1. The Report of Monographic Study
Karhunen-Loeve Transform
指導教授 : 尤信程 老師
報 告 者 : 侯侑成
報告地點 : 資工所辨公室
報告時間 : 90年1月3日(星期四) AM 1:30~3:00

2. Outline Introduction Derivation
The properties of the Karhunen-Loeve transform
Application of Karhunen-Loeve transform
Conclusions
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3. Introduction The objectives of transform coding :
- Removes correlation
- Packs energy in as few transform coefficients as possible
Consider N sampled points of a zero mean random vector x, then x expanded in terms of Фi is We seek the best representation of a given random function in the MSE sense. ( ). The KLT is said to be an optimal transform because it has the following properties
- It completely decorrelates the signal in the transform domain.
- It minimizes the MSE in bandwidth reduction or data compression.
- It contains the most variance (energy) in the fewest number of transform
coefficients.
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4. Derivation(1)
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5. Derivation(2)
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6. The properties of the KLT
Optimally decorrelating transform
find matrix such that
is diagonal
Optimal energy packing
Data dependent, need to notify decoder what T is
Requires estimation of autocorre- lation function
Fig.1. Energy packing
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7. Application of Karhunen-Loeve transform
A modified MPEG Advanced Audio Coding (ACC) scheme based on the Karhunen-Loeve transform (KLT) to remove inter-channel redundancy.
Fig.2. Inter-channel de-correlation via KLT
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8. Application (cont’d)
This method utilizes the KLT in the preprocessing stage for the powerful multichannel audio compression tool, i.e. ACC, to remove inter-channel redundancy and further improve the coding performance. However, as described in these papers, each element of the covariance matrix, from which the KLT matrix is derived, is scalar quantized to 16 bits. The results in 240 bits overhead for each KLT matrix for typical 5 channel audio contents.
It is shown in this work that, with vector quatization, the new method reduce the overhead from 200 bits to less than 3 bits per KLT while maintaining comparable coding performance
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9. Conclusion Transform coding : - Removes correlation
- Packs energy in few transform coefficients
KLT’s Propertys:
- The KLT’s MSE is minimum
- Optimally decorrelating transform
- Optimal energy packing
- Data dependent
- Requires estimation of autocorrelation function
Application:
- Adaptive KLT
- codebook
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