SC_I_02_Transform-Intro_4

SC_I_02_Transform-Intro_4 - Signal Compression 28 Signal...

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Signal Compression 28 Copyright © 2005 – 2008 Hayder Radha The Transform A wide range of transforms have been developed for multimedia applications, and more specifically, for media compression and coding. Some of these transforms are of theoretical importance; but they may not have been widely used mainly because of their complexity and related factors. One particular transform that has a great Signal Compression 29 Copyright © 2005 – 2008 Hayder Radha deal of theoretical significance is the Karhunen-Loeve Transform (KLT) . In addition to its theoretical and analytical significance, concepts that stem from the KLT represent the foundation of other (highly popular) transforms such as the Discrete Cosine Transform (DCT), which is employed in virtually all image and video Signal Compression 30 Copyright © 2005 – 2008 Hayder Radha coding standards (except JPEG200 which is based on wavelets). We first motivate the KLT and related transforms by highlighting some of the basic characteristics and desired features of popular signal transforms: Signal Compression 31 Copyright © 2005 – 2008 Hayder Radha ± Invertibility In general, when a transform () yTx = is used, one can attempt to recover the original signal x through another transform U . In other words, () () Uy UTx = .
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Signal Compression 32 Copyright © 2005 – 2008 Hayder Radha Clearly, it will be desirable to have: () () 1 UT ⋅= . In this case: () 1 Uy UTx T Tx x == = . Signal Compression 33 Copyright © 2005 – 2008 Hayder Radha ± Independence Ideally, the transform T is capable of transforming a (generally) dependent random source X x = into an independent random output Yy = . In other words, if YTX = , and when Signal Compression 34 Copyright © 2005 – 2008 Hayder Radha ,, ij i j XX i j X i X j f xx f x f x i j ≠∀ , then it will be desirable to have: i j YY i j Y i Y j f yy f y f y =∀ . Independence greatly facilitates compression and coding since, for one thing, the encoder can compress and code each individual element of the Signal Compression 35 Copyright © 2005 – 2008 Hayder Radha random sequence independently of each other. Furthermore, it enables the encoder to focus on few of the resulting (independent) symbols, which may have some particular significance (e.g., in terms of perceived signal quality by the human visual system).
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Signal Compression 36 Copyright © 2005 – 2008 Hayder Radha ± Generating uncorrelated output In practical systems and signals, it is virtually impossible to transform a random input into an independent random output. Hence, one would resort to the next best thing and that is generating an uncorrelated sequence. In this case, the objective is to generate an output Y such that: Signal Compression 37 Copyright © 2005 – 2008 Hayder Radha [ ] , ij i j Eyy EyEy i j ⎡⎤ =∀ ⎣⎦ .
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SC_I_02_Transform-Intro_4 - Signal Compression 28 Signal...

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