Lossy Compression Algorithms

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    Lossy Compression

    AlgorithmsThe lossless compression algorithms like Huffmancoding, dictionary based coding, arithmetic coding etc

    are mainly used to compress text

    When it comes to a huge data arrangement like images,speech, video etc., these algorithms fail to provide

    sufficient compression in the data

    The term Lossy indicates that there is a loss of

    information (mostly intentional) while encoding the data,which is mostly unnoticeable by the human eyes

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    Distortion Measure

    A distortion measure is a mathematicalquantity that specifies how close is an

    approximation to its original, using some

    distortion criteria

    Difference between original and

    approximated data?

    In Images Perceptual distortion is

    important and not the numerical distortion

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    Numerical distortion measures

    Mean Squared Error2 = (1/N) n=1

    N (xn - yn)2

    (Average Pixel Difference)

    Signal to Noise RatioSNR = 10 log (x2 / 2)

    (Size of error w.r.t the Input signal)

    Peak Signal to Noise RatioPSNR = 10 log (xpeak

    2 / 2)

    (Size of error w.r.t peak value of input signal)

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    Rate Distortion theory

    Lossy compression =

    Trade off between

    Rate and Distortion

    R specifies the ratefor given distortion

    When D = 0, the bit

    rate is maximum and

    follows lossless

    compression

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    Quantization A Basic Lossy

    scheme

    Depending on whether the input is scalar

    or vector, we have Scalar Quantization

    (SQ) and Vector Quantization (VQ)

    When input source is uniformly distributed,

    an Uniform Quantizer (UQ) can be used

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    Non Uniform SQ

    Here smaller step size is used where the

    signal occurs predominantly and enlarged

    step size is used where signal sparsely

    occurs

    Two common approaches Lloyd Max Quantizer

    Companding

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    Lloyd-Max Quantizer

    Begin

    Choose initial level set (y0,y1)

    i = 0;

    RepeatCompute bi using its equation

    i=i+1

    Compute yi using its equationUntill |yi yi-1| <

    End

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    Companding

    In companded qunatization, input is

    mapped by compressor function C(x)

    Then it is quantized by an uniform

    quantizer

    After Transmission the quantized values

    are mapped back using an expander

    function C-1(x)

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    Vector Quantization

    Encoding of frequently occurring words inEnglish like the or and etc. providebetter compression than encoding each

    letters t h e a o etc The same concept can be extended to

    speech, images as segments in speech ora collection of pixels in image and thesegroups of samples are called asVECTORS

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    Vector Quantization

    A compression system performs better if itis operated in a group of vectors(collection of data)

    VQ is same as SQ but extended to

    multiple dimensions In VQ an n-component code vector

    represents vectors that lie within a region

    in n dimensional vector space. A collectionof these code vectors form the codebookfor the vector quantizer

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    Basic VQ procedure

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    Transform Coding

    In case of data like speech and images, the

    different bit encoding should not be done to

    highly correlated samples. But however, majority

    of the samples are correlated to each other! Eg in text : THINK and THING, THROUGH and

    THOUGH, MIGHT and FIGHT

    Transform coding helps in removal of these

    correlation, so that effective encoding can be

    done

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    Transform Coding

    Let X =[x1,x2,x3..xk]T be vector of samples

    X -> T{x} -> Y -> T-1{x} -> X

    Y is the resultant of linear transformation of X

    and vector Ys elements are made lesscorrelated, so that the encoding is done better

    Less correlated = More Othogonal

    Transform T itself doesnt compress the data, ithelps the compression algorithm to perform

    better (Better compression ratio)

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    Discrete Cosine Transform

    Definition (DFT?) In Fourier Analysis, any signal can be represented in

    terms of its Sine and Cosine components at variousamplitudes and freqeucies

    For instance if we take the signal as an electrical signal,the process of determining the amplitudes of AC and DCcomponents of the signal is called as Cosine Transformand in particular if the indices are integers, it is called asDCT

    From definition if u=0, it gives the DC component andwhen u=1,2 it gives the first, second ACcomponents

    The IDCT is reconstruction of original signal from theseDC and AC components

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    DCT (Continued)

    For the 1D DCT defined, there will be a

    set of 8 different basis functions and their

    coefficent matrix is as shown

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    Characteristics of DCT

    DCT produces frequency spectrum F(u) correspondingto the spatial signal f(i)

    The first coefficient is the DC component of the signal,which gives the average magnitude of the signal

    The other seven AC components (basis functions) havecharacteristics like the first component completes halfcycle, second completes one cycle, third completes oneand half seventh completes three and half cycles

    DCT coefficients can also take negative values (Forimage DC component is always positive)

    DCT is a linear Transformation

    DCT (pA + qB) = p DCT(A) + q DCT(B)

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    Two Dimensional 8x8 DCT

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    Comparision between DCT and DFT

    The DFT coefficients will be exponential

    values

    By using Eulers formula,

    eix = cos x + isin x

    We can guess that DCT is likely a

    transform that forms the real part of DFT