Let p_k = Probability that symbol a_k is generated (transmitted) by the source
  
• Entropy: H(S)=-(p₁log p₁ + p₂log p₂ + ... + p_nlog p_n)
  
• Theorem (Shannon): H(S) is the minimum average number of bits/symbol possible
  That is, no matter which lossless compression is ever invented, its bitrate can never be better (smaller) than H(S) for any memoryless source S.
• Sources with Memory: Presence of inter-symbol correlation
  
• Their entropy is still the min average number of bits/symbol
  
• Adjoint Source of Order N
  • Treat each possible block A of N symbols as a macrosymbol, and compute the probability P_A
  • Treat the source as a memoryless source consisting of the macrosymbols A's and their probabilities P_A
  • The entropy
  
  
• Theorem (Shannon): For any source S with memory, as N
  • This implies that for any source S with memory (i.e., with inter-symbol correlation/redundancy), if we divide it into blocks of large enough size and then block-code it without taking advantage of inter-block correlation, then we can approximate the performance of any other coder for the source.